Description
Jacques
Bélair· IanA.FrigaardHerbKunze· RomanMakarovRoderickMelnik· RaymondJ.Spiteri Editors
Mathematicaland ComputationalApproachesin AdvancingModern Scienceand Engineering Mathematical and Computational Approaches inAdvancing Modern Science and Engineering Jacques Bélair • Ian A. Frigaard • Herb KunzeRoman Makarov • Roderick MelnikRaymond J. SpiteriEditors
Mathematical andComputational Approachesin Advancing ModernScience and Engineering
123 EditorsJacques Bélair Ian A. FrigaardDepartment of Mathematics and Statistics Department of MathematicsUniversity of Montreal University of British ColumbiaMontreal, QC Vancouver, BCCanada Canada
Herb Kunze Roman MakarovDepartment of Mathematics and Statistics Department of MathematicsUniversity of Guelph Wilfrid Laurier UniversityGuelph, ON Waterloo, ONCanada Canada
Roderick Melnik Raymond J. SpiteriMS2Discovery Institute Department of Computer ScienceWilfrid Laurier University University of SaskatchewanWaterloo, ON Saskatoon, SKCanada Canada
ISBN 978-3-319-30377-2 ISBN 978-3-319-30379-6 (eBook)DOI 10.1007/978-3-319-30379-6
Library of Congress Control Number: 2016943639
Mathematics Subject Classification (2010): 00A69, 00A71, 00A79, 92-XX, 35Qxx, 81T80, 97M10,47N60, 49-xx, 91Axx, 62Pxx, 97Pxx, 70-xx
© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.
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This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AG Switzerland Preface
This book consists of five parts covering a wide range of topics in appliedmathematics, modeling, and computational science (AMMCS). It resulted from twohighly successful meetings held jointly in Waterloo (Canada) on the main campusof Wilfrid Laurier University. It is the oldest university in the Cambridge-Kitchener-Waterloo-Guelph area, a beautiful part of Canada, just west of the city of Toronto.The main campus of the university is located in a comfortable driving distancefrom some of North America’s most spectacular tourist destinations, including theNiagara Escarpment, a UNESCO World Biosphere Reserve. Over the years, this uni-versity has become a traditional venue for the International Conference on AppliedMathematics, Modeling and Computational Science, and in 2015 it was held jointlywith the annual meeting of the Canadian Applied and Industrial Mathematics(CAIMS) from June 7–12, 2015. The AMMCS interdisciplinary conference seriesruns biannually. Focusing on recent advances in applied mathematics, modeling,and computational science, the 2015 AMMCS-CAIMS Congress drew some of thetop scientists, mathematicians, engineers, and industrialists from all over the worldand was a true celebration of interdisciplinary research and collaboration involvingmathematical, statistical, and computational sciences within a larger internationalcommunity. The book clearly demonstrates the importance of interdisciplinary interactionsbetween mathematicians, scientists, engineers, and representatives from other dis-ciplines. It is a valuable source of the methods, ideas, and tools of mathematicalmodeling, computational science, and applied mathematics developed for a varietyof disciplines, including natural and social sciences, medicine, engineering, andtechnology. Original results are presented here on both fundamental and appliedlevels, with an ample number of examples emphasizing the interdisciplinary natureand universality of mathematical modeling. The book contains 70 articles, arranged according to the following topicsrepresented by five parts:• Theory and Applications of Mathematical Models in Physical and Chemical Sciences
v vi Preface
Fig. 1 Participants of the 2015 International AMMCS-CAIMS Congress, Canada (Photo taken byTomasz Adamski on the Waterloo Campus at Wilfrid Laurier University)
• Mathematical and Computational Methods in Life Sciences and Medicine• Computational Engineering and Mathematical Foundation, Numerical Methods, and Algorithms• Mathematics and Computation in Finance, Economics, and Social Sciences• New Challenges in Mathematical Modeling for Scientific and Engineering Applications These chapters are based on selected refereed contributions made by theparticipants of both meetings. The AMMCS-CAIMS Congress featured over 30special and contributed sessions with mini-symposia ranging from mathematicalmodels in nanoscience and nanotechnology to statistical equilibrium in economicsand to mathematical neuroscience, the embedded Conference of the ComputationalFluid Dynamics Society of Canada, and the 2nd Canadian Symposium on ScientificComputing and Numerical Analysis, as well as larger sessions around such sci-entific themes as applied analysis and dynamical systems, industrial mathematics,mathematical biology, financial mathematics, and much more. Over 600 participantsfrom all continents attended the Congress and shared the latest achievements,ideas, insights, and theories about modern problems in science, engineering, andsociety that can be approached with new advances in mathematical modeling andmathematical, computational, and statistical methods. This book presents a selected sample of the above topics and can serve as areference to some of the state-of-the-art original works on a range of such topics. It Preface vii
Fig. 2 Members of the local organizing committee and student volunteers (Photo taken by Dr.Shyam Badu on the Waterloo Campus at Wilfrid Laurier University)
has a strong multidisciplinary focus, supported by fundamental theories, rigorousprocedures, and examples from applications. Furthermore, the book provides amultitude of examples accessible to graduate students and can serve as a sourcefor graduate student projects. Taking this opportunity, we would like to thank our colleagues on the AMMCS-CAIMS Congress organizing team, as well as our sponsors and partners, in particu-lar the Fields Institute and PIMS, and the Centre de Recherches Mathématiques, aswell as Wilfrid Laurier University, NSERC, and the Government of Ontario. Amongothers, traditional supporters of the AMMCS Interdisciplinary Conference serieswere Maplesoft and SHARCNET, as well as Springer, De Gruyter, and CRC Press.The Congress was held under the auspices of the MS2Discovery InterdisciplinaryResearch Institute based at Wilfrid Laurier University and in cooperation withthe Society of Industrial and Applied Mathematics and the American Institute ofMathematical Sciences. The Congress scientific committee included 15 internationally knownresearchers. We would like to thank them, as well as the Congress referees whosehelp in the refereeing process was invaluable. Among them we had some of theleading researchers from all parts of the world, and their assistance was decisive incompleting this project. Our technical support committee and students’ team wereexemplary, and we are truly grateful for their efforts. Last but not least, we are viii Preface
also grateful to the editorial team at Springer, in particular Martin Peters and RuthAllewelt, whose continuous support during the entire process was at the highestprofessional level. We believe that the book will be a valuable addition to the libraries, as wellas to private collections of university researchers and industrialists, scientists andengineers, graduate students, and all of those who are interested in the recentprogress in mathematical modeling and mathematical, computational, and statisticalmethods applied in interdisciplinary settings.
Montreal, Canada Jacques BélairVancouver, Canada Ian FrigaardGuelph, Canada Herb KunzeWaterloo, Canada Roman MakarovWaterloo, Canada Roderick MelnikSaskatoon, Canada Raymond Spiteri Contents
Part I Theory and Applications of Mathematical Models in Physical and Chemical SciencesCompressibility Coefficients in Nonlinear Transport Modelsin Unconventional Gas Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3Iftikhar Ali, Bilal Chanane, and Nadeem A. MalikSolutions of Time-Fractional Diffusion Equationwith Reflecting and Absorbing Boundary Conditions Using Matlab . . . . . . 15Iftikhar Ali, Nadeem A. Malik, and Bilal Chananehom*oclinic Structure for a Generalized Davey-Stewartson System . . . . . . . 27Ceni Babaoglu and Irma HacinliyanNumerical Simulations of the Dynamics of Vortex RossbyWaves on a Beta-Plane .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 35L.J. CampbellOn the Problem of Similar Motions of a Chain of CoupledHeavy Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 47Dmitriy ChebanovOn Stabilization of an Unbalanced Lagrange Gyrostat... . . . . . . . . . . . . . . . . . . . 59Dmitriy Chebanov, Natalia Mosina, and Jose SalasApproximate Solution of Some Boundary Value Problemsof Coupled Thermo-Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71Manana ChumburidzeSymmetry-Breaking Bifurcations in Laser Systemswith All-to-All Coupling .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81Juancho A. Collera
ix x Contents
Effect of Jet Impingement on Nano-aerosol Soot Formationin a Paraffin-Oil Flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 89Masoud Darbandi, Majid Ghafourizadeh,and Mahmud AshrafizaadehNormalization of Eigenvectors and Certain Propertiesof Parameter Matrices Associated with The Inverse Problemfor Vibrating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101Mohamed El-Gebeily and Yehia KhuliefComputational Aspects of Solving Inverse Problems for EllipticPDEs on Perforated Domains Using the Collage Method . . . . . . . . . . . . . . . . . . . 113H. Kunze and D. La TorreDynamic Boundary Stabilization of a Schrödinger EquationThrough a Kelvin-Voigt Damped Wave Equation . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121Lu Lu and Jun-Min WangMolecular-Dynamics Simulations Using Spatial Decompositionand Task-Based Parallelism .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 133Chris M. Mangiardi and R. MeyerModelling of Local Length-Scale Dynamics and IsotropizingDeformations: Formulation in Natural Coordinate System .. . . . . . . . . . . . . . . . 141O. Pannekoucke, E. Emili, and O. ThualPost-Newtonian Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 153Erik I. Verriest
Part II Mathematical and Computational Methods in Life Sciences and MedicineA Quantitative Model of Cutaneous Melanoma DiagnosisUsing Thermography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167Ephraim Agyingi, Tamas Wiandt, and Sophia MaggelakisTime-Dependent Casual Encounters Games and HIV Spread.. . . . . . . . . . . . . 177Safia Athar and Monica Gabriela CojocaruModelling an Aquaponic Ecosystem Using OrdinaryDifferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189C. Bobak and H. KunzeA New Measure of Robust Stablity for Linear OrdinaryImpulsive Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197Kevin E.M. ChurchCoupled Lattice Boltzmann Modeling of Bidomain TypeModels in Cardiac Electrophysiology .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 209S. Corre and A. Belmiloudi Contents xi
Dynamics and Bifurcations in Low-Dimensional Modelsof Intracranial Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 223D. Evans, C. Drapaca, and J.P. CusumanoPersistent hom*ology for Analyzing Environmental LakeMonitoring Data.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233Benjamin A. Fraser, Mark P. Wachowiak,and Renata Wachowiak-SmolíkováEstimating Escherichia coli Contamination Spread in GroundBeef Production Using a Discrete Probability Model . . . .. . . . . . . . . . . . . . . . . . . . 245Petko M. Kitanov and Allan R. WillmsThe Impact of Movement on Disease Dynamics in a Multi-cityCompartmental Model Including Residency Patch . . . . . .. . . . . . . . . . . . . . . . . . . . 255Diána KniplA Chemostat Model with Wall Attachment: The Effectof Biofilm Detachment Rates on Predicted Reactor Performance .. . . . . . . . . 267Alma Mašić and Hermann J. EberlApplication of CFD Modelling to the Restoration of Eutrophic Lakes . . . . 277A. Najafi-Nejad-Nasser, S.S. Li, and C.N. MulliganOn the Co-infection of Malaria and Schistosomiasis . . . . .. . . . . . . . . . . . . . . . . . . . 289Kazeem O. Okosun and Robert Smith?A Discrete-Continuous Modeling Framework to Study the Roleof Swarming in a Honeybee-Varroa destrutor-Virus System.. . . . . . . . . . . . . . . . 299Vardayani Ratti, Peter G. Kevan, and Hermann J. EberlTo a Predictive Model of Pathogen Die-off in Soil FollowingManure Application.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 309Andrew Skelton and Allan R. WillmsMathematical Modeling of VEGF Binding, Production,and Release in Angiogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 319Nicoleta TarfuleaA Mathematical Model of Cytokine Dynamics Duringa Cytokine Storm.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 331Marianne Waito, Scott R. Walsh, Alexandra Rasiuk,Byram W. Bridle, and Allan R. WillmsExamining the Role of Social Feedbacks and Misperceptionin a Model of Fish-Borne Pollution Illness . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 341Michael Yodzis, Chris T. Bauch, and Madhur Anand xii Contents
Part III Computational Engineering and Mathematical Foundation, Numerical Methods and AlgorithmsStability Properties of Switched Singular Systems Subjectto Impulsive Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 355Mohamad S. Alwan, Humeyra Kiyak, and Xinzhi LiuInput-to-State Stability and H1 Performance for StochasticControl Systems with Piecewise Constant Arguments . . .. . . . . . . . . . . . . . . . . . . . 367Mohamad S. Alwan and Xinzhi LiuSwitched Singularly Perturbed Systems with Reliable Controllers . . . . . . . . 379Mohamad S. Alwan, Xinzhi Liu, and Taghreed G. SugatiApplication of an Optimized SLW Model in CFD Simulationof a Furnace .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 389Masoud Darbandi, Bagher Abrar, and Gerry E. SchneiderNumerical Investigation on Periodic Simulation of FlowThrough Ducted Axial Fan .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 401Seyedali Sabzpoushan, Masoud Darbandi, Mohsen Mohammadi,and Gerry E. SchneiderNumerical Analysis of Turbulent Convective Heat Transferin a Rotor-Stator Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 413D.-D. Dang and X.-T. PhamDetermining Sparse Jacobian Matrices Using Two-SidedCompression: An Algorithm and Lower Bounds . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 425Daya R. Gaur, Shahadat Hossain, and Anik SahaAn h-Adaptive Implementation of the Discontinuous GalerkinMethod for Nonlinear Hyperbolic Conservation Lawson Unstructured Meshes for Graphics Processing Units . . . . . . . . . . . . . . . . . . . . 435Andrew Giuliani and Lilia KrivodonovaExtending BACOLI to Solve the Monodomain Model. . . .. . . . . . . . . . . . . . . . . . . . 447Elham Mirshekari and Raymond J. SpiteriAn Analysis of the Reliability of Error Control B-SplineGaussian Collocation PDE Software .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 459Paul Muir and Jack PewOn the Simulation of Porous Media Flow Using a NewMeshless Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 469S. Hossein Musavi and Mahmud AshrafizaadehA Comparison Between Two and Three-DimensionalSimulations of Finite Amplitude Sound Waves in a Trumpet . . . . . . . . . . . . . . . 481Janelle Resch, Lilia Krivodonova, and John Vanderkooy Contents xiii
A Dual-Rotor Horizontal Axis Wind Turbine In-House Code(DR_HAWT) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 493K. Lee Slew, M. Miller, A. Fereidooni, P. Tawagi, G. El-Hage,M. Hou, and E. MatidaNumerical Study of the Installed Controlled Diffusion Airfoilat Transitional Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 505Hao Wu, Paul Laffay, Alexandre Idier, Prateek Jaiswal,Marlène Sanjosé, and Stéphane Moreau
Part IV Mathematics and Computation in Finance, Economics, and Social SciencesFinancial Markets in the Context of the General Theoryof Optional Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 519M.N. Abdelghani and A.V. MelnikovA Sufficient Condition for Continuous-Time Finite Skip-FreeMarkov Chains to Have Real Eigenvalues . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 529Michael C.H. Choi and Pierre PatieBifurcations in the Solution Structure of Market EquilibriumProblems . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 537F. Etbaigha and M. CojocaruPricing Options with Hybrid Stochastic Volatility Models . . . . . . . . . . . . . . . . . . 549Glynis Jones and Roman MakarovDelay Stochastic Models in Finance .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 561Anatoliy SwishchukSemi-parametric Time Series Modelling with Autocopulas . . . . . . . . . . . . . . . . . 573Antony Ware and Ilnaz AsadzadehOptimal Robust Designs of Step-Stress Accelerated LifeTesting Experiments for Proportional Hazards Models .. . . . . . . . . . . . . . . . . . . . 585Xaiojian Xu and Wan Yi HuangDetecting Coalition Frauds in Online-Advertising . . . . . . .. . . . . . . . . . . . . . . . . . . . 595Qinglei Zhang and Wenying Feng
Part V New Challenges in Mathematical Modeling for Scientific and Engineering ApplicationsCircle Inversion Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 609B. Boreland and H. KunzeComputation of Galois Groups in magma . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 621Andreas-Stephan Elsenhans xiv Contents
Global Dynamics and Periodic Solutions in a SingularDifferential Delay Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 629Anatoli F. Ivanov and Zari A. DzalilovLocalized Spot Patterns on the Sphere for Reaction-DiffusionSystems: Theory and Open Problems. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 641Alastair Jamieson-Lane, Philippe H. Trinh, and Michael J. WardContinuous Dependence on Modeling in Banach Space Usinga Logarithmic Approximation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 653Matthew Fury, Beth Campbell Hetrick, and Walter HuddellSolving Differential-Algebraic Equations by SelectingUniversal Dummy Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 665Ross McKenzie and John D. PryceOn a Topological Obstruction in the Reach Control Problem . . . . . . . . . . . . . . 677Melkior Ornik and Mireille E. BrouckeContinuous Approaches to the Unconstrained BinaryQuadratic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 689Oksana Pichugina and Sergey YakovlevFixed Point Techniques in Analog Systems. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 701Diogo Poças and Jeffery ZuckerA New Look at Dummy Derivatives for Differential-AlgebraicEquations . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 713John D. Pryce and Ross McKenzieNew Master-Slave Synchronization Criteria of Chaotic Lur’eSystems with Time-Varying-Delay Feedback Control . . .. . . . . . . . . . . . . . . . . . . . 725Kaibo Shi, Xinzhi Liu, Hong Zhu, and Shouming ZhongRobust Synchronization of Distributed-Delay Systemsvia Hybrid Control .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 737Peter Stechlinski and Xinzhi LiuRegularization and Numerical Integration of DAEs Basedon the Signature Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 749Andreas SteinbrecherSymbolic-Numeric Methods for Improving Structural Analysisof Differential-Algebraic Equation Systems. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 763Guangning Tan, Nedialko S. Nedialkov, and John D. PrycePinning Stabilization of Cellular Neural Networkswith Time-Delay Via Delayed Impulses. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 775Kexue Zhang, Xinzhi Liu, and Wei-Chau Xie Contents xv
Convergence Analysis of the Spectral Expansion of StableRelated Semigroups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 787Yixuan Zhao and Pierre Patie
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 799 Part ITheory and Applications of MathematicalModels in Physical and Chemical Sciences Compressibility Coefficients in NonlinearTransport Models in Unconventional GasReservoirs
Iftikhar Ali, Bilal Chanane, and Nadeem A. Malik
Abstract Transport models for gas flow in unconventional hydrocarbon reservoirspossess several model parameters such as the density ./, the permeability .K/,the Knudsen number .Kn /, that are strongly dependent upon the pressure p. Eachphysical parameter, say , in the system has an associated compressibility factor D . p/ (which is the relative rate of change of the parameter with respect tochanges in the pressure, Ali I et al. (2014, Time-fractional nonlinear gas transportequation in tight porous media: an application in unconventional gas reservoirs.In: 2014 international conference on fractional differentiation and its applications(ICFDA), Catania, pp 1–6, IEEE)). Previous models have often assumed that D Const, such as Cui (Geofluids 9(3):208–223, 2009), and Civan (TranspPorous Media 86(3):925–944, 2011). Here, we investigate the effect of selectedcompressibility factors (real gas deviation factor .Z /, gas density . /, gas viscosity. /, permeability .K /, and the porosity . / of the source rock) as functions ofthe pressure upon rock properties such as K and . We also carry out a sensitivityanalysis to estimate the importance of each model parameter. The results arecompared to available data.
1 Introduction
Unconventional gas reservoirs include tight gas, coalbed methane, and shale gas.Shale gas is distributed over large areas and is found in discrete largely unconnectedgas pockets. Different methods are applied to induce fractures inside the rocks torelease the gas, such as hydraulic fracturing, but this is very expensive. Hence,an initial guess is required before drilling. Reservoir simulations can be crucial in
I. Ali () • B. Chanane • N.A. MalikDepartment of Mathematics & Statistics, King Fahd University of Petroleum and Minerals,P. O. Box 5046, Dhahran 31261, Saudi Arabiae-mail: [emailprotected]; [emailprotected]; [emailprotected];[emailprotected]
© Springer International Publishing Switzerland 2016 3J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_1 4 I. Ali et al.
assisting this process for economical recovery. This requires accurate determinationof fluid and rock properties, and a realistic transport model, [2, 5, 11, 15]. Unconventional gas reservoirs are characterized by extremely low permeability,in the nano- to micro-Darcy range, and low porosity, in the 4 %–15 % range. Thegas extraction process is very complex and involves new technologies, and takesa lot of time, money and human resources, [18]. The science and technology oftight gas transport and extraction is still in its infancy, and field data urgentlyrequired especially from shale gas reservoirs in order to test the newly emergingtheories. Reservoir simulations typically solve model transport equations in the form ofadvection-diffusion partial differential equations (PDE). Some of the latest modelsare highly non-linear, where the apparent diffusivity D. p/ and the apparent velocityU. p; px/ are strongly non-linear functions of the pressure and its derivative, [7]. Dand U involve compressibility factors of various physical parameters,
@ ln 1 @ D D : (1) @p @p
and these must be known as functions of p and px . However, most applications todate have been simplified by assuming constant compressibility factors. The impactof this important assumption has not been assessed to date. The aim here is to assess the importance of using fully pressure dependent modelparameters. This is done through numerical simulations of the transport equationand matching the results against the data from Pong et al. [17]. A sensitivity analysisis also carried out to assess the importance of each physical parameter in thesystem.
2 Physical Properties of Shale Gas Reservoirs
Various flow regimes occur in the gas transport process through tight shale rockformations [10]. They are classified by a Knudsen number, see Table 1 and [17, 19],which is the ratio of mean free path of gas molecules q () to the radius (R) of theflow channels, Kn D =R. is given by [13], D 2Rg T , where is gas density,T is temperature, Rqg is universal gas constant, and is gas viscosity. R is given by, p[4, 6], R D 2 2 K , where is the tortuosity and is the porosity of porousmedia and K is intrinsic permeability. Several recent works have focused transporton the so-called four flow regimes, Table 1. Nonlinear Transport Models in Unconventional Gas Reservoirs 5
Table 1 Classification of Knudsen number Flow regimesflow regimes based onKnudsen number, [19] Kn < 0:01 Continuous flow 0:01 < Kn < 0:1 Surface diffusion or slip flow 0:1 < Kn < 10 Transition flow Kn > 10 Knudsen diffusion or free molecular flow
The correlation between porosity and intrinsic permeability is given by theKozeny-Carman equation [8] s ˇKC K D KC ; (2) ˛KC
where < ˛KC 1, 0 ˇKC < 1 and KC 0. ˛KC ; ˇKC , and KC are empiricalconstants which must be determined, or estimated, before hand. For the simulation purposes, we use the following porosity-pressure correlation,
D a exp.b pc /; (3)
where a , b and c are model constants. Tortuosity is related to porosity by,
D 1 C a .1 /; (4)
where a is also a model constant. There is a difference between the intrinsic permeability, K, and the apparentpermeability, Ka . K is the measured permeability from rock samples, but dueto various physical effects such as slip flow, the quantity appearing in transportequations is Ka . Beskok [3] has derived an formula that relates the two quantities,
Ka D Kf .Kn / (5)
where f .Kn / is the flow condition function given by
f .Kn / D .1 C Kn / .1 C .4 bSF /Kn / .1 bSF Kn /1 ; (6)
where is called the Rarefaction Coefficient Correlation [6] given by 1 D o 1 C A KnB ; (7)
where A and B are empirical constants and bSF in Eq. 6 is the slip factor. 6 I. Ali et al.
Some of the gas adheres (clings) to pore surfaces due to the diffusion of gasmolecules. Cui [9] and Civan [7] developed a formula for estimating the amount ofadsorbed gas based on Langmuir isotherms and is given by
s Mg s Mg qL p qD qa D ; (8) Vstd Vstd pL C p
where s (kg/m3 ) denotes the material density of the porous sample, q (kg/m3) isthe mass of gas adsorbed per solid volume, qa (std m3 /kg) is the standard volumeof gas adsorbed per solid mass, qL (std m3 /kg) is the Langmuir gas volume, Vstd(std m3 /kmol) is the molar volume of gas at standard temperature (273.15 K) andpressure (101,325 Pa), p (Pa) is the gas pressure, pL (Pa) is the Langmuir gaspressure, and Mg (kg/kmol) is the molecular weight of gas. Gas density (kg/m3 ) is given by the real-gas equation of state,
Mg p D (9) ZRg T
where Z (dimensionless) is the real gas deviation factor [12] and it can be found byusing the correlation developed by Mahmoud [14] and it is given by
Z D ap2r C bpr C c (10) a D 0:702 exp.2:5tr / (11) b D 5:524 exp.2:5tr / (12) c D 0:044Tr2 0:164tr C 1:15 (13)
where pc is the critical pressure and tc is the critical temperature, and pr D p=pc andtr D t=tc are the reduced pressure and temperature respectively. Mahmoud [14] also gave correlations for determining the gas viscosity,
D Sc exp.AB / (14) A D 3:47 C 1588T 1 C 0:0009Mg B D 1:66378 0:04679A 3 4 1=6 1 M pc Sc D .10:5/4 Tc 0:807Tr0:618 0:357 exp.0:449Tr / C 0:34 exp.4:058Tr / C 0:018 Nonlinear Transport Models in Unconventional Gas Reservoirs 7
3 Mathematical Formulation
The ultra low permeability and the occurrence of various flow regimes are keyfeatures of unconventional gas reservoirs (UGR). The PDE’s that are used todescribe transport process in conventional gas reservoirs (CGR) are based onDarcy’s law u D .K=/dp=dx and continuity equation .u/x D 0, where K, ,and are constants, but such models do not produce satisfactory results in UGRs.Civan [7] has proposed a transport model for gas flow through tight porous mediawhich incorporates all flow regimes that occur in the reservoirs. Civan’s model is anon-linear advection-diffusion PDE for the pressure field p.x; t/, which is given by,
@p @p @2 p C U. p; px / D D. p/ 2 : (15) @t @x @x
The apparent diffusivity D (m2 /s) is given by,
Ka DD f1 . p/ C .1 /q2 . p/g1 ; (16)
and the apparent convective flux (velocity) U (m/s) is given by,
@p U D 3 . p/D : (17) @x
where the 1 , 2 and 3 appearing in D and U are given by
1 . p/ D . p/ C . p/; (18) 2 . p/ D q . p/ . p/; (19) 1
3 . p/ D Π. p/ C Ka . p/ . p/ : (20)
where Ka D K C f which is obtained from Eq. (5). A numerical solver for the system equations (15), (16), (17), (18), (19), and (20)has been developed. We use a finite volume implicit method with constant grid sizeand constant time step. The system is linearised and iterated to convergence beforeadvancing to the next time step. The implicit nature of the solver gives stability tothe solver which is essential for such a highly non-linear system. The solver can alsobe applied to the steady state system, see below. 8 I. Ali et al.
4 Model Validation Under Steady State Conditions
The steady state solution for the pressure field is obtained by solving, (see [1, 7]), @p @2 p La D ; 0 x L; (21) @x @x2
where @p La D . p/ C K . p/ C f . p/ . p/ ; (22) @x
with boundary conditions, p.0/ D pL and p.L/ D pR ; pL and pR assumed known. Sixteen different models were considered, Table 2. An entry of ‘0’ means thatthe compressibility factor is zero, D 0; an entry of ‘p’ means that 6D 0 and theassociated physical parameter is a function of pressure, D . p/. The final columnshows the relative error between the simulated values and the experimental valuesof Pong et al. [16], given by,
N cal X 2 p pmeas Relative Error D i i : (23) iD1 pcal i
where the summation is over the N D 30 data-points in [16]. Case 1 in Table 2corresponds to the Darcy law where all the physical parameters are constant and
Table 2 List of models Cases K f Errorconsidered. In columns 2–5,an entry of 0 means that the 1 0 0 0 0 2.69e02compressibility factor is zero; 2 p 0 0 0 2.68e02an entry of p means that it is 3 0 p 0 0 4.05e03nonzero and the associated 4 0 0 p 0 3.16e01physical parameter is function 5 0 0 0 p 2.69e02of pressure p. The finalcolumn shows the relative 6 p p 0 0 1.17e01error from simulations using 7 p 0 p 0 3.19e02Eq. (23) 8 p 0 0 p 2.68e02 9 0 p p 0 1.84eC00 10 0 p 0 p 4.05e03 11 0 0 p p 3.17e01 12 p p p 0 1.37e04 13 p p 0 p 1.17e01 14 0 p p p 3.19e02 15 p 0 p p 1.84eC00 16 p p p p 1.36e04 Nonlinear Transport Models in Unconventional Gas Reservoirs 9
D 0. Case 16 is the fully pressure-dependent case. An additional case, fromCivan [8] with constant factors for K , , , and , was also carried out. Figure 1 shows the comparisons of the simulated results (solid lines) for thepressure against the distance for selected models (see captions) against the dataof Pong et al. [16] (symbols). The inlet pressures for the different simulationsare, respectively from bottom to top, 135, 170, 205, 240, and 275 kPa. Figure 1acompares with Darcy’s Law Case 1, and it shows significant errors. Figure 1bcompares with Civan’s case, and Fig. 1c compares with Case 16, which is the bestfit to the data. Figure 1d shows the relative error on log-scale for the 16 cases inTable 2. We refer to Case 16 as the Base Case henceforth (Table 3). It is important to note that although the Civan case Fig. 1b and the Base CaseFig. 1c appear to yield similar results, the rock properties obtained in the two casesare quite different. Civan used D 0:2 independent of pressure and he predicted
a Darcy’s Law b Civan (2011) 280 280 Pin Pin 135 kPa 135 kPa 240 240 170 kPa 170 kPaPressure [kPa]
Pressure [kPa]
205 kPa 205 kPa 200 240 kPa 200 240 kPa 275 kPa 275 kPa
160 160
120 120
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 −3 −3 Distance [m] x 10 x 10 Distance [m] New Approachc 280 Pin d 5 10 135 kPa 240 170 kPa Pressure [kPa]
205 kPa Relative Error
200 240 kPa 0 275 kPa 10
160
120 −5 10
0 0.5 1 1.5 2 2.5 3 0 5 10 15 −3 Distance [m] x 10 Cases
Fig. 1 Pressure, p against the distance along the core sample, x, from numerical solutions of theSteady State Model, Eqs. (21), (22), for different inlet pressures, Pin , as indicated by color. Solidslines are from the simulations, and symbols are data from Pong et al. [16]. (a) Darcy’s law, Case 1in Table 2, with compressibility factors, D 0. (b) Civan’s model with constant compressibilityfactors, D Const, for some parameters, see [7]. (c) Case 16 in Table 2 (new model) with pressuredependent parameters and non-constant compressibility factors, . p/. (d) Relative errors for the16 cases in Table 2 10 I. Ali et al.
Table 3 Reservoir model Parameter Parameterparameters used in the BaseCase, Case 16 in Table 2 L (m) 0:003 ˛KC 1 Nx 100 ˇKC 1 Rg (J kmol1 K1 ) 8314:4 KC 1 Mg (kg kmol1 K1 ) 16 a 0:2 T (K) 350 b 1 106 pc (kPa) 3:1 103 c 1:96 tc (K) 125 0 10 bSF 1 A 0:2 a 1:5 B 0:4
Table 4 The range of Cases ˛KC ˇKC KC a b cparameters that are used todetermine the values of 1 1:0 1:0 1.0 0:20 1e6 1:96permeability, and porosity in 2 1:0 0:65 1e7 0:08 1e6 2:09Fig. 3, from Eqs. (2) and (3) 3 0:75 0:66 1e7 0:15 1e6 2:09 4 0:25 0:4 1e8 0:15 1e6 1:96 5 0:5 0:5 0.1 0:10 1e6 2:1112 6 0:5 1:5 1.0 0:05 1e8 2:90 7 0:45 0:65 1e6 0:01 1e8 2:88
K D 1 1015 m2 . From the present calculations the porosity is pressure dependentand in the range 0:01 0:2, and the permeability is also pressure dependentand in the range 1020 K 103 m2 , which are more realistic (Table 4).
5 Sensitivity Analysis and Estimation of Model Parameters
It is important to determine how much the results and predicted rock propertieschange due to small changes in model parameters. A sensitivity analysis was carriedout by adjusting one model parameter at a time by factors of 2 and 1=2, startingwith Case 16 as the base case – One-at-a-Time (OAT) methodology. Sensitivity ismeasured by monitoring the changes in the model output. Figure 2 shows sensitivity to selected parameters: (a) pc (critical pressure), (b)T (temperature), (c) a (constant in the tortuosity model), (d) a (constant in theporosity model). Except for the temperature, Fig. 2b, all results show significantsensitivity to changes in the selected parameter especially at higher inlet pressures. Figure 3 illustrates the sensitivity of the calculated permeability, and porosityagainst the pressure, for different combinations of ˛KC , ˇKC , and KC , Eq. (2). Nonlinear Transport Models in Unconventional Gas Reservoirs 11
a 300 b 300 135 kPa 135 kPa 170 kPa 170 kPa 250 250 205 kPa 205 kPa 240 kPa 240 kPa
p, [kPa]p, [kPa]
275 kPa 275 kPa 200 200
150 150
100 100 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 −3 −3 x, [m] x 10 x, [m] x 10
c 300 d 300 135 kPa 135 kPa 170 kPa 170 kPa 250 250 205 kPa 205 kPa 240 kPa 240 kPa p, [kPa]
p, [kPa]
275 kPa 275 kPa 200 200
150 150
100 100 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 −3 −3 x, [m] x 10 x, [m] x 10
Fig. 2 OAT sensitivity analysis of the new model. Symbols are the data from Pong et al. [16] (seeFig. 2 for details). Sensitivity to the following parameters: (a) Critical pressure pc , (b) TemperatureT, (c) Tortuosity parameter a in Eq. (3), (d) porosity parameter a in Eq. (4). Red lines are theBase Case parameter values in Table 3. Blue lines: the specific parameter is divided by 2. Greenlines: the specific parameter is multiplied by 2
0 10 0.25 1 6 1 −5 0.20 10 5 3 4 0.15 K, [m ] 2
−10 10 φ
5 7 0.10 2
10 −15 2 6 4 0.05
3 7 −20 10 0 100 150 200 250 300 100 150 200 250 300 p, [kPa] p, [kPa]
(a) Permeability (b) Porosity
Fig. 3 Permeability (K), and porosity (), against the pressure p, based upon the parameter valuesin Table 3. Seven cases for each parameter, shown in Table 4, are considered and are indicatedon the plots. (Case 1 is the Base Case, shown as black dashed line.) (a) permeability curves areobtained using data from columns 2 to 4 of Table 4. (b) Porosity curves are obtained using datafrom columns 5 to 7 12 I. Ali et al.
6 Summary
The Base Case 16 in Table 2, which is the fully pressure-dependent non-linearmodel, performs better than other models giving the smallest error against availabledata, Fig. 1c. Darcy’s Law performs the worst illustrating its limitations for gastransport in tight porous media. A OAT sensitivity analysis shows that rockproperties such as the porosity, and permeability, are very sensitive to most of themodel parameters, Figs. 2 and 3. In the future, the sensitivity analysis for all of themodel parameters will be completed.
Acknowledgements The authors would like to acknowledge the support provided by KingAbdulaziz City for Science and Technology (KACST) through the Science Technology Unit atKing Fahd University of Petroleum and Minerals (KFUPM) for funding this work through projectNo. 14-OIL280-04.
References
1. Ali, I., Malik, N.A., Chanane, B.: Time-fractional nonlinear gas transport equation in tight porous media: an application in unconventional gas reservoirs. In: 2014 International Con- ference on Fractional Differentiation and Its Applications (ICFDA), Catania, pp. 1–6. IEEE (2014) 2. Aziz, K., Settari, A.: Petroleum Reservoir Simulation, vol. 476. Applied Science Publishers, London (1979) 3. Beskok, A., Karniadakis, G.E.: Report: a model for flows in channels, pipes, and ducts at micro and nano scales. Microsc. Thermophys. Eng. 3(1), 43–77 (1999) 4. Carman, P.C., Carman, P.C.: Flow of Gases Through Porous Media. Butterworths Scientific Publications, London (1956) 5. Chen, Z.: Reservoir Simulation: Mathematical Techniques in Oil Recovery. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 77. SIAM, Philadelphia (2007) 6. Civan, F.: Effective correlation of apparent gas permeability in tight porous media. Transp. Porous Media 82(2), 375–384 (2010) 7. Civan, F., Rai, C.S., Sondergeld, C.H.: Shale-gas permeability and diffusivity inferred by improved formulation of relevant retention and transport mechanisms. Transp. Porous Media 86(3), 925–944 (2011) 8. Civan, F., et al.: Improved permeability equation from the bundle-of-leaky-capillary-tubes model. In: SPE Production Operations Symposium, Oklahoma City. Society of Petroleum Engineers (2005) 9. Cui, X., Bustin, A., Bustin, R.M.: Measurements of gas permeability and diffusivity of tight reservoir rocks: different approaches and their applications. Geofluids 9(3), 208–223 (2009)10. Cussler, E.L.: Diffusion: Mass Transfer in Fluid Systems. Cambridge University Press, Cambridge/New York (2009)11. Darishchev, A., Rouvroy, P., Lemouzy, P.: On simulation of flow in tight and shale gas reservoirs. In: 2013 SPE Middle East Unconventional Gas Conference & Exhibition, Muscat (2013)12. Kumar, N.: Compressibility factors for natural and sour reservoir gases by correlations and cubic equations of state. M.Sc. Thesis, Texas Tech University (2004)13. Loeb, L.B.: The Kinetic Theory of Gases. Courier Dover Publications (2004) Nonlinear Transport Models in Unconventional Gas Reservoirs 13
14. Mahmoud, M.: Development of a new correlation of gas compressibility factor (z-factor) for high pressure gas reservoirs. J. Energy Resour. Technol. 136(1), 012903 (2014)15. Peaceman, D.W.: Fundamentals of Numerical Reservoir Simulation. Elsevier, New York (1977)16. Pong, K.C., Ho, C.M., Liu, J., Tai, Y.C.: Non-linear pressure distribution in uniform microchan- nels. ASME public. FED 197, 51–51 (1994)17. Rathakrishnan, E.: Gas Dynamics. PHI Learning, New Delhi (2013)18. Wang, Z., Krupnick, A.: A retrospective review of shale gas development in the United States. What led to the boom? Pub. Resources, Washington (2013)19. Ziarani, A.S., Aguilera, R.: Knudsen’s permeability correction for tight porous media. Transp. Porous Media 91(1), 239–260 (2012) Solutions of Time-Fractional Diffusion Equationwith Reflecting and Absorbing BoundaryConditions Using Matlab
Iftikhar Ali, Nadeem A. Malik, and Bilal Chanane
Abstract The main objective of this work is to develop Matlab programs forsolving the time-fractional diffusion equation (TFDE) with reflecting and absorbingboundary conditions on finite and infinite domains. Essentially, there are threemajor codes, one for finding the exact solution of the TFDE and other two arefor finding the numerical solution of the TFDE. The code for finding the exactsolutions is based on the fundamental solution of the TFDE, whereas the codesfor finding the numerical solutions are based on the explicit and the implicit finitedifference schemes, respectively. Finally, we illustrate the effectiveness of the codesby applying them to TFDEs with sharp initial data and for various reflecting andabsorbing boundary conditions both on finite and infinite domains. The results showthe difference of solutions between the standard diffusion equation and the time-fractional diffusion equation.
1 Introduction
Many physical processes evolve in spaces that are heterogeneous in nature, such as,crowded system, protein diffusion within cells, anomalous diffusion through porousmedia, see [3, 4, 6, 17]. Mathematical models, based on standard calculus, havefailed to describe such intricate processes whereas mathematical models, based onfractional calculus techniques, have proven their effectiveness in explaining suchcomplex processes, [1, 2, 5, 10, 15]. Time-fractional diffusion equation have been derived in the framework of Con-tinuous Time Random Walk (CTRW) model. It is based on the idea of consideringthe transport processes as the flow of particles in the form of packets and thenassigning a probability of locating a packet at position x at time t. Law of TotalProbability is used to determine probability P.x; t/. Luchko has derived the time-fractional diffusion equation by using these concept, see the details in [8, 9]. The
I. Ali () • N.A. Malik • B. ChananeKing Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabiae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 15J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_2 16 I Ali et al.
equation is given by,
@t P.x; t/ D @t1˛ Œv@x P.x; t/ C k2 @2x P.x; t/ (1)
as t ! 1 and jxj ! 1. Equation (1) is called time-fractional advection-diffusionequation and in the case v D 0 it reduces to time-fractional diffusion equation. Formore detailed derivation, see [9, 11]. In this work, we develop Matlab programs for finding exact and numericalsolutions of the time fractional diffusion equation (TFDE) on finite and infinitedomains, and also with various boundary conditions. The manuscript is organizedas follows; in Sect. 2, procedure for finding the fundamental solution of the TFDE isexplained; in Sect. 3, the numerical schemes are discussed; in Sect. 4, Matlab codesare provided; in Sect. 5, several examples are given to illustrate the effectiveness ofMatlab programs; finally, in Sect. 6, conclusions are given.
2 Fundamental Solution of Time Fractional Diffusion Equation
Consider the time fractional diffusion equation, in Caputo form, over the whole realline with given initial data,
@˛ @2 u.x; t/ D u.x; t/; 0<˛1 (2) @t˛ @x2
u.x; 0/ D f .x/: (3)
Equation (2) can be written in the integral form as follows, Z 1 t u.x; t/ D f .x/ C .t /˛1 uxx .x; /d : (4) .˛/ 0
Application of Laplace transform yields a second order linear differential equation
uQ xx .x; p/ p˛ uQ .x; p/ D f .x/p˛1 : (5)
The solution of Eq. (5) is given by Z 1 uQ .x; p/ D Q yj; p˛=2 /p˛1 f .y/dy; k.jx (6) 1 Time-Fractional Diffusion Equations 17
where k.jxj; / D p 1 jxj1=2 k1=2 .jxj/ is modified Bessel function of second kind 2[16]. Furthermore, Eq. (6) can be expressed as Z 1 uQ .x; p/ D Q ˛ .jx yj; p/f .y/dy; G (7) 1
where G Q ˛ .jxj; p/ D k.jxj; Q p˛=2 /p˛1 : Note that directly taking the inverse Laplace transform is not feasible, so we usethe relationship between the Laplace and Mellin transforms to obtain Z 1 Q ˛ .jxj; s/ D 1 G ps G Q ˛ .jxj; p/dp .1 s/ 0 Z 1 jxj1=2 D p p3˛=4s1 kQ 1=2 .jxjp˛=2 /dp: (8) 2 .1 s/ 0Using the results, 1 sC sC M Œx f .ax / D a b fQ b ; b b hs i s C kQ .s/ D 2s2 ; 2 2Equation (8) becomes
Q ˛ .jxj; s/ D 1 Œ1 s=˛ Œ1=2 s=˛ G p 22s=˛ jxj2s=˛1 : (9) ˛ Œ1 s
Taking the inverse Mellin transform and using Fox function, we obtain 2=˛ ˛ 1 1 20 jxj .1; 1/ G .jxj; t/ D p jxj H12 : (10) ˛ 22=˛ t .1=2; 1=˛/; .1; 1=˛/
The general solution of the time fractional diffusion equation is given by Z 1 u.x; t/ D G˛ .jx yj; t/f .y/dy: (11) 1
If the initial data is given as delta potential, that is, u.x; 0/ D ı.x/, then thesolution (11) becomes 2=˛ 1 1 20 jxj .1; 1/ u.x; t/ D p jxj H12 : (12) ˛ 22=˛ t .1=2; 1=˛/; .1; 1=˛/
For more details, readers are referred to Wyss [18] and Schneider & Wyss [14]. 18 I Ali et al.
3 Numerical Solutions
In this section, we provide two numerical schemes based on finite differencemethods, one is explicit and other one is implicit. First, we give an explicit finitedifference scheme which was derived by Yuste [19]. A uniform grid is placed on thespace-time domain, that is, xj D j x and tm D m t. The numerical approximationof the unknown function u.x; t/ at the point .xj ; tm / is denoted by Ujm and it isobtained by Ujm umj u.xj ; tm /. The time-fractional Riemann-Liouville derivativeis discretized by Grunwald-Letnikov derivative [13]. The explicit finite differencescheme is given by the following formula
X m .1 / umC1 j D um j C S !k mk uj1 2ujmk C ujC1 mk ; (13) kD0
where .1 / .1 / k2C .1 / !0 D 1; !k D !k1 : (14) k
The numerical scheme (13) is stable on the interval 0 S 1=2.2 /, whereS D k t = x2 , for more details about the stability condition, see [19]. Note thatin this work, we only develop Matlab codes for uniform grid; for the case of non-uniform grid, see for example [20]. We use the implicit finite difference scheme derived by Langlands and Henry [7].The numerical scheme is given by
j1 C .1 C 2S /uj S ujC1 D uj S um m m m1
X m .1 / mk C S !k uj1 2ujmk C ujC1 mk : (15) kD1
The above scheme (15) is unconditionally stable, see Theorem 2.1 in [12]. For moretechnical details, readers are referred to [7, 12].
3.1 Discretization of Boundary Conditions
Dirichlet boundary conditions, u.x; t/ D 0, are discretized in the standard wayand implemented naturally, where x represents left or right boundary. However,Neumann boundary conditions, ux .x; t/ D 0, are discretized by the second orderdifference formula, that is, .um jC1 uj1 /=2 x, where x represents left or right m
boundary. Time-Fractional Diffusion Equations 19
In the case of left Neumann boundary condition, we have um m 1 D u1 and lettingj D 0 in the explicit scheme (13), we obtain
X m .1 / mk umC1 0 0 C 2S D um !k u1 u0mk : (16) kD0
On the other hand, the implicit scheme (15) yields
X m .1 / mk .1 C 2S /um 0 2S u1 D u0 m m1 C 2S !k u1 u0mk : (17) kD1
Similarly, we can obtain the schemes at the right boundary.
4 Matlab Codes
The following Matlab code is used for the computation of the exact solution (12).
Listing 1 Matlab code for exact solution 1 clc, clear all 2 format long e 3 syms x xx c_n 4 n = 55; L = 10; T = [0.1 1 5]; 5 dx = 0.1; x1 = (0:dx:L); 6 x0 = 1; gma = 0.5; K_gma = 1; v = 1; 7 ExactSol = zeros(length(x1), length(T)); 8 for k = 1:length(T) 9 a1 = 1/sqrt(4*K_gma*T(k)^gma); 10 gama = (-1).^((1:n)-1) ... 11 ./(factorial((1:n)-1).*gamma(1-0.5*gma*(1:n))); 12 xx = ((x-x0).^2/(K_gma*T(k)^gma)).^(0.5*((1:n)-1)) ... 13 - ((x+x0).^2/(K_gma*T(k)^gma)).^(0.5*((1:n)-1)); 14 c_n = a1*sum(gama.*xx); 15 ExactSol(:,k) = subs(c_n, x1); 16 end 17 plot(x1,ExactSol), 18 axis([0 5 0 0.72])
The following Matlab code is used for the computation of the numerical solutionswhich is based on the explicit finite difference scheme (13). The code can be easilymodified for various initial and boundary conditions. 20 I Ali et al.
Listing 2 Matlab code for explicit finite difference scheme 1 clc, clear all 2 gma = 0.5; S_gma = 0.33; 3 K_gma = 1; L = 10; 4 dt = 0.001; T = 5; 5 t = (0:dt:T); M = length(t); 6 dx = sqrt(K_gma*dt.^gma./S_gma); 7 x1 = (0:dx:1); x2 = (1+dx:dx:L); 8 x = [x1 1 x2]; J = length(x); 9 for a = 1:J 10 if x(a)==1; aa = a; break, end 11 end 12 w = ones(1,M); 13 for k = 2:M 14 w(k) = (1-(2-gma)/(k-1))*w(k-1); 15 end 16 V = zeros(J,M); V(aa,1) = 1/dx; 17 for m = 2:M 18 for j = 2:J-1 19 V(j,m) = V(j,m-1) + ... 20 S_gma*sum(flipud(w(1:m-1)')'.*(V(j-1, 1:m-1)... 21 - 2*V(j,1:m-1) + V(j+1,1:m-1))); 22 end 23 end 24 plot(x,V(:,t==0.1),'o', x, V(:,t==1),'o', x, V(:,t==5),'o') 25 axis([0 5 0 0.72])
The following Matlab code is used for the computation of the numerical solutionswhich is based on the implicit finite difference scheme (15). The code can be easilymodified for various initial and boundary conditions.
Listing 3 Matlab code for implicit finite difference scheme 1 function ImplicitSolution 2 clc, clear all 3 alfa = 0.5; K_alfa = 1; L = 10; 4 dt = 0.01; T = 5; 5 t = (0:dt:T); M = length(t); 6 dx = 0.1; x = (0:dx:L); J = length(x); 7 S_alfa = K_alfa*dt^alfa/dx^2; 8 for a = 1:J 9 if x(a)==1; aa = a; break, end 10 end 11 w = ones(1,M); 12 for k = 2:M 13 w(k) = (1-(2-alfa)/(k-1))*w(k-1); 14 end 15 V = zeros(J,M); V(aa,1) = 1/dx; 16 d = S_alfa*ones(1,J-2); 17 E = diag(1+2*d) + diag(-d(1:J-3),1) + diag(-d(2:J-2),-1); 18 for m = 2:M Time-Fractional Diffusion Equations 21
19 V(2:J-1, m) = BTCS(E, S_alfa, V(:, 1:m), w(2:m));20 end21 plot(x,V(:,t==0.1), x, V(:,t==1), x, V(:,t==5))22 axis([0 5 0 1])23 end24 %########################################################25 function vnew = BTCS(E, s, U, v)26 v = flipud(v'); [J, m] = size(U);27 uold = U(2:J-1,m-1);28 V1 = U(1:J-2, 1:m-1);29 V2 = U(2:J-1, 1:m-1);30 V3 = U(3:J, 1:m-1);31 V4 = V1 - 2*V2 + V3;32 lhs = uold + s*(V4*v);33 vnew = E\lhs;34 end
5 Numerical Experiments
In this section, we provide several examples which are solved by using above Matlabcodes on an Intel Core-i7 machine. The computation time is given for each problem.Example 1 Consider the TFDE @u 1 @2 u D K 0 Dt ; (18) @t @x2
on the domain x > 0 and t > 0 with the initial data is taken as a delta function atx D 1. Absorbing and reflecting boundary conditions are taken (one by one) at leftboundary x D 0, where as u.x; t/ ! 0 as x ! 1. Exact solution is given by
u.x; t/ D W.x x0 ; t/ ˙ W.x C x0 ; t/; (19)
where W.x; t/ is the solution of TFDE (18) over the whole real line with decayingboundary conditions when jxj becomes large. In series form, u.x; t/ is expressed as 1 1 X .1/n .x x0 / n=2 .x C x0 / n=2 u.x; t/ D ˙ 4K t nD0 nŠ .1 .1 C n/=2/ K t K t (20)Note the minus sign is taken in the case of absorbing BC and plus sign is taken in thecase of reflecting BC. Figure 1 shows the numerical solutions at times t D 0:1; 1; 5.The data used for numerical computation is t D 0:001, T D 5, D 0:5, S D0:33; and the computational time is 9:91 s. 22 I Ali et al.
Fig. 1 Solutions of time fractional diffusion equation 18 with absorbing and reflecting boundaryconditions at the left boundary. Initial condition is taken as delta function at x D 1, where D 0:5.(a) Absorbing boundary condition. (b) Reflecting boundary condition
Fig. 2 Solutions of time fractional diffusion equation 18 with absorbing boundary conditions atleft and right boundaries. Initial condition is taken as delta function at x D 0. Cusp shape is thedistinct feature of the curve in the case D 0:5. (a) Standard diffusion D 1:0. (b) Anomalousdiffusion D 0:5
Example 2 Consider the TFDE (18) on a box 1 x 1 and t > 0 with absorbingboundaries at x D 1 and 1. Initial data is given by delta function at x D 0. Exactsolution of the problem is given by 1 X u.x; t/ D ŒW.x C 4n; t/ W.4n x C 2; t/ : (21) nD1
Figure 2 shows the numerical solutions at times t D 0:05; 0:1 for D 1 andat times t D 0:006; 0:1 for D 0:5. The data used for numerical computation is Time-Fractional Diffusion Equations 23
Fig. 3 Solutions of time fractional diffusion equation 18 with reflecting boundary conditions atleft and right boundaries. Initial condition is taken as delta function at x D 0. Cusp shape is thedistinct feature of the curve in the case D 0:5. (a) Standard diffusion D 1:0. (b) Anomalousdiffusion D 0:5
x D 0:1, T D 0:1, D 1:0, S D 0:49; and the computational time is 9:5 s. In thecase of D 0:5, we choose S D 0:33, and the computational time is 14:71 s.Example 3 We solve TFDE (18) on a box 1 < x < 1 and t > 0 with reflectingboundaries at x D 1 and 1. Initial data is given by delta function at x D 0. Exactsolution of the problem is given by 1 X u.x; t/ D ŒW.x C 4n; t/ C W.4n x C 2; t/ : (22) nD1
Figure 3 shows the numerical solutions at times t D 0:05; 0:1; 0:2 for D 1 andat times t D 0:006; 0:1; 0:2 for D 0:5. The data used for numerical computation is x D 0:01, T D 0:2, D 1:0, S D 0:49; and the computational time is 50:12ṡ. Inthe case of D 0:5, we choose S D 0:33, x D 0:1, and the computational timeis 99:97 s.
6 Conclusions
Time fractional advection-diffusion equation is derived under the CTRW framework[8, 9]. The fundamental solution of the time-fractional diffusion equation is obtainedby applying Fourier and Laplace transforms and which is finally expressed in termsof Fox’s function by using Mellin transform [14, 18]. Matlab codes are developed for obtaining the exact and the numerical solutionsof time-fractional diffusion equation. The code for the exact solution is based on the 24 I Ali et al.
work of Wyss [18] and Schneider & Wyss [14], whereas the code for finding thenumerical solution is based on the work of Yuste & Acedo (explicit case) [19] andLanglands & Henry (implicit case) [7]. We have given several examples that illustrate the effectiveness of the codes.Essentially, numerical solution are found both on finite and infinite domainswith absorbing and reflecting boundary conditions. Initial conditions are takenas nonsmooth functions (delta functions) and it is observed that in the case ofanomalous diffusion the cusp shape remains in the solutions compared to standarddiffusion where the solution smooths up as time increases.
Acknowledgements The author would like to acknowledge the support provided by KingAbdulaziz City for Science and Technology (KACST) through the Science Technology Unit atKing Fahd University of Petroleum and Minerals (KFUPM) for funding this work through projectNo. 14-OIL280-04.
References
1. Ali, I., Malik, N.A.: Hilfer fractional advection–diffusion equations with power-law initial condition; a numerical study using variational iteration method. Comput. Math. Appl. 68(10), 1161–1179 (2014) 2. Ali, I., Malik, N.A., Chanane, B.: Time-fractional nonlinear gas transport equation in tight porous media: an application in unconventional gas reservoirs. In: 2014 International Con- ference on Fractional Differentiation and Its Applications (ICFDA), Catania, pp. 1–6. IEEE (2014) 3. Chen, W., Sun, H., Zhang, X., Korošak, D.: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59(5), 1754–1758 (2010) 4. Fedotov, S., Iomin, A.: Migration and proliferation dichotomy in tumor-cell invasion. Phys. Rev. Lett. 98(11), 118,101 (2007) 5. Jeon, J., Tejedor, V., Burov, S., Barkai, E., Selhuber-Unkel, C., Berg-Sørensen, K., Oddershede, L., Metzler, R.: In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. Phys. Rev. Lett. 106(4), 048,103 (2011) 6. Koch, D., Brady, J.: Anomalous diffusion in heterogeneous porous media. Phys. Fluids (1958– 1988) 31(5), 965–973 (1988) 7. Langlands, T., Henry, B.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205(2), 719–736 (2005) 8. Luchko, Y.: Anomalous diffusion: models, their analysis, and interpretation. Adv. Appl. Anal. 115–145 (2012) 9. Luchko, Y., Punzi, A.: Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. GEM Int. J. Geomath. 1(2), 257–276 (2011)10. Malik, N., Ali, I., Chanane, B.: Numerical solutions of non-linear fractional transport models in unconventional hydrocarbon reservoirs using variational iteration method. In: 5th International Conference on Porous Media and Their Applications in Science (Engineering and Industry, Eds, ECI Symposium Series, Volume) (Hawaii, 2014). http://dc.engconfintl.org/ porous_media_V/4311. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)12. Murio, D.A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56(4), 1138–1145 (2008) Time-Fractional Diffusion Equations 25
13. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Frac- tional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic, New York (1998)14. Schneider, W., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30(1), 134– 144 (1989)15. Tabei, S., Burov, S., Kim, H., Kuznetsov, A., Huynh, T., Jureller, J., Philipson, L., Dinner, A., Scherer, N.: Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl. Acad. Sci. 110(13), 4911–4916 (2013)16. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge/New York (1995)17. Weiss, M., Elsner, M., Kartberg, F., Nilsson, T.: Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J. 87(5), 3518–3524 (2004)18. Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986)19. Yuste, S., Acedo, L.: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005)20. Yuste, S.B., Quintana-Murillo, J.: A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput. Phys. Commun. 183(12), 2594–2600 (2012) hom*oclinic Structure for a GeneralizedDavey-Stewartson System
Ceni Babaoglu and Irma Hacinliyan
Abstract In this study, we analyze the hom*oclinic structure for the generalizedDavey-Stewartson system with periodic boundary conditions. This system involvesthree coupled nonlinear equations and describes (2 C 1) dimensional wave prop-agation in a bulk medium composed of an elastic material with coupled stresses.We first provide linearized stability analysis of the plane wave solutions of thegeneralized Davey-Stewartson system. Then, give an analytic description of thecharacteristics of hom*oclinic orbits near the fixed point by finding soliton typesolutions. These solutions are derived via Hirota’s bilinear method. We also showthat two of these solutions form a pair of symmetric hom*oclinic orbits and all thesesymmetric hom*oclinic orbit pairs construct the hom*oclinic tubes.
1 Introduction
hom*oclinic orbits are important for the study of chaos in deterministic nonlineardynamics. In a neighborhood of such an orbit, an extended knowledge of geometricstructures lead one to a better understanding of chaotic dynamics. The hom*oclinicstructure of nonlinear Schrodinger (NLS) equation is considered by Ablowitz andHerbst in [1]. They have shown how this structure associated with the cubic NLSequation may be obtained from the N-soliton solutions of the defocusing NLSequation. We follow a similar approach and first observe that the fixed point inthe GDS system is hyperbolic. Then, we analyze the hom*oclinic structure for thegeneralized Davey-Stewartson (GDS) equations.
C. Babaoglu () • I. HacinliyanFaculty of Science and Letters, Department of Mathematics, Istanbul Technical University, 34469Maslak-Istanbul, Turkeye-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 27J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_3 28 C. Babaoglu and I. Hacinliyan
The generalized Davey-Stewartson system has been introduced in [2] to study(2 C 1) dimensional wave propagation in a bulk medium composed of an elasticmaterial with coupled stresses. They are given by
Q 2A C k2 iAQ C pAQ C rAQ D qjAj .3 Q 1; C 1 Q2; /A; Q 2! .c2g c21 /Q 1; c22 Q 1; .c21 c22 /Q 2; D 3 k2 .jAj Q 2 / ;
.c2g c22 /Q 2; c21 Q2; .c21 c22 /Q 1; D 1 k2 .jAj Q 2 / ; (1)
where and are spatial coordinates and is time; AQ is the complex amplitude of theshort transverse wave mode, and Q1 and Q2 are long longitudinal and long transversewave modes, respectively. The coefficients that appear in (1) can be given as follows
C 2 c21 D ; c22 D ; cg D c22 .k C 8m2 k3 /=!; 0 0 B A C 2B 1 D c21 2c22 C ; 3 D c21 C ; 0 20 1 2 c2 k6 32 pD .cg c22 24m2 c22 k2 /; r D 2 .1 C 8m2 k2 /; q D ; 2! 2! !D1 .2k; 2!/
where k is the wave number, ! is the frequency, m, , , A and B are materialconstants, cg is the group speed of transverse waves whereas c1 and c2 are phasespeeds of longitudinal and transverse waves, respectively. Also the frequency andthe dispersion relation is as follows 1 ! D c2 k.1 C 4m2 k2 / 2 ; D1 .k; !/ D ! 2 c21 k2 4.1 C /c22 m2 k4 : (2)
A simple algebra shows that the coefficients p, q and r are all positive. In terms ofdimensionless variables, AQ D u; Q1 D '1 ; Q2 D '2 ; and
12 .c2g c21 /2 c2g c21 1 .c2g c21 / D t; D x; D y; (3) 34 k4 r 3 k 2 32 k2
the three coupled evolution equations take the form
iut C ıuxx C uyy D juj2 u C b.'1;x C '2;y /u; '1;xx C m2 '1;yy C n'2;xy D .juj2 /x ; '2;xx C m1 '2;yy C n'1;xy D .juj2 /y ; (4) hom*oclinic Structure for a Generalized Davey-Stewartson System 29
where the non-dimensional coefficients are
p 1 2 q12 .c2g c21 /2 c2g c21 1 2 ıD . / ; D ; bD . /; r 3 r34 k4 2!r 3 c21 3 2 c22 3 2 c22 c2g c21 c22 3 m1 D . / ; m 2 D . / ; D ; n D . /; c21 c2g 1 c21 c2g 1 c21 c2g c21 c2g 1
so .1 /.m1 m2 / D n2 , m1 > m2 and < 1. It should be also noted that
c2g c22 D 4m2 c42 k4 .3 C 16m2 k2 /=! 2 (5)
is always positive. There exists a critical wave number 1 kc2 D Œc21 4c22 C c1 .c21 C 8c22 / 2 =.32c22 m2 / (6)
such that c21 c2g < 0 if k > kc and c21 c2g > 0 if k < kc because c21 > c22 . (Thecase where k D kc corresponds to long-wave short-wave resonance since the phasespeed of longitudinal wave, c1 , is equal to the group speed of the transverse wave,cg .) Thus, depending on the wave number k chosen, the coefficients of the secondand third equations of the GDS system may change their sign. For example, therespective sign of .m1 ; m2 ; / is .; ; C/ if k > kc and is .C; C; / if k < kc . Now the GDS system will be classified according to the values of parameters. Infact, since ı > 0 the first equation is always elliptic. The classification of the lasttwo coupled equations of (4) is based on eigenvalues of the coefficient matrix of afirst order linear system with four equations equivalent to the second-order linearsystem, (4)2 and (4)3 . Therefore, system (4) can be classified as elliptic-elliptic-elliptic, elliptic-elliptic-hyperbolic, and elliptic-hyperbolic -hyperbolic according tothe respective sign of .m1 ; m2 ; /: .C; C; C/, .C; C; /, and .; ; C/ [3]. Thedescriptions given above lead that m1 , m2 and cannot be positive at the same time.Thus, the last two cases correspond to physical cases.
2 Linearized Stability Analysis
In this section, the GDS system (4) is considered with the following boundary
u.x; y; t/ D u.x C l1 ; y C l2 ; t/; 'i .x; y; t/ D 'i .x C l1 ; y C l2 ; t/; i D 1; 2; (7) 30 C. Babaoglu and I. Hacinliyan
and initial conditions
u.x; y; 0/ D u0 .x; y/; u0 .x; y/ D u0 .x; y/; 'i .x; y; 0/ D 'i 0 .x; y/; 'i 0 .x; y/ D 'i 0 .x; y/; i D 1; 2; x; y 2 ˝: (8)
The starting point for developing hom*oclinic-type solutions is to find a suitable fixed 2point. In this case, for .u; '1 ; '2 /, the fixed point will be chosen as .a eijaj t ; 0; 0/,where a is any complex number. Next, we investigate the stability of fixed point by considering small perturba-tions of the form, 2 u D a eijaj t .1 ".x; y; t//; '1;x D '1" .x; y; t/; '2;y D '2" .x; y; t/: (9)
Substituting (9) into (4) and keeping linear terms leads one to
i"t C "xx C "yy D jaj2 ." C " / b.'1" C '2" /; " " " '1;xx C m2 '1;yy C n'2;xx D jaj2 ." C " /xx ; " " " '2;xx C m1 '2;yy C n'1;yy D jaj2 ." C " /yy : (10)
Assuming the following form of solutions for the linearized system we obtain
".x; y; t/ D AŒei.n xCNn y/Cn t C BŒei.n xCNn y/Cn t ; '1" .x; y; t/ D CŒei.n xCNn y/Cn t C ei.n xCNn y/Cn t ; (11) '1" .x; y; t/ D DŒei.n xCNn y/Cn t C ei.n xCNn y/Cn t ;
where the growth rate n of the nth mode is
2jaj2 .m1 1/ n˙ D ˙jı2n C Nn 2 jŒ 1 1=2 : (12) .ı 1/.2n C m1 Nn 2 /
For the proper choice of the parameters we see that the fixed point is hyperbolic.
3 hom*oclinic Structure of the GDS System
hom*oclinic structure of the GDS system will be studied in this section. One of thetwo physical cases, i.e. the respective sign of .m1 ; m2 ; / is .C; C; / with m1 > m2 ,will be given in detail, since they have similar approach and result. hom*oclinic Structure for a Generalized Davey-Stewartson System 31
In order to write the GDS system in Hirota bilinear form we need to make thefollowing restrictions on the parameter values
.1 ım1 / m2 D m1 n; D 1 n; bD : (13) ı1We have to note that under these conditions imposed on the parameters generalizedDavey-Stewartson equations are isomorphic to that of the standard integrableDavey-Stewartson equations. Since we are interested in a hom*oclinic orbit of the fixed point, the followingsubstitutions will be made for (4)
G ia2 t 2.1 ı/ 2.1 ı/ uDa e ; '1 D .log F/x ; '2 D .log F/y ; (14) F .1 m1 / .1 m1 /
where G is complex, a and F are real. After straightforward calculations we writethe system in Hirota bilinear form
.iD t C ıD 2x C D 2y C a2 /G F D ˇG F; 2ˇ .D 2x C m1 D 2y /F F 2a2 jGj2 D F F: (15)
Now, Eqs. (15) are assumed to possess the following solution functions:
G D 1 C Œb1 ei. p1 xCp2 y/ C b2 ei. p1 xCp2 y/ e˝tC C b3 e2.˝tC / ; F D 1 C b4 Œei. p1 xCp2 y/ C ei. p1 xCp2 y/ e˝tC C b5 e2.˝tC / ; (16)
where p1 ; p2 ; ˝; ; b4 ; b5 are real and b1 ; b2 ; b3 are complex. Collecting coefficientsleads to the following relations among the constants
ıp21 C p22 C i˝ b1 D b2 D b4 ; ˇ D a2 ıp21 C p22 i˝ .ıp21 C p22 C i˝/3 2 ıp21 C p22 2 2 b3 D b ; b5 D Œ1 C . / b4 ; ˝ 2 .ıp21 C p22 i˝/ 4 ˝ 2a2 .m1 1/ ˝˙ D ˙jıp21 C p22 jŒ 1 1=2 : (17) .ı 1/. p21 C m1 p22 /
Here the result obtained for ˝ coincides with the result in the linearized stabilityanalysis. At this step, p1 and p2 will be chosen as s s .m1 1/ .m1 1/ p1 D a sin ; p2 D a sin ; (18) .ı 1/ .ı 1/m1 32 C. Babaoglu and I. Hacinliyan
where 0 is a constant. Then, we get the following solutions for the GDSsystem (4) as
2t 1 C 2b˙ cos. p1 x C p2 y/e˝˙ tC C b2˙ e2.˝˙ tC / sec2 u˙ D aeia ; 1 C 2 cos. p1 x C p2 y/e˝˙ tC C e2.˝˙ tC / sec2
˙ 4.ı 1/p21 2e˝˙ tC C cos. p1 x C p2 y/.1 C e2.˝˙ tC / sec2 / '1;x D e˝˙ tC ; .1 m1 / .1 C 2 cos. p1 x C p2 y/e˝˙ tC C e2.˝˙ tC / sec2 /2
˙ 4.ı 1/p22 2e˝˙ tC C cos. p1 x C p2 y/.1 C e2.˝˙ tC / sec2 / '2;y D e˝˙ tC ; .1 m1 / .1 C 2 cos. p1 x C p2 y/e˝˙ tC C e2.˝˙ tC / sec2 /2 (19)
where b˙ D cos.2/ sin.2/.
4 Conclusions
We will conclude by discussing the hom*oclinic structure of the GDS system (4).The solutions (19) have the characteristics of a hom*oclinic orbit having spatiallyperiodic fixed points with periods p1 D 2m=L1 and p2 D 2m=L2 . The solution C C 2.uC ; '1;x ; '2;y / leaves the ring of fixed points .a eia t ; 0; 0/ as t ! 1 2and returns to the ring .a eia t b2 ; 0; 0/ as t ! 1. On the other hand, 2the solution .u ; '1;x ; '2;y / leaves the ring of fixed points .a eia t ; 0; 0/ as 2t ! 1 and returns to the ring .a eia t b2C ; 0; 0/ as t ! 1. Moreover, C C phase shift is seen between the solutions .uC ; '1;x ; '2;y / and .u ; '1;x ; '2;y /. C C C CBesides, while .uC ; '1;x ; '2;y /.x0 ; y0 ; t/ forms hom*oclinic orbit .uC ; '1;x ; '2;y /.x0 C2m=p1 ; y0 C 2m=p2 ; t/ forms hom*oclinic orbit as well and this result also C Cholds for .u ; '1;x ; '2;y /. Finally, we can say that the solutions .uC ; '1;x ; '2;y / and .u ; '1;x ; '2;y / form a pair of symmetric hom*oclinic orbits and all of theseorbit pairs construct hom*oclinic tubes. As an illustration of the dynamical behaviorseparated by the hom*oclinic orbits, the amplitude C D juC j for D 10 and D ju j for D 10 are shown in Fig. 1. hom*oclinic Structure for a Generalized Davey-Stewartson System 33
Fig. 1 The solution of (4) for (a) ı D 0:5, D =3, D 0:5, D 1, a D 1, m1 D 1:75,m2 D 0:25 and n D 1:5 for (b) ı D 1:5, D =3, D 0:1, D 1, a D 1, m1 D 1:2, m2 D 0:1and n D 1:1
References
1. Ablowitz, M.J., Herbst, B.M.: On hom*oclinic structure and numerically induced chaos for the nonlinear Schrodinger equation. SIAM J. Appl. Math. 50, 339–351 (1990)2. Babaoglu, C., Erbay, S.: Two-dimensional wave packets in an elastic solid with couple stresses. Int. J. Non-Linear Mech. 39, 941–949 (2004)3. Babaoglu, C., Eden, A., Erbay, S.: Global existence and nonexistence results for a generalized Davey-Stewartson system. J. Phys. A Math. Gen. 37, 11531–11546 (2004) Numerical Simulations of the Dynamicsof Vortex Rossby Waves on a Beta-Plane
L.J. Campbell
Abstract Observational analyses of hurricanes in the tropical atmosphere indicatethe existence of spiral rainbands which propagate outwards from the eye and affectthe structure and intensity of the hurricane. These disturbances may be described asvortex Rossby waves. It has been suggested that vortex Rossby waves may playa role in the eyewall replacement cycle observed in tropical cyclones in whichconcentric rings of high-intensity wind develop and propagate in towards the centreof the cyclone. In previous work with Nikitina, we investigated the dynamics ofvortex Rossby waves in a cyclonic vortex in a two-dimensional configuration ona beta-plane and derived analytical solutions. In the current work, some results ofnumerical simulations are presented.
1 Introduction
A tropical cyclone is an axisymmetric vortex of rotating fluid flow that originatesover the tropical ocean and is driven by heat transfer from the ocean. An intensetropical cyclone with high sustained wind speed tends to develop an eye, anarea of relative calm at the centre of the circulation surrounded by a ring ofintense rainbands with maximum tangential wind speed called the eyewall. A largepercentage of tropical cyclones with wind speeds of at least 60 ms1 undergoan eyewall replacement cycle [8] in which an outer ring of rainbands called asecondary eyewall develops outside the original eyewall [25], strengthens andcontracts inwards, while the original inner eyewall weakens and is replaced by thesecondary eyewall. This process repeats over a time frame of up to 2 days [20]. Vortex Rossby waves are oscillations that occur within cyclonic vortices as aresult of the radial gradient of cyclonic vorticity. In the context of hurricanes ortropical cyclones, vortex Rossby waves have been observed as spiral rainbandspropagating outwards from the eye. It has been suggested that vortex Rossbywave mean-flow interactions may play a role in the secondary eyewall formation[3, 13, 14], although this role appears to be less significant [1, 10] to that played
L.J. Campbell ()Carleton University, Ottawa, ON, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 35J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_4 36 L.J. Campbell
by the dominant system-scale processes such as surface friction and boundary layereffects [15, 21]. Vortex Rossby wave mean-flow interactions take place primarily in the vicinityof the critical radius where the mean angular velocity of the vortex is equal tothe phase speed of the waves. This suggests that vortex Rossby wave critical layertheory could be helpful in advancing our understanding of the mechanisms by whichthese waves could contribute to the development of secondary eyewalls in tropicalcyclones. The theory for the analogous problem of barotropic planetary Rossbywaves in a rectangular domain on a beta-plane is well-developed and that problemhas been studied extensively using analytical and numerical methods, e.g. [2, 4–6, 22–24]. Recent analytical investigations with Nikitina [16–18] of the barotropicvortex Rossby wave configuration indicate several qualitative similarities betweenthe two problems. Nikitina and Campbell [17] presented analytical solutions for a configurationinvolving a cyclonic vortex with angular velocity ˝.r/ N on a horizontal plane definedin terms of polar coordinates r and . The f -plane approximation was made inthis preliminary investigation, i.e., the Coriolis parameter was approximated by aconstant. A wave of the form cos .k !t/ was forced at a fixed radius r D r1representing the location of the primary eyewall and the linearized barotropicvorticity equation was solved to determine the amplitude of the forced wave asa function of the radial variable r and time t. For a special quadratic profile of˝.r/, N exact analytical solutions were obtained in terms of hypergeometric functionsfor waves with steady amplitude and these solutions were then used to find late-time asymptotic solutions for waves with time-dependent amplitude, first in theouter region away from the critical radius and then in the inner region in thevicinity of the critical radius. The solutions obtained show that the wave amplitudeis greatly attenuated at the critical radius as the wave propagates outwards fromthe eyewall. This can be interpreted as wave absorption by the mean flow. In thelimit of infinite time, the time-dependent solution in the outer region approachesthe corresponding steady solution, but the inner solution grows with time. Theseconclusions are consistent with the situation that is attained in the case of forcedplanetary Rossby waves in a rectangular domain on a ˇ-plane where the Coriolisparameter is considered to be a linear function of latitude. Nikitina and Campbell [18] extended the investigation of [17] to include thenonlinear terms in the governing equations, as well as the terms arising from thelatitudinal gradient of the Coriolis parameter which give the so-called ˇ-effect.The wave amplitude was then considered as an expansion in powers of two smallparameters representing nonlinearity and the ˇ-effect with the leading-order termsin the expansion being given by the outer and inner solutions derived in [17]. It wasfound that nonlinearity gives rise to higher wavenumbers in multiples of the forcedwavenumber k, a zero wavenumber component which represents a divergence ofmomentum flux into the mean flow, and an inward displacement of the instantaneouscritical radius. The variation of the Coriolis force gives rise to wave modes withwavenumbers .k ˙1/. In the case where the forced wavenumber k D 1, the variation Numerical Simulations of the Dynamics of Vortex Rossby Waves on a Beta-Plane 37
of the Coriolis force thus introduces an additional zero wavenumber component andhence contributes to the evolution of the mean flow. The model employed by Nikitina and Campbell [17, 18] is highly idealizedconsidering that it is two-dimensional and does not take into account effectssuch as diabatic heating and boundary layer friction which form the basis of theconventional theory of tropical cyclone secondary eyewall generation. But on theother hand, the simplicity of the model allowed us to examine Rossby wave mean-flow interaction mechanisms in isolation of other effects and the analytical solutionsobtained gave us some insight into the temporal evolution of the solution. Themain features of the solutions, namely the critical layer absorption of the waves,the development of concentric rings of high wave activity and the changes in thelocation of these rings with time, are consistent with the hypothesis that vortexRossby waves contribute to the secondary eyewall replacement cycle. However, theweakly-nonlinear analysis of [18] is valid only for finite time since the higher-orderterms in the perturbation expansion for the solution grow with time. Multiple-time-scale asymptotic analyses would be needed to continue the analytical investigationto later time and to higher orders in the expansion parameters. In analogy with theclassical rectangular configuration [22, 24], it can be anticipated that there would bewave reflection at the critical radius at late time, possibly leading to eventual wavebreaking and instabilities. The purpose of the current investigation is to use numerical methods to furtherelucidate the critical layer behaviour of forced vortex Rossby waves propagatingoutwards in a cyclonic vortex and investigate the effects of the critical layerinteraction on the evolution of the mean vortex. The model used is described inSect. 2 and some preliminary numerical results are presented in Sect. 3.
2 Configuration and Equations
The numerical simulations presented here make use of a two-dimensional model forbarotropic vortex Rossby waves in a cyclonic vortex on a ˇ-plane, a horizontal planeon which the Coriolis parameter is approximated by a linear function of latitude.The flow is described in terms of a streamfunction and is represented by thenondimensional barotropic vorticity equation (see, e.g., [9]). This can be written inpolar coordinates r and as [17]
1 1 ˇ r 2 t r 2 r C r r 2 sin C ˇr cos D 0; (1) r r rwhere the subscripts denote partial differentiation with respect to r and . The non-dimensional parameter ˇ is the latitudinal gradient of the Coriolis parameter f . Thecorresponding dimensional quantity is [9] 2˝Earth ˇ D cos 0 ; (2) REarth 38 L.J. Campbell
where 0 is the latitude of the centre of the vortex, ˝Earth 7 105 s1 is theangular velocity of the Earth’s rotation, and REarth 6:3106 km is the radius of theEarth. Near the equator where 0 is close to zero, ˇ is close to 2.2 1011 m1 s1 .For a typical tropical cyclonic vortex radius L of about 2 3 105 m, and a typicaltangential wind speed U of about 30-100 ms1 , ˇ D L2 ˇ =U 102 and canthus be considered as a small parameter in the nondimensional problem. The limitof ˇ ! 0 gives the f -plane approximation, in which the Coriolis parameter isapproximated by a constant. The vortex wave is represented as a small-amplitude perturbation to the basicflow of the cyclonic vortex. The streamfunction N .r/, the angular velocity ˝.r/ N andthe azimuthal component v.r/ N of the velocity of the basic flow are related by
v.r/ N D N 0 .r/; v.r/ N D r˝.r/; N (3)
where the prime denotes differentiation with respect to r. The total streamfunctionis written as
.r; ; t/ D N .r/ C " .r; ; t/; (4)
where the parameter " is the ratio of the dimensional magnitude of the perturbationto that of the basic flow. Observations of the asymmetric spiral bands that aredescribed as vortex Rossby waves in hurricanes indicate small deviations fromcircular symmetry. For example, analyses [7, 19] of Doppler wind and Omegadropwindsonde data from Hurricane Gloria (1985) show that within 500 km of thehurricane centre, the asymmetric components of the tangential wind correspondingto azimuthal wavenumbers 1, 2, 3 and 4 are much smaller than the symmetrictangential wind. This would suggest that " can be considered as a small parameterin the nondimensional problem and thus justifies the use of linear [14, 17] andweakly-nonlinear analyses [18] as a means to provide insight into a fully nonlinearconfiguration that can only be examined numerically. Substituting (4) into (1) gives a nonlinear equation for the perturbation @ vN @ 1 d vN C r2 vN 0 C @t r @ r dr r ˇ ˇ " 2 2 sin C ˇ r cos C vN cos D . rr r r /: (5) r " r
This equation is examined in the annular region r1 r < 1; 0 < 2 shownin Fig. 1 for t 0. The waves are forced at r D r1 by a boundary condition of theform .r1 ; ; t/ D cos .k !t/, where k is the azimuthal wave number and ! isthe circular frequency of the wave. 2 The mean angular velocity profile is taken to be either ˝.r/N D ˝0 e˛r , assuggested by Martinez et al. [11, 12] as a reasonable representation of typical flows Numerical Simulations of the Dynamics of Vortex Rossby Waves on a Beta-Plane 39
Fig. 1 Configuration: the waves are forced by a sinusoidal boundary condition at r D r1 andpropagate outwards [17]
– Ω Ω0 ω/k
– Ω=Ω0 exp(–αr2) – Ω=Ω0 (1–αr2)
r1 rc √2rc r
Fig. 2 The mean flow angular velocity profiles used in the numerical simulations and the intervalsin which the analytical solutions obtained in [17, 18] are valid
observed in tropical cyclones, or ˝.r/ N D ˝0 .1 ˛r2 /, as suggested by Brunetand Montgomery [3] as an approximation for the exponential profile of [11, 12].These functions are shown in Fig. 2. The positive constants ˛ and ˝0 are chosenso that there is a critical radius within the domain at some value r D rc where˝.r N c / D !=k, as shown in Fig. 2. The numerical method used for the solution of (5)involves a pseudo-spectral approximation and is based on that described in [4]. 40 L.J. Campbell
3 Results
The results presented here are for the quadratic profile shown in Fig. 2. The wavesare forced at r D r1 D 2 with a frequency of ! D 1 and a wavenumber of k D 2or k D 1. The nonlinear parameter is set to " D 0:05 and the gradient of planetaryvorticity is ˇ D 0 or ˇ D 0:05. In each case, the nondimensional parameters ˛ and˝0 are chosen so that the critical radius is at r D rc D 12. Three configurations are examined here: ˇ D 0, k D 2 ( f -plane); ˇ D 0:05,k D 2 (ˇ-plane); ˇ D 0:05, k D 1 (ˇ-plane). In each case, contour plots ofthe perturbation streamfunction and the perturbation vorticity and a graph of theFourier or wavenumber spectrum of the perturbation streamfunction are presentedat nondimensional time t D 100. The perturbation streamfunction is written as 1 X .r; ; t/ D .r; ; t/ei (6) D1
and j.r; ; t/j is shown as a function of wavenumber at r D rc D 12 and t D 100.The zero wavenumber component in the sum (6) gives the change in the total meanflow. The total mean angular velocity at location r and time t is ˝.r; t/ D ˝.r/N C ˝.r; t/, where the change in angular velocity is ˝.r; t/ D "r .r; 0; t/=r. A plotof ˝.r; t/ at r D rc D 12 and t D 100 is presented for each configuration. Fromthe asymptotic analysis of [18], it is expected that, at least within the early timeframe in which the analysis is valid, the total mean angular velocity decreases asa result of nonlinearity and, in the case where k D 1, the ˇ-effect also affects themean angular velocity. Figures 3, 4, 5, and 6 show results obtained with ˇ D 0 and k D 2. In each ofthe contour plots shown, there are 20 equally-spaced levels. The analytical resultsof [17, 18] tell us that the steady term in the solution varies like rk D r2 betweenthe inner boundary and the critical radius. This attenuation of the waves and theirabsorption by the mean flow at the critical radius is seen in the contour plot of thewave streamfunction in Fig. 3a. A ring of high vorticity is seen in the vicinity of thecritical radius in Fig. 3b. Figure 4 shows the Fourier spectrum of the perturbationstreamfunction at the critical radius and time t D 100. The black circles indicatethe forced wave modes and the unfilled circles indicate the modes generated by thenonlinear effects. There is a zero wavenumber component of the solution ( D 0)as well as higher harmonics ( D ˙4; ˙6; : : :), which are nonzero but of smallamplitude. According to Fig. 5, the mean flow decreases initially and then starts toincrease until the change becomes positive indicating a decrease in the extent ofwave absorption at late time and an evolution towards a reflecting state. This is ananalogous result to that obtained in the classical case of forced planetary Rossbywaves in a rectangular domain on a ˇ-plane [2, 4]. Numerical Simulations of the Dynamics of Vortex Rossby Waves on a Beta-Plane 41
Fig. 3 Nonlinear numerical simulations on an f -plane with k D 2: (a) Wave streamfunction; (b)Wave vorticity
|φ |
κFig. 4 Nonlinear numerical simulations on an f -plane with k D 2: Fourier spectrum of thestreamfunction amplitude at r D rc D 12
Figures 6, 7, and 8 show results obtained with ˇ D 0:05 and k D 2. Withthe addition of the ˇ-effect, Fig. 6a, b indicate that there is a greater degree oftransmission beyond the critical radius than is seen in the corresponding plotsfor the f -plane configuration. The graph of the Fourier spectrum given in Fig. 7shows that there are modes corresponding to D ˙1; ˙3; : : : (indicated by 42 L.J. Campbell
ΔΩ
Fig. 5 Nonlinear numerical simulations on an f -plane with k D 2: change in the mean flow as afunction of time at r D rc D 12
Fig. 6 Nonlinear numerical simulations on a ˇ-plane with k D 2: (a) Wave streamfunction; (b)Wave vorticity
asterisks), which are generated by the ˇ-effect, in addition to those correspondingto D 0; ˙2; ˙4; : : :. The presence of these additional modes results in a largerwave amplitude at the critical radius than that in the f -plane case shown in Fig. 4.Figure 8 indicates again a decrease in the extent of wave absorption at late time. Numerical Simulations of the Dynamics of Vortex Rossby Waves on a Beta-Plane 43
|φ |
κFig. 7 Nonlinear numerical simulations on an ˇ-plane with k D 2: Fourier spectrum of thestreamfunction amplitude at r D rc D 12
ΔΩ
Fig. 8 Nonlinear numerical simulations on a ˇ-plane with k D 2: change inthe mean flow as afunction of time at r D rc D 12
Figures 9, 10, and 11 show results obtained with ˇ D 0:05 and k D 1. Withthis smaller value of k D 1, the wave amplitude decreases less rapidly with radialdistance from the centre followed by a more rapid attenuation at the critical radius.This is seen in Fig. 9a, b. From Fig. 10 it is also seen that the ˇ-effect generatesmodes corresponding to D ˙.k 1/; ˙.k C 1/ D 0; ˙2; : : :, which coincide 44 L.J. Campbell
Fig. 9 Nonlinear numerical simulations on a ˇ-plane with k D 1: (a) Wave streamfunction; (b)Wave vorticity
|φ |
κFig. 10 Nonlinear numerical simulations on an ˇ-plane with k D 1: Fourier spectrum of thestreamfunction amplitude at r D rc D 12
with the modes generated by the effect of nonlinearity to produce terms with greatermagnitude than those shown in Fig. 7. Figure 11 shows that the mean flow evolutiontakes the form of oscillations with an amplitude that increases slowly with time. Numerical Simulations of the Dynamics of Vortex Rossby Waves on a Beta-Plane 45
ΔΩ
Fig. 11 Nonlinear numerical simulations on a ˇ-plane with k D 1: change in the mean flow as afunction of time at r D rc D 12
4 Concluding Remarks
Numerical methods were used to investigate vortex Rossby wave mean-flow inter-actions in a barotropic model of a tropical cyclone. Consistent with the analyticalsolutions presented in [17, 18], it was found that nonlinearity gives rise to higherwavenumbers, momentum flux divergence into the mean flow and an (initial) inwarddisplacement of the critical radius. The ring of high vorticity around the criticalradius resembles a developing secondary wall. The variation of the Coriolis forcegives rise to wave modes with wavenumbers ˙.k ˙ 1/ and affects the mean flowif k D 1. In each case, the dominant contributions to the vortex perturbation comefrom the zero wavenumber and low wavenumber components, at least within thetime frame shown. This is consistent with observations of Hurricane Gloria (1998)[7] analyzed and discussed by Shapiro and Montgomery [19]. These numerical results are preliminary; further numerical experimentation andanalyses are needed. Ultimately, vertical variation, diabatic heating and other effectsneed to be included for a more realistic representation, but the observations fromthe results obtained with this simplified barotropic configuration can be used as astarting point for further studies.
Acknowledgements The author would like to thank Dr. L. Nikitina for drawing the schematicdiagrams shown in Figs. 1 and 2 and the anonymous reviewer for helpful comments. 46 L.J. Campbell
References
1. Abarca, S.F., Montgomery, M.T.: Essential dynamics of secondary eyewall formation. J. Atmos. Sci. 70, 3216–3230 (2013) 2. Béland, M.: Numerical study of the nonlinear Rossby wave critical level development in a barotropic zonal flow. J. Atmos. Sci. 33, 2066–2078 (1976) 3. Brunet, G., Montgomery, M.T.: Vortex Rossby waves on smooth circular vortices: part 1. Theory. Dyn. Atmos. Oceans 35, 135–177 (2002) 4. Campbell, L.J.: Wave mean-flow interactions in a forced Rossby wave packet. Stud. Appl. Math. 112, 39–85 (2004) 5. Campbell, L.J., Maslowe, S.A.: Forced Rossby wave packets in barotropic shear flows with critical layers. Dyn. Atmos. Oceans 28, 9–37 (1998) 6. Dickinson, R.E.: Development of a Rossby wave critical level. J. Atmos. Sci. 27, 627–633 (1970) 7. Franklin, J.L., Lord, S.J., Feuer, S.E., Marks, Jr., F.D.: The kinematic structure of Hurricane Gloria (1985) determined from nested analyses of dropwindsonde and Doppler radar data. Mon. Weather Rev. 121, 2433–2451 (1993) 8. Hawkins, J.D., Helveston, M.: Tropical cyclone multiple eyewall characteristics. In: 28th Conference of Hurricanes and Tropical Meteorology, Orlando, 28 April – 2 May. American Meteorological Society (2008) 9. Holton, J.R.: An Introduction to Dynamic Meteorology, 3rd edn. Academic, San Diego (1992)10. Huang, Y.-H., Montgomery, M.T., Wu, C.-C.: Concentric eyewall formation in Typhoon Sinlaku (2008). Part II: axisymmetric dynamical processes. J. Atmos. Sci. 69, 662–674, (2012)11. Martinez, Y., Brunet, G., Yau, M.K.: On the dynamics of two-dimensional hurricane-like vortex symmetrization. J. Atmos. Sci. 67, 3559–3580 (2010)12. Martinez, Y., Brunet, G., Yau, M.K.: On the dynamics of two-dimensional hurricane-like concentric rings vortex formation. J. Atmos. Sci. 67, 3253–3268 (2010)13. Martinez, Y., Brunet, G., Yau, M.K., Wang, X.: On the dynamics of concentric eyewall genesis: space-time empirical normal modes diagnosis. J. Atmos. Sci. 68, 457–476 (2011)14. Montgomery, M.T., Kallenbach, R.J.: A theory of vortex Rossby-waves and its application to spiral bands and intensity changes in hurricanes. Q. J. R. Meteorol. Soc. 123, 435–465 (1997)15. Montgomery, M.T., Smith, R.K.: Paradigms for tropical cyclone intensification. Aust. Meteo- rol. Oceanograph. J. 64, 37–66 (2014)16. Nikitina, L.: Dynamics of vortex Rossby waves in tropical cyclones. Ph.D. thesis, Carleton University (2013)17. Nikitina, L.V., Campbell, L.J.: Dynamics of vortex Rossby waves in tropical cyclones, part 1: linear steady-state exact solutions on an f -plane. Stud. Appl. Math. 135, 377–421 (2015)18. Nikitina, L.V., Campbell, L.J.: Dynamics of vortex Rossby waves in tropical cyclones, part 2: nonlinear time-dependent asymptotic analysis on a ˇ-plane. Stud. Appl. Math. 135, 422–446 (2015)19. Shapiro, L.J., Montgomery, M.T.: A three-dimensional balance theory for rapidly rotating vortices. J. Atmos. Sci. 50, 3322–3335 (1993)20. Sitkowski, M., Kossin, J.P., Rozoff, C.M.: Intensity and structure changes during hurricane eyewall replacement cycles. Mon. Weather Rev. 139, 3829–3847 (2011)21. Smith, R.K., Montgomery, M.T., Nguyen, V.S.: Tropical cyclone spin up revisited. Q. J. R. Meteorol. Soc. 135, 1321–1335 (2009)22. Stewartson, K.: The evolution of the critical layer of a Rossby wave. Geophys. Astro. Fluid Dyn. 9, 185–200 (1978)23. Warn, T., Warn, H.: On the development of a Rossby wave critical level. J. Atmos. Sci. 33, 2021–2024 (1976)24. Warn, T., Warn, H.: The evolution of a nonlinear critical level. Stud. Appl. Math. 59, 37–71 (1978)25. Willoughby, H.E., Clos, J.A., Shoreibah, M.G.: Concentric eyewalls, secondary wind maxima, and the evolution of the hurricane vortex. J. Atmos. Sci. 39, 395–411 (1982) On the Problem of Similar Motions of a Chainof Coupled Heavy Rigid Bodies
Dmitriy Chebanov
Abstract This chapter contributes to the study of dynamic properties of a chain ofn heavy Hess tops coupled by ideal spherical joints by constructing a new class ofparticular solutions of the equations of chain’s motion. The new class describes thechain’s motions under which all bodies move similar to each other. We establishconditions for existence of the new solutions and show how the equations of motioncan be reduced to quadratures in the case when these conditions are fulfilled.
1 Introduction
One of the classic directions of investigation in the theory of motion of a systemof several coupled rigid bodies is concerned with particular cases of integrabilityof the equations of system’s motion [2, 9, 16]. In comparison with the Euler andPoisson equations describing the dynamics of a single rigid body about a fixed point,the analytical study of mathematical models for a system of hinge-connected rigidbodies is a much more complicated problem; as a consequence, there are very fewknown cases of integrability in the dynamics of many-body systems [1, 15]. In this chapter we study dynamic properties of a mechanical system consisting ofan arbitrary number of heavy rigid bodies coupled by ideal spherical joints so thatthe system constitutes a chain of rigid bodies. The chain rotates about a fixed point.Under assumption that, for every chain’s link, the axis connecting its attachmentpoints passes through its center of mass, we construct a class of particular solutionsof the equations of chain’s motion that describes a rotational motion of the chainwith the following property: while the chain is in motion, its skeleton composed ofthe segments of the abovementioned axes bounded by the corresponding attachmentpoints belongs to a vertical plane rotating about the vertical line while the skeleton’ssegments change their position with respect to the plane identically in time, i.e., allthe bodies move similarly. The similar motions were first discovered for a chainof n heavy Lagrange tops [14, 18]. Some properties of these motions and theirgeneralizations can be found in the works [4, 6, 7, 14]. Recently, paper [8] suggested
D. Chebanov ()City University of New York – LaGCC, Long Island City, NY, USAe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 47J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_5 48 D. Chebanov
a generic approach for establishing the existence of such motions under no initialassumption regarding the mass distribution in the bodies. We present a new particular solution of the problem described above that isanalogous to the solution of the Euler and Poisson equations found by Hess [10].We describe the structure of the new solution, establish conditions for its existence,prove compatibility of these conditions, and, in the case when the conditions arefulfilled, reduce the equations of motion to quadratures. Then, we explore someaspects of the geometry of the chain’s motion described by the new solution.
2 Description of the Model
In this section, we recall the terminology and notations of [8] complementing themwith the ones needed for the purpose of this contribution. We consider a mechanical system S consisting of n heavy rigid bodies B1 ; B2 ; : : : ;Bn . The bodies Bi and BiC1 .i D 1; 2; : : : ; n 1/ are coupled by an ideal sphericaljoint at a common point OiC1 so that the system S constitutes a chain of rigid bodies.One of the chain’s end links, B1 , is absolutely fixed at one of its points O1 .¤ O2 /.It is assumed that the line li .i D 1; 2; : : : ; n 1/ connecting the attachment pointsOi and OiC1 of the body Bi passes through its center of mass Ci . For the body Bn ,ln denotes the line passing through the body’s attachment point On and its center ofmass Cn . If the position of the points Ci and OiC1 relative to Oi are determined bythe vectors ci and si , respectively, then, due to the above assumptions, ci D ci ei andsi D si ei , where ei is a unit vector directed along li . While studying the motion of system S, it is convenient to deal with masscharacteristics of so-called augmented body Bi instead of mass characteristics of Bi .Let us denote the mass of Bi by mi . By definition, the body Bi consists of the body Bi Pnand the point mass mi D mj which is rigidly attached to Bi at point OiC1 . Since jDiC1it has been assumed that the points Oi ; Ci , and OiC1 lie on li .i D 1; 2; : : : ; n1/, thisdefinition implies that the mass center Ci of body Bi lies on li as well. It also followsfrom the definition of an augmented body that the absolute angular velocities ofbodies Bi and Bi are equal, while the first-order mass momentum ai and the tensorof inertia Ii of Bi at Oi can be expressed as
ai D .mi C mi /ci D mi ci C mi si D ai ei ; (1) Ii D Ii C mi s2i ı si ˝ si D Ii C mi s2i .ı ei ˝ ei / ; (2)
where ci D Oi Ci , ai D mi ci C mi si , Ii is the tensor of inertia of Bi at Oi , ı is the3 3 identity matrix, and ˝ denotes a dyadic product of two tensors. Since the bodyBn has no descendant chain links connected to it, we have mn D 0 implying that thebodies Bn and Bn coincide. Therefore, an D mn cn and In D In . On the Problem of Similar Motions of a Chain of Coupled Heavy Rigid Bodies 49
The vector equations of motion for the system S under consideration can bewritten as follows [5, 13]:
X i1 X n .Ii !i / C ai sj ei !j ej C si aj ei !j ej C ai gei D 0; jD1 jDiC1
(3)
where !i is the absolute angular velocity of body Bi , is the upward vertical unitvector, and the dot denotes absolute derivative. Let ˙ D fO1 ; 1 2 3 g be a Cartesian reference frame whose vectors are fixed .i/ .i/ .i/in inertial space so that 3 D . Let also ė i D fOi ; eQ 1 eQ 2 eQ 3 g be an orthonormalmoving frame whose axes are the principal axes xof yBi at Oi . Then,x the inertia matrixeIi of Bi in this frame takes the form e Ii D diag eI i ;e I i ;e I zi , where e I i ;e I i , and e y I zi are theprincipal moments of inertia of Bi at Oi . In the rest of the chapter, we study a case when, for every i, the center of mass ofBi belongs to one of its principal planes. In this case, without loss of generality, we .i/ .i/assume that Ci is located in the plane formed by the vectors eQ 1 and eQ 3 , implying .i/ .i/ .i/that ei D .exi ; 0; ezi / in the principal axes frame, and choose ˙i D fOi ; e1 e2 e3 g tobe a so-called “special” frame [11, 12] whose base vectors are given by .i/ .i/ ei ei eQ 3 .i/ .i/ eQ ei e3 D ei ; e2 D ˇ 3 ˇ D eQ 2.i/ ; .i/ .i/ e1 D e2 e3 .i/ D ˇ ˇ ; ˇ .i/ ˇ ˇ .i/ ˇ ˇei eQ 3 ˇ ˇei eQ 3 ˇ
i.e., the special frame ˙i can be obtained from the principal axes frame ė i by .i/rotating the latter about its second axis. Hence, e2 is a unit vector of a principal axisof Bi at Oi and the rotation matrix Ri describing the above frame transformation isgiven by 0 1 ezi 0 exi Ri D @ 0 1 0 A : exi 0 ezi
Then, in the special frame, ei D .0; 0; 1/ and the inertia tensor Ii can be representedas 0 2 2 x 1 0 1 I xi ezi C e e I zi exi 0 eIi e I zi exi ezi Iix 0 Iixz B C @ Ii D Rie Ii RTi D @ 0 ey Ii 0 A D 0 Ii 0 A : y x 2 2 Ii e e I zi exi ezi 0 e I xi exi C e I zi ezi Ii 0 Iiz xz
In order to determine the position of body Bi with respect to the reference frame,we use Euler angles i ; i ; and 'i , where i .0 i < / is the angle of nutation, 50 D. Chebanov
i .0 i < 2/ is the angle of precession, and 'i .0 'i < 2/ is the angleof proper rotation. The motion of system S is a superposition of the motion of itsskeleton O1 O2 : : : On Cn , that is composed of the segments of axes li bounded by thecorresponding attachment points, and the pure rotation of each body about li . Theformer motion is completely determined by all angles i ; i , while the rotation of Biabout li is described by the angle 'i .
3 Formulation of the Problem
We will say that system S performs similar motions if it moves so that its skeletonbelongs to a vertical plane ˘ rotating about the vertical axis defined by inaccordance with a non-stationary law .t/ while the skeleton’s segments changetheir position with respect to ˘ identically in time, i.e., all the bodies movesimilarly. For such motions, it is fulfilled that
i D .t/; i D .t/ C ıi ; (4)
where .t/ and .t/ are functions of time to be determined, ıi 2 f1; 0; 1g, andi D 1; 2; : : : ; n. The similar motions (4) were first discovered in [14, 18] for a chain of n heavyLagrange tops. Below we follow the strategy for analyzing the problem on similarmotions suggested in [8]. Projecting equations (3) onto the axes of the corresponding body-fixed framesand substituting (4) into the equations so obtained yields the following overde-termined system of 3n second-order differential equations with respect to n C 2unknowns .t/; .t/, and 'i .t/: h i y Jix pP i C Iixz rPi C Iiz Ji qi ri C Iixz pi qi ai g C Hi .cos /R sin cos 'i D 0; h i Ji qP i C Jix Iiz pi ri C Iixz ri2 p2i C ai g C Hi .cos /R sin sin 'i D 0; y
y Iixz pP i C Iiz rPi C Ii Iix pi qi Iixz qi ri D 0; (5)
where pi D P cos 'i C P sin sin 'i ; qi D P sin 'i C P sin cos 'i ; ri D 'Pi CP cos ;
Ji˛ D Ii˛ C mi s2i C Li .˛ D x; y/; (6) X i1 X n X i1 X n Li D ai sj "ij C si aj "ij ; Hi D a i sj ij C si aj ij ; (7) jD1 jDiC1 jD1 jDiC1
"ij is equal to 1, if i D j, and 1, otherwise, and ij D 1 "ij . On the Problem of Similar Motions of a Chain of Coupled Heavy Rigid Bodies 51
In what follows, we construct a new class of particular solutions of the system ofequations (5).
4 Structure of the Solution
Let us assume that each body Bi is a Hess top [10], i.e. a rigid body whose center ofmass lies on the perpendicular to the circular cross-section of the gyration ellipsoid.In the principal axes frame ė i , the condition on the mass parameters of a Hess topBi is given by x 2 x y 2 z x y ei e Ii eIi e Ii e I zi D ezi e Ii e Ii : (8)
When expressed in ˙i , it can be written as 2 y Iixz D Iiz Iix Ii : (9)
Below we construct a solution of (5) that, as the Hess solution [10] of the Euler andPoisson equations, is characterized by the invariant relation
Iixz pi C Iiz ri D 0: (10)
By virtue of (9), (10), equations (5) reduce to the system of 2n equations h 1 i h i pi qi ai g C Hi .cos /R sin cos 'i D 0; y Ji pP i C Iixz Iiz (11) h 1 2 i h i pi C ai g C Hi .cos /R sin sin 'i D 0; y Ji qP i Iixz Iiz (12)
which possesses the following first integrals Ji P 2 C P 2 sin2 C Hi P 2 sin2 C 2ai g cos D hi ; Ji P sin2 D ki ; y y (13)
where hi and ki are integration constants. Solving (13) for P 2 and P yields
P 2 D i ./; P D i ./; (14) y y y where i ./ D J .h 2ai g cos / sin2 ki2 = Ji Ji CHi sin2 sin2 ; y 2 i ii ./ D ki = Ji sin : To ensure that, for every i D 1; 2; : : : ; n, equations (11), (12) reduce to the set ofintegrals (14) with the same right-hand sides, we require
1 ./ 2 ./ : : : n ./; 1 ./ 2 ./ : : : n ./: (15) 52 D. Chebanov
Each of the relations i ./ 1 ./ and i ./ 1 ./ .i D 2; 3; : : : ; n/ can betransformed to the form P.cos / 0; where P.cos / is a polynomial in cos which needs to be satisfied identically in cos . Equating the coefficients of allpowers of cos to zero in the identities obtained from (15) leads to the conditions y Ji Hi ai hi ki y D D D D : (16) J1 H1 a1 h1 k1
Proposition 1 If the conditions (9) and (16) are fulfilled, the system of equations (3)has a class of exact solutions with properties (4). Indeed, we infer from the previous discussion that, under the assumptions of thisproposition, the system (5) is compatible. To find the dependence of the variables; , and 'i on time, one can proceed as follows. From the first equation in (14), wefind Z cos s y J i C Hi Hi 2 t D t0 C y d: (17) cos 0 .hi 2ai g/ .1 2 / ki2 =Ji
Thus, cos can be obtained as the inverse of the hyperelliptic integral (17), in theform of an hyperelliptic function. Let cos D F.t t0 /. We can get .t/ by solvingthe last equation for . Then, we find .t/ from (14): Z t ki d .t/ D 0 C y : Ji t0 1 F 2 . t0 /
Finally, we observe that, for each i, the relation (10) can be transformed, by virtueof (14), (17), into a Ricatti equation with respect to the variable ui .t/ D tan.'i .t/=2/: .1/ .2/ .3/ .1/ .2/ uP i D i .t/ i .t/ u2i i .t/ui i .t/ i .t/; (18)
where s .hi 2ai gF.t t0 // .1 F 2 .t t0 // ki2 =Ji y .1/ I xz i .t/ D iz y ; 2Ii Ji C Hi Hi F 2 .t t0 / .1 F 2 .t t0 //
.2/ ki F.t t0 / .3/ I xz ki i .t/ D y ; i .t/ D p i : 2Ji .1 F 2 .t t0 // y Iiz Ji 1 F 2 .t t0 / h i .2/ .1/Using the change of variables ui .t/ D wP i .t/= i .t/ i .t/ wi .t/ , equa-tion (18) can be reduced to the second order linear differential equation
.1/ .2/ R i Pi .t/wP i C Pi .t/wi D 0; w (19) On the Problem of Similar Motions of a Chain of Coupled Heavy Rigid Bodies 53
where .1/ P i P i .2/ 2 2 .1/ .3/ .2/ .2/ .1/ Pi .t/ D .1/ .2/ i ; Pi .t/ D i i : i i
Given a solution wi .t/ of (19), one can find 'i .t/ from n h io .2/ .1/ 'i .t/ D 2 tan1 wP i .t/= i .t/ i .t/ wi .t/ :
Note, however, that, in a general case, equation (19) cannot be integrated inquadratures.
5 On Compatibility of the Conditions (9) and (16)
In this section we show that there exist physically meaningful values of themultibody chain parameters making the conditions (9), (16) compatible in the casewhen si ¤ 0 and ai ¤ 0: For the sake of brevity, we consider a simplest case ofa two-body system assuming that 2 D 1 C in (4). Then, due to (2), (6), (7),"12 D 1; 12 D 2; J1 D I1 C m2 s21 C L1 ; J2 D I2 C L2 ; L1 D L2 D a2 s1 ; H1 D y y y y
H2 D 2a2 s1 , the conditions (9) are equivalent to (8), and the conditions (16) become y y J2 D J1 ; a2 D a1 ; h2 D h1 ; k2 D k1 : (20)
Taking into account (1) and (6), we observe that the relations (8), (20) form an j 2algebraic system of 6 equations with respect to 19 parameters e j I i .> 0/; ei . exi C z 2 yc yc ei D 1/; Ii .> 0/; mi .> 0/; ci ; hi ; ki .i D 1; 2I j D x; z/; s1 . Here Ii is the central .i/moment of inertia of body Bi with respect to the axis defined by vector eQ 2 , andhence
Ii D e I i D Ii C mi c2i : y y yc (21)
In the rest of this section we solve the following problem: if the parameters definingthe mass distribution in the bodies are known, find possible ways for coupling thebodies as well as the initial conditions of their motion. From the second equation in (20), we obtain that m1 c2 D c1 C s1 : (22) m2 54 D. Chebanov
Substituting the expression for c2 into the first equation in (20) and solving theequation so obtained for s1 yields yc yc I1 I2 m2 m1 s1 D C c1 : (23) 2m1 c1 2m2
Using (23), the formula (22) for determining c2 becomes yc yc I1 I2 m2 C m1 c2 D C c1 : (24) 2m1 c1 2m2
In order to satisfy the system of relations (9), (16), one can now select theparameters of system S as follows. Assuming that the masses mi and parameter c1have been chosen arbitrarily, the inertia moments e I xi ;e yc I zi ; Ii and parameters exi ; ezi canbe selected to comply with (8). Then, the values for s1 and c2 can be found from (23)and (24), respectively. Finally, the values of hi and ki are to be determined to makethe last two relations in (20) true. Knowing the values of the integration constants,one can obtain the initial conditions of motion, using (4) and (14). As follows from the above analysis, in the case under consideration, there is aunique way of coupling the tops B1 and B2 that guarantees the existence of the yc ycmotion of interest. If c1 > 0; m2 > m1 ; and I1 I2 , then s1 > 0 and c2 > 0, i.e.the conditions (9), (16) can q be satisfied by positive values of c1 ; c2 ; and s1 . Moreover, 1 yc yc s1 > c1 , when 0 < c1 < m1 m2 .m1 C m2 / 1 I1 I2 . Thus, we have proven that it is possible to select physically meaningful values ofparameters characterizing the chain of rigid bodies under consideration so that theconditions (9), (16) are fulfilled. This completes our proof of the existence of thesimilar motions for the chain S of Hess tops.
6 Geometry of the Motion of the Chain’s Skeleton
The class of particular solutions of equations (3) that is constructed in the previoussections of this chapter describes a relatively complex motion of system S; therefore,a complete analysis of geometry of the system’s motion in this case is a quitecomplicated problem. Since the motion of S is a superposition of the motion of itsskeleton O1 O2 : : : On Cn and the pure rotation of each body about li , then, in orderto understand how the system moves, one can study the rotational motion of eachof the above components of its motion and then put them together to get a completepicture. In this section, we give some properties of the skeleton’s motion. As was noted before, the skeleton’s motion is completely determined by theangles i ; i . For the similar motions (4), however, it is sufficient to analyze therotation of any of the skeleton’s segments Oi OiC1 about Oi in order to get anunderstanding of the nature of the skeleton’s motion. Once such an analysis is On the Problem of Similar Motions of a Chain of Coupled Heavy Rigid Bodies 55
complete, one can get a clear idea of how the plane ˘ rotates about the vertical linethrough O1 as well as how the barycentric axes li move relative to ˘ . A generaldescription of the motion of any segment can be obtained from the propertiesof differential equations (14) without integration, employing the methods that areusually used for studying the rotation of the symmetry axis of a symmetric top abouta fixed point [17]. In what follows, we analyze the rotation of O1 O2 about O1 . y Equations (14) can be somewhat simplified by taking D cos ; a D ai g=Ji ; h D y y yhi =Ji ; H D Hi =Ji ; and k D ki =Ji . Then, the equations become 2 .h 2a / 1 2 k2 2af1 . / k P D 2 D ; P D : (25) 1 C H H f2 . / 1 2
We leave it to the reader to verify that, due to (6), (7), (16), (21), f2 . / ¤ 0 atany moment of time, i.e. there are no singularities in the right-hand side of the firstequation in (25) and thus .t/P is a bounded function of time. The polynomial f1 . / in (25) is negative for D 1; 1, and C1, and positivefor D C1. Since D cos and for real motion is real, there should be two realroots, 2 and 3 , of f1 . / between 1 and C1, and a third root, 1 , is greater thanC1. The former can be achieved, for example, by requiring h > k2 . We thereforeconclude that oscillates between the values 2 and 3 . From the second equation in (25), we observe that P has same sign as k atany moment of time. Hence, when k ¤ 0, the angle of precession monotonicallyincreases/decreases over time implying that, while the system S performs the similarmotions, the plane ˘ always rotates in the same direction. Let us consider a unit sphere drawn at O1 as a center. The l1 -axis intersects theunit sphere in a point P. As the body B1 moves, this point describes a curve C onthe sphere. Let 2 and 3 be the angles corresponding to 2 and 3 , respectively.If two cones are constructed so that the apex of each cone is at O1 and the cones’generating angles are 2 and 3 , respectively, these two cones intersect the spherein two circles, C2 and C3 . Since 3 < 2 , then 3 > 2 and therefore the circle C2lies above the circle C3 . The curve C lies on the sphere between these two circles,touching the first one and then the other. Let O1 X; O1 Y, and O1 Z be the coordinate axes of ˙ and let O1 x; O1 y, and O1 zbe the coordinate axes of ˙1 . Following [17], we represent the position of the pointP on the unit sphere by the arc vector ZP D and the angle XZP, which is thelongitude, , of the pole of the xy-plane. Then, P D P and p d k f2 . / D p ; (26) d .1 2 / 2af1 . /
which is the differential equation of the curve C. 56 D. Chebanov
Fig. 1 Trajectory of the pointP of the barycentric axis l1 onthe unit sphere
Let ˛ be the angle which the curve C makes with the arc vector ZP. Using theresults of [17] and (26), we derive that p d d k f2 . / tan ˛ D sin D 1 2 D p : (27) d d 2af1 . /
As follows from (26) and (27), tan ˛ is infinite only when is equal to 2 or 3 andtherefore the curve C is tangent to the circle C2 or C3 whenever takes on one ofthese values. A typical curve C is shown in Fig. 1. By analogy with [3], one canshow that the time it takes for P to reach C2 starting from C3 is equal to the time ittakes to get back from C3 to C2 . Thus, while chain S of Hess tops rotates about O1 so that its bodies move similarto each other, the plane ˘ rotates about the vertical line through O1 in the samedirection and the barycentric axis of each body moves along a spherical curve Coscillating between C2 and C3 .
Acknowledgements Support for this project was provided by a PSC-CUNY Award, jointlyfunded by The Professional Staff Congress and The City University of New York (Award # 67306-00 45).
References
1. Bolgrabskaya, I., Lesina, M., Chebanov, D.: Dynamics of Systems of Coupled Rigid Bodies. Naukova Dumka, Kyiv (2012) 2. Borisov, A., Mamaev, I.: Dynamics of a Rigid Body. Hamiltonian Methods, Integrability, Chaos. Institute of Computer Science, Moscow/Izhevsk (2005) 3. Bukhgolts, N.: An Elementary Course in Theoretical Mechanics, vol. 2. Nauka, Moscow (1969) On the Problem of Similar Motions of a Chain of Coupled Heavy Rigid Bodies 57
4. Chebanov, D.: On a generalization of the problem of similar motions of a system of Lagrange gyroscopes. Mekh. Tverd. Tela 27, 57–63 (1995) 5. Chebanov, D.: New dynamical properties of a system of Lagrange gyroscopes. Proc. Inst. Appl. Math. Mech. 5, 172–182 (2000) 6. Chebanov, D.: Exact solutions for motion equations of symmetric gyros system. Multibody Syst. Dyn. 6(1), 30–57 (2001) 7. Chebanov, D.: A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the system’s skeleton. Mekh. Tverd. Tela 41, 244–254 (2011) 8. Chebanov, D.: New class of exact solutions for the equations of motion of a chain of n rigid bodies. Discret. Cont. Dyn. Syst. Supplement, 105–113 (2013) 9. Gashenenko, I., Gorr, G., Kovalev, A.: Classical Problems of the Rigid Body Dynamics. Naukova Dumka, Kyiv (2012)10. Hess, W.: Über die Eulerschen Bewegungsgleichungen und über eine neue particulare Lösung des Problems der Bewegung eines starren Körpers un einen festen Punkt. Math. Ann. 37(2), 178–180 (1890)11. Kharlamov, P.: On the Equations of Motion for a Heavy Rigid Body with a Fixed Point. J. Appl. Math. Mech. 27(4), 1070–1078 (1963)12. Kharlamov, P.: Lectures on Rigid Body Dynamics. Novosibirsk University, Novosibirsk (1965)13. Kharlamov, P.: The equations of motion of a system of rigid bodies. Mekh. Tverd. Tela 4, 52–73 (1972)14. Kharlamov, P.: Some classes of exact solutions of the problem of the motion of a system of Lagrange gyroscopes. Mat. Fiz. 32, 63–76 (1982)15. Kharlamova, E.: A survey of exact solutions of problems of the motion of systems of coupled rigid bodies. Mekh. Tverd. Tela 26(II), 125–138 (1998)16. Leimanis, E.: The General Problem of the Motion of Coupled Rigid Bodies About Fixed Point. Springer, Berlin (1965)17. MacMillan, W.: Dynamics of Rigid Bodies. Dover Publications, New York (1960)18. Savchenko, A., Lesina, M.: A particular solution for motion equations of Lagrange gyroscopes system. Mekh. Tverd. Tela 5, 27–30 (1973) On Stabilization of an Unbalanced LagrangeGyrostat
Dmitriy Chebanov, Natalia Mosina, and Jose Salas
Abstract In this contribution we consider a chain of two coupled Lagrangegyrostats moving about a fixed point in a non-hom*ogeneous gravitational field andshow the existence of the following stabilization effect: for a gyrostat that is in itsunstable equilibrium position, there is another gyrostat such that, when the two ofthem are coupled to form this chain, the rotation of the latter gyrostat can be used tostabilize the equilibrium of the former one. We establish and analyze stabilizationconditions in the space of mechanical parameters characterizing the chain.
1 Introduction
In this contribution we investigate dynamic properties of a chain of two coupledgyrostats rotating about a fixed point in a gravitational field. By definition, a gyrostatis a mechanical system consisting of a rigid carrier and other bodies connected toit such that their motion relative to the carrier does not alter the distribution ofmasses of the mechanical system. Examples of such systems include a rigid body towhich axes of several symmetric rotors are connected, or a rigid body with cavitiescompletely filled with a hom*ogeneous fluid [12, 18]. Mechanical models involvinggyrostats have a number of applications; for instance, they are used for controllingthe attitude dynamics of spacecraft and for stabilizing its rotations [1, 8], play animportant role in studying fluid dynamical systems [4, 5], can be employed in theanalysis of the mechanical models of DNA molecules [16], open-loop dynamiccharacteristics of smooth air gap brushless dc motors [7], instabilities associatedwith the operation of lasers [6], microscopic dynamics of deterministic chemicalchaos [10], etc. Assuming that, for each gyrostat, its mass distribution is analogous to the oneof a Lagrange top, its attachment point(s) lie on its dynamic symmetry axis, andits gyrostatic moment is directed along this axis, we study a problem of stabilityof the state of chain’s motion with the following property: one of the gyrostatspermanently rotates about a vertical axis, whereas the other one is at rest. While
D. Chebanov () • N. Mosina • J. SalasCity University of New York – LaGCC, Long Island City, NY, USAe-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 59J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_6 60 D. Chebanov et al.
studying this problem, we show the existence of an interesting stabilization effect:for any Lagrange gyrostat that is in its unstable equilibrium position, there is anotherLagrange gyrostat such that, when the two gyrostats are coupled to form a chain,the rotation of the latter gyrostat can be used to stabilize the former one. A similarstabilization effect was discovered for a system of two heavy Lagrange tops movingabout a fixed point [9, 15, 17]. This contribution extends the results of [9, 15, 17]to a case of a system of two gyrostats moving in a non-hom*ogeneous gravitationalfield.
2 Description of the Model
We consider a mechanical system S consisting of two gyrostats G1 and G2 . Eachgyrostat Gk is composed by a rigid body Bk and other (variable or rigid) bodies B ekwhich are connected to it. The motion of these bodies relative to Bk does not changethe distribution of masses of the gyrostat Gk . We assume that the body B1 is attachedto an immovable base at one of its points O1 , while the bodies B1 and B2 are coupledat a common point O2 .¤ O1 / by an ideal spherical joint. Let mk denote the mass of Gk . Let also B1 denote an augmented body [2]consisting of B1 and the mass point m2 located at O2 . Similarly, the augmentedgyrostat G1 consists of G1 and the mass point m2 located at O2 . For our convenience,we use the notation B2 D B2 and G2 D G2 . Note that the absolute angular velocitiesof bodies Bk and Bk are equal, while the first-order mass momentum ak and thetensor of inertia Ik of Gk at Ok can be expressed as
a1 D .m1 C m2 /b D m1 c1 C m2 s; a 2 D m2 c 2 ; (1) I1 D J1 C m2 s2 ı s ˝ s ; I2 D J2 ; (2)
where b D O1 C1 ; C1 is the barycenter of G1 , ck D Ok Ck , Ck is the center of massof Gk ; s D O1 O2 , Jk is the tensor of inertia of Gk at point Ok , ı is the 3 3 identitymatrix, and ˝ denotes a dyadic product of two tensors. We further assume that each gyrostat Gk has the mass distribution analogous tothe one of a Lagrange top, the attachment point(s) of Bk lie on the axis of dynamicsymmetry of Gk , and the gyrostatic moment k of Gk is constant relative to Bk anddirected along the Gk ’s symmetry axis, i.e., O1 ; O2 ; C1 ; C1 2 l1 ; O2 ; C2 2 l2 , andk jjlk , where lk denotes the dynamic symmetry axis of Gk . Suppose that system S moves under the gravitational attraction of a point mass located at a fixed point P whose position with respect to O1 is defined by the vectorR where is the unit vector in the direction of O1 P and R is the distance betweenO1 and P. We introduce a Cartesian reference frame fO1 ; 1 2 3 g whose axes are fixed .k/ .k/ .k/in inertial space such that 3 D and a Cartesian frame fOk ; e1 e2 e3 g that isrigidly embedded in body Bk such that its coordinate axes are the principal axes of On Stabilization of an Unbalanced Lagrange Gyrostat 61
.k/inertia of Gk and e3 jj lk . We also assume that, in the corresponding moving frame, .1/ .k/ .k/Ik D diagfIk; Ik ; Ikz g; Jk D diagfJk ; Jk ; Jkz g; s D se3 ; ak D ak e3 , and k D k e3 . We determine the position of gyrostat Gk with respect to the reference frame byBryan-Krylov angles ˛k ; ˇk ; and k [11, 18]. The equations of motion of system Sin terms of the Bryan-Krylov angles can be written in the form [3]
.m/ .m/ .m/ Fk C a2 sGkj C "a2 sHkj D 0; Rk ˛R k sin ˇk ˛P k ˇPk cos ˇk D 0; (3)
where " D g=R; m D 1; 2; j; k D 1; 2 with j ¤ k; .1/ Fk D Ik ˇRk C ˛P k2 sin ˇk cos ˇk C Ikz .Pk ˛P k sin ˇk / ˛P k cos ˇk C k ˛P k cos ˇk Cak g cos ˛k sin ˇk C 3" Ik Ikz cos2 ˛k sin ˇk cos ˇk ; .1/ Gkj D ˇRj cos ˇj ˇPj2 sin ˇj cos ˇk C 2˛P j ˇPj sin ˇj ˛R j cos ˇj sin ˇk sin.˛k ˛j / C ˇRj sin ˇj C ˛P j2 C ˇPj2 cos ˇj sin ˇk cos.˛k ˛j /; .1/ Hkj D 2 cos ˛k cos ˛j sin ˛k sin ˛j sin ˇk cos ˇj C sin ˇj cos ˇk ; .2/ Fk D Ik ˛R k cos ˇk 2˛P k ˇPk sin ˇk Ikz .Pk ˛P k sin ˇk / ˇPk
k ˇPk C ak g sin ˛k C 3".Ik Ikz / sin ˛k cos ˛k cos ˇk ; .2/ Gkj D ˛R j cos ˇj C 2˛P j ˇPj sin ˇj cos.˛k ˛j / C ˇRj sin ˇj C ˛P j2 C ˇPj2 cos ˇj sin.˛k ˛j /; .2/ Hkj D cos ˇj 2 sin ˛k cos ˛j C sin ˛j cos ˛k :
3 The Problem of Interest
The motion of a gyrostat is a permanent rotation if, while the gyrostat is in motion,its angular velocity vector is constant. This vector defines an axis of permanentrotation which is fixed in both an inertial space and the gyrostat’s carrier. When thepermanent axis is a vertical line, the gyrostat permanently rotates about a verticalaxis. In this contribution we investigate a problem of the stability of the state of motionof system S with the following property: G1 permanently rotates about its dynamicsymmetry axis coinciding with the vertical axis passing through O1 , whereas G2 is 62 D. Chebanov et al.
.2/at rest so that e3 jj. One can check that equations (3) have a particular solution
˛k 0; ˇk 0; 1 D !t; 2 0; ! D const (4)
that describes the motion of interest; the angular velocity ! can be chosen in (4) .2/arbitrarily. Below we discuss a case when the parameters c1 ; s, and c2 D c2 e3are negative. In this case, a1 < 0 and a2 < 0. The solution (4) is a special case of a more general solution (˛k 0; ˇk 0; k D !k t; !k D const) of equations (3) that describes permanent rotationsof system S about a vertical axis. The necessary conditions for stability of suchrotations have been recently established in [3]. For the motion of interest, theseconditions assume the form 2 22 1 3 > 0; 12 22 1 3 21 1 5 42 4 C 323 > 0; (5)
3 2 1 5 42 4 C 323 27 1 3 5 C 22 3 4 1 24 22 5 33 > 0;
where 1 D I1 I2 a22 s2 ; 2 D .I1 2 C I2 w/ =4; 3 D w2 I1 d2 I2 d1 C 2"a22 s2=6; 4 D .wd2 C 2 d1 / =4; 5 D d1 d2 "2 a22 s2 ; w D I1z ! C 1 ; dk D 2"a2s Cak g C 3" Ik Ikz : While analyzing the conditions (5) in the rest of the contribution, we aimto establish the existence of the following stabilization effect. Suppose that theparameters of G2 are selected so that, if G2 is decoupled from the chain S andthen coupled to an immovable base at point O2 , its equilibrium position would beunstable. (Using Lyapunov’s indirect method, it is possible to show that this is thecase when 22 C 4J2 m2 c2 g C 3" J2 J2z < 0 or, by virtue of (1), 22 C 4I2 a2 g C 3" I2 I2z < 0: (6)
Special cases of (6) are given in [1, 13, 14].) We seek to justify that the parametersof G1 can be chosen so that, when both gyrostats form chain S, the motion of Sdescribed by (4) is stable, i.e., the permanent rotation of G1 stabilizes G2 .
4 Analysis of Necessary Stability Conditions
The conditions (5) depend on 12 parameters: angular velocity !, mass characteris-tics Ik ; Ikz ; ak , the magnitudes of the gyrostatic moments k , parameters " and g, andthe distance s between the gyrostats’ attachment points. Recall that ak < 0; s < 0and assume additionally that Ik < Ikz . Then, for sufficiently small ", we have dk < 0.In order to reduce the number of parameters in the problem of study, we introduce On Stabilization of an Unbalanced Lagrange Gyrostat 63
dimensionless parameters x; 1 ; 2 ; 3 ; 4 by the formulas: s jd2 j p w D I1 x ; I2 d1 D 1 I1 d2 ; a22 s2 D 2 I1 I2 ; "I2 D 3 d2 ; 2 D 4 I2 jd2 j: I2 (7)The conditions (5) now deduce to the form
j > 0; j D 1; 2; 3; (8)
where
1 D 3x2 C 2 .42 1/ 4 x C 342 8 .1 2 / .1 22 3 C 1/ ; 2 D 3x4 C 4 .42 1/ 4 x3 C 2 822 C 1 42 16 .1 2 / .1 22 3 C 2 / x2 C4 .42 1/ 42 4 .1 2 / .1 C 1/ .32 1/ 422 3 4 x C344 16 .1 2 / .1 2 22 3 C 1/ 42 C16 .1 2 /2 .1 1 /2 C 42 .1 3 / .1 3 / ;
3 D 6 x6 C 5 x5 C 4 x4 C 3 x3 C 2 x2 C 1 x C 0 :
The expressions for 6 ; 5 ; : : : ; 0 in terms of the parameters 1 ; 2 ; 3 ; 4 are givenin Appendix. The conditions (8) form a system of three algebraic inequalities (with respect tox) whose left-hand sides, 1 .x/; 2 .x/; and 3 .x/, are polynomials of the second,fourth, and six order in x, respectively. The coefficients of these polynomials dependon four parameters 1 ; 2 ; 3 , and 4 . It is quite difficult to conduct a thoroughanalytical study of the conditions (8) due to their high nonlinearity and tediousness.Therefore, we have studied them numerically, seeking to find the values of theparameters under which the system (8) is compatible. Note that due to the assumptions stated at the beginning of this section and (7),the parameters 1 ; 2 are positive, while 3 is negative and j3 j is relatively small.Moreover, 2 < 1. (Indeed, by (7), this inequality is equivalent to I1 I2 a22 s2 > 0.In order to prove the latter, we observe that, due to (1), (2), a2 D m2 c2 ; I1 D J1 Cms2 ; I2 D J2 . Furthermore, J2 D J2c C m2 c22 , where J2c is the central equatorialmoment ofinertia of G2. Then, I1 I2 a22 s2 D J1 C m2 s2 J2c C m2 c22 m22 c22 s2 DJ1 J2c C m2 J1 c22 C J2c s2 > 0:) Since each of the polynomials j is invariant withrespect to the simultaneous replacement of x and 4 with x and 4 , respectively,it is enough to study only a case when 4 > 0. We also note that the inequality (6) can be written in terms of 2 ; 3 , and 4 asfollows 4 q < 2: (9) p p 1 23 2 I1 =I2 64 D. Chebanov et al.
Since 3 is p negative and relatively small and 0 < 2 < 1, the expression p1 23 2 I1 =I2 is positive for all possible values of the parameters it containsand its second term is negligible. Moreover, 4 provides an upper bound for theleft-hand side in (9). Hence, requiring
4 < 2 (10)
guarantees the fulfillment of (9). Thus, based on the above discussion, the suitable values of 1 ; 2 ; and 4 belongto the set D D f.1 ; 2 ; 4 /j1 > 0; 0 < 2 < 1; 0 < 4 < 2g. Setting 3 D 0:01,we have conducted a numerical analysis of the conditions (8) for the members ofD. Our analysis reveals that there exists a simply connected region D D in theparameter space where these conditions are fulfilled. The existence of the interval(s) of x for which the conditions (8) are fulfilleddepends on the number of real zeros of j .x/ and their relative position along thex-axis. Our study indicates that, when parameters 1 ; 2 ; and 4 are selected fromD , the number of zeros of 1 .x/ can vary from 0 to 2, while 2 .x/ and 3 .x/have ether two zeros or no zeros at all. (The number of zeros of 1 .x/ is completelydetermined by the sign of the expression L D .22 C 1/42 3.1 22 3 C 1/. Inmore details, 1 .x/ has two zeros, when L < 0, no zeros, when L > 0, and one zero,otherwise). We also observe that the leading coefficients of 1 .x/; 2 .x/, and 3 .x/take on positive, positive, and negative values, respectively, for any .1 ; 2 ; 4 / 2 D.Figures 1 and 2 demonstrate various cases of the relative position of the intervalsof positiveness of i .x/ that have been encountered in our analysis. It is interestingto note that for any set of parameters from D , we have been able to find only onestabilization interval Œx ; x whose endpoints are always the zeros of 3 .x/.
60 Λ(x) Λ(x) 60
40 40
20 20
x* x–5 –4 –3 x * –2 –1 0 1 2 : x* x* –4 –3 –2 –1 0 x 1 –20
–40 –20
–60 –40
Fig. 1 The graphs of 1 .x/ (dashed), 2 .x/ (dotted), 3 .x/ (dash-dotted), and the intervalŒx ; x : Left: 1 D 2; 2 D 0:6; 3 D 0:01; 4 D 1:9; x D 3:6608; x D 2:4547. Right:1 D 1; 2 D 0:5; 3 D 0:01; 4 D 1:9; x D 3:4564; x D 1:8527 On Stabilization of an Unbalanced Lagrange Gyrostat 65
3 1 Λ(x ) Λ(x )
2 0.5
1 x* x 0 –1.4 –1.2 x –1.0 –0.8 –0.6 x x * *–2 –1 x* 0
–0.5 –1
2 –1
Fig. 2 The graphs of 1 .x/ (dashed), 2 .x/ (dotted), 3 .x/ (dash-dotted), and the intervalŒx ; x D Œ1:0816; 0:6153 for 1 D 0:1; 2 D 0:9; 3 D 0:01, and 4 D 1:35. Left:zoom on 2 x 1. Right: zoom on 1:5 x 0:4
Table 1 Numerical 4 1 2 Œx ; x estimates for 1 ; 2 ; x , andx for some values of 4 1.40 0.11 0.75 Œ0:6668; 0:6473 1.40 0.17 0.80 Œ0:7914; 0:8154 1.40 0.25 0.85 Œ0:9535; 0:9378 1.65 0.20 0.60 Œ0:9651; 0:8665 1.65 0.40 0.70 Œ1:2547; 1:1815 1.65 0.70 0.80 Œ1:4966; 1:4746 1.90 2.00 0.50 Œ2:5674; 2:5324 1.90 2.50 0.60 Œ3:2193; 2:7011 1.90 3.80 0.70 Œ3:1720; 3:0646
We have also observed that, for a fixed 4 , there are some values k such thatonly the values k satisfying 0 < 1 < 1 and 2 < 2 < 1 belong to D ; moreover,k depends on both 4 and i .i; k D 1; 2; i ¤ k/. Numerical estimates for k aswell as for the interval Œx ; x of the fulfillment of (8) are given in Table 1. When4 is close to its upper bound (10), there is a relatively wide range of values for 1and 2 in D . As 4 starts getting smaller, this range narrows down significantly at ahigh rate. In particular, we have observed that in this case 1 tends to drop its valuesignificantly for the same 2 . Thus, it is possible to select the dimensionless parameters x and 1 through 4to achieve the desired stabilization effect. Once the mass characteristics and thegyrostatic moment of G2 are given, and the dimensionless parameters are chosen tocomply with (8), the distance s, the angular velocity, mass parameters, and gyrostaticmoment of G1 can be determined from (7). A similar analysis can be conducted tojustify that it is possible to stabilize an unstable equilibrium position of G1 by apermanent rotation of G2 . 66 D. Chebanov et al.
5 Conclusion
In this contribution, we have conducted a dynamic analysis of a multibody chainconsisting of two Lagrange gyrostats and moving about a fixed point in a non-hom*ogeneous gravitational field. We have established and analyzed the conditionsfor stability of the state of chain’s motion when the gyrostat connected to the fixedpoint permanently rotates about a vertical axis while the other one is at rest. Itfollows from our analysis that that permanent rotation of the former gyrostat canbe used to stabilize an unstable equilibrium position of the latter one.
Acknowledgements Support for this project was provided by a PSC-CUNY Award, jointlyfunded by The Professional Staff Congress and The City University of New York (Award # 68091-00 46). The work of the second and third authors was supported by the NASA New York SpaceGrant CCPP Program.
Appendix
The coefficients 6 ; 5 ; : : : ; 0 of 3 in (8) admit the following representation interms of the dimensionless parameters 1 ; 2 ; 3 ; 4 .
6 D 42 4; 5 D 24 1 22 1 C 32 C 1 42 C 92 1 C 32 2 .21 2 3 C 2/ ; 4 D 82 32 .22 1/ C 1 .1 82 C 4/ C 1 44 2 1222 33 C 31 2 32 C4022 32 81 2 3 C 1222 3 192 32 C 12 191 2 82 3 C121 C 32 C 1 42 2722 34 C 3622 33 C 361 2 32 222 32 4012 3 C 3622 3 122 32 812 121 2 2722 402 3 C321 C 362 8; 3 D 24 12 C 1 212 2 22 32 44 C 6423 33 C 81 22 32 4022 33 C212 2 3 40122 3 C 71 2 32 C 822 32 C 13 C 712 2 C 241 2 3 C72 32 912 C 71 2 C 22 3 91 C 1 42 C 4823 34 C 61 22 33 1602333 1822 34 C 312 2 32 381 22 32 C 4823 32 C 1302233 212 2 3 C 130122 3 261 2 32 3822 32 413 C 712 2 181 22 921 2 3 C 622 3 C 72 32 C 2012 261 2 22 3 C 201 C32 4 ; 2 D 12 46 2 1212 22 3 C 401 22 32 C 1222 33 C 313 2 812 2 3 On Stabilization of an Unbalanced Lagrange Gyrostat 67
191 2 32 C 13 C 3812 2 81 2 3 C 32 32 C 1212 C 1 44 C 25624 34 641 23 33 3202334 6612 22 32 320123 32 C761 22 33 6423 33 222 34 413 2 3 C 7612 22 3 C 4012 2 32 C696122 32 C 762233 C 14 C 4013 2 212 22 10812 2 3 C 761 22 3 19212 32 6622 32 413 19212 2 10812 3 C 40232 C 10212 C401 2 42 3 41 C 1 42 C 4 6424 35 241 23 34 12824 34 7223 35 612 22 33 561 23 33 C 361 22 34 C 6424 33 C 15223 34 C13 2 32 C 4012 22 32 C 1521 23 32 C 521 22 33 562333 1222 34 C613 2 3 1412 22 3 4912 2 32 721 23 3 1921 22 32 2423 32 1422 33 14 1513 2 1212 22 C 26122 3 C 521 22 3 C 311 2 32 C4022 32 C 1213 C 3112 2 C 361 22 C 261 2 3 622 3 152 32 2212 491 2 C 62 3 C 121 C 2 1 ; 1 D 24 913 2 C 212 2 3 C 91 2 32 413 412 44 C 4812 23 32 160123 33 C 4823 34 C 613 22 3 3812 22 32 C 1301 22 33 1822 34 C314 2 1813 22 213 2 3 C 1301222 3 C 712 2 32 381 22 32 C622 33 414 2613 2 9212 2 3 261 2 32 C 2013 C 712 2 C 32 32 21 2 3 C 2012 41 42 C 4 812 23 33 321 23 34 C 823 35 C 213 22 32 3212 23 32 212 22 33 C 96123 33 C 201 22 34 3223 34 131 2 32 213 22 3 313 2 32 C 812 23 3 C 5412 22 32 321 23 32 921 22 33 C823 33 C 2022 34 314 2 C 2013 22 413 2 3 9212 22 3 1312 2 32 C541 22 32 222 33 C 414 1313 2 C 201222 C 7212 2 3 21 22 3 C222 32 413 1312 2 41 2 3 32 32 412 31 2 C 41 ; 0 D 413 46 C 2714 22 C 3613 22 3 212 22 32 C 36122 33 2722 34 C3614 2 40132 3 1212 2 32 814 1213 2 4012 2 3 C361 2 32 C 3213 812 44 C 2561224 33 5121 24 34 C 2562435 961323 32 22412 23 33 C 608123 34 28823 35 2414 22 3 2881323 3 C 1601322 32 C 60812 23 32 5612 22 33 2241 23 33 481 22 34 9623 34 C 415 2 C 14414 22 C 2414 2 3 C 20813 22 3 60132 32 76812 22 32 C 2081 22 33 C 14422 34 415 196142 68 D. Chebanov et al.
4813 22 C 10413 2 3 5612 22 3 C 12412 2 32 C 1601 22 32 2422 33 C4814 C 12413 2 C 104122 3 19612 32 8813 6012 2 C241 2 3 C 42 32 C 4812 41 42 C 2561224 34 5121 24 35 C2562436 12813 23 33 5121224 33 1281223 34 C 10241 24 34 C5121 23 35 5122435 2562336 C 1614 22 32 1281323 32 C12813 22 33 C 2561224 32 C 115212 23 33 1281222 34 5121 24 33 1536123 34 C 2562434 C 51223 35 C 12814 22 3 1614 2 32 C51213 23 3 6413 22 32 15361223 32 6401222 33 C 11521 23 33 C5121 22 34 1282334 1615 2 12814 22 128142 3 25613 23 6401322 3 C 19213 2 32 C 51212 23 3 C 16321222 32 1281 23 32 640122 33 12823 33 1282234 C 1615 C 192142 C 51213 22 C12813 2 3 6401222 3 35212 2 32 641 22 32 C 12822 33 6414 352132 12812 22 C 12812 2 3 C 1281 22 3 C 1921 2 32 C 1622 32 C9613 C 19212 2 12812 3 162 32 6412 161 2 C 161:
References
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11. Lurie, A.: Analytical Mechanics. Springer, Berlin (2002)12. Moiseyev, N., Rumyantsev, V.: Dynamic Stability of Bodies Containing Fluid. Springer, New York (1968)13. Rubanovskii, V.: On bifurcation and stability of stationary motions in certain problems of dynamics of a solid body. J. Appl. Math. Mech. 38(4), 573–584 (1974)14. Rumyantsev, V.: On the stability of motion of gyrostats. J. Appl. Math. Mech. 25(1), 9–19 (1961)15. Savchenko, A., Bolgrabskaya, I., Kononyhin, G.: Stability of Motion of Systems of Coupled Rigid Bodies. Naukova Dumka, Kyiv (1991)16. Starostin, E.: Three-dimensional shapes of looped DNA. Meccanica 31(3), 235–271 (1996)17. Varkhalev, I., Savchenko, A., Svetlichnaya, N.: On the stabilization of an unbalanced Lagrange gyroscope at rest. Mekh. Tverd. Tela 14, 105–109 (1982)18. Wittenburg, J.: Dynamics of Multibody Systems. Springer, Berlin (2008) Approximate Solution of Some Boundary ValueProblems of Coupled Thermo-Elasticity
Manana Chumburidze
Abstract We consider a non-classical model of a pseudo oscillation system ofpartial differential equations of coupled thermo-elasticity in the Green-Lindsayformulation. The matrices of fundamental and singular solutions for isotropichom*ogeneous elastic materials have been obtained. We propose and justify atechnique of approximate method for the solution of boundary value problems withmixed boundary conditions. The tools applied in this development are based onsingular integral equations, the potential method and the generalized Fourier seriesanalysis.
Mathematics Subject Classifications (2010) 26A33 60G22 35R60 34K37
1 Introduction
As we know, non-classical theories of thermo-elasticity have been developed inorder to remove the paradox of physically impossible phenomenon of infinitevelocity of thermal signals in the conventional coupled thermo-elasticity. Lord-Shulman [17, 23] theory and Green-Lindsay [14] theory are important generalizedtheories of thermo-elasticity that become centre of interest of recent research in thisarea. In Lord-Shulman theory, a flux rate term into the Fourier’s law [2] of heatconduction is incorporated (with one relaxation time) and formulated a generalizedtheory admitting finite speed for thermal signals. Green-Lindsay theory called astemperature rate-dependent is included among the constitutive variables with twoconstants that act as two relaxation times, which does not violate the classicalFourier law of heat conduction when the body under consideration has a center ofsymmetry. There are special classes of thermo-elasticity problems such as coupledthermo-elasticity, which require entirely different mathematical approaches andmeans of analysis [11, 20–22].
M. Chumburidze ()Ak.Tsereteli State University, Kutaisi, Georgiae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 71J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_7 72 M. Chumburidze
The dynamical problems for a conjugated system of two-dimensional lineartheory coupled thermo-elasticity (CPTE) [4, 12, 16] in the Green-Lindsay formu-lation for isotropic hom*ogeneous elastic materials with a center of symmetry areinvestigated in work [4]. Green’s tensors for basic boundary value problems arederived. The boundary integral method in combination with the harmonic potentialstheory and Laplace transform [3, 7, 16] are applied to solve the problems. This paper is devoted to the development of approximate methods for theconstruction of solutions in the form that admit efficient numerical evaluation [3,6, 8, 16]. In particular, we consider two-dimensional stationary problems of CPTEin Green-Lindsay formulation. The formulation is given for isotropic hom*ogeneouselastic materials with a center of symmetry. Boundary value problems for a finitedomain have been derived, when the couple-stresses components, displacementcomponents, rotation, heat flux and temperature are given on the surface of Holderclass [19]. Introduce the notations: R2 is two-dimensional Euclidean space, x D xj . j D1; 2/ is point of this space. ˝ R2 is finite domain, bounded by the closed surface@˝ of Holder class. We have [4]:
. C ˛/ u .x; / C . C ˛/ graddivu C 2˛rotu3 gradu4 2 u D h.1/ .x/ (1) . C ˇ/ u3 .x; / C 2˛rotu 4˛u3 I 2 u3 D h3 .x/ u4 .x; / < u4 divu D h4 .x/
where u D .u1 ; u2 / is the displacement vector, u3 is a characteristic of the rotation, 1u4 is the temperature variation, D .1 C 2 /; D <1 .1 C 1 /, 2 > 1 > 0 < are constants of relaxation [3], > 0; ˛ > 0; > 0; 3 C 2 > 0; >0; ˇ > 0; > 0; < > 0; I > 0 are constants of elasticity [4, 16, 20], is two-dimensional Laplacian operator [13], D C iq; > 0 corresponds to the general Cdynamic problems [4], H D h.1/ ; h3 ; h4 D .h1 ; h2 ; h3 ; h4 / 2 C0;˛ .˝ /rotu3 D
T @u3 @u3 @x2 ; @x1 ; rotu D @u @u1 @x1 @x2 : 2
Let us construct the matrix of differential operators: ˇ ˇ ˇ kT.@x; n.x//k3x3 k N k3x1 ˇ P.q/ .@x; n.x// D ˇˇ ˇ ; q D 0; 3 kı4k Œ.ı1q C ı3q / C .ı0q C ı3q / @n k1x4 ˇ @
ˇ ˇ ˇ kıjk k3x4 ˇ ˇ Q.q/ .@x; n.x// D ˇ ˇ ; q D 0; 3 kı4k Œ.ı0q C ı3q / C .ı1q C ı3q / @n k1x4 ˇ @ Approximate Solution of Some Boundary Value Problems 73
where N.x/ D .n; 0/; n D .n1 ; n2 /; T.@x; n.x// D kTjk .@x; n.x//k3x3 is the matrix ofstress operator on the plain of couple-stress elasticity:
@ @ @ Tjk .@x; n.x// D nj .x/ C . ˛/nk .x/ . C ˛/ıkj ; j; k D 1; 2 @xk @xj @nx 2 X Tjk .@x; n.x// D 2˛ jkp np .x/; j D 1; 2; k D 3 pD1
@ Tjk .@x; n.x// D . C ˇ/ıkj ; j D 3; k D 1; 3 @nx
Differential equations seem to be well suited as models for systems. Thus anunderstanding of differential equations are at least as important as an understandingof matrix equations. Allow us introduce the matrix L.@x; / D jLjk .@x; /j4x4 of differential operatorof pseudo oscillation: ˇ ˇ ˇ kL.1/ k2x2 kL.2/ k2x1 k GT .@x/k2x1 ˇˇ ˇ ˇ ˇ L.@x; / D ˇ kL.3/ k1x2 L.4/ 0 ˇ ˇ ˇ ˇk G.@x/k1x2 0 < ˇ
where
.1/ @2 Lij .@x/ D ıij . C ˛/ 2 C . C ˛/ ; i; j D 1; 2 @xi @xj 2 X .2/ .3/ @ Lij .@x/ D Lij .@x/ D 2˛ ijp .x/; i D 1; 2j D 3I j D 1; 2; i D 3 pD1 @xp
L.4/ .@x/ D . C ˇ/ 4˛ I 2 ; G.@x/ D .@x1 ; @x2 /
where ıij is Kronecker’s symbol, ijp is Levi-Chivita’ s symbol. Now the system (1) can be written in the form:
L.@x; /U.x; / D H.x/ (2)
In view of this the conjugated with L.@x; / operator will be considered: ˇ ˇ ˇ kL.1/ k2x2 kL.2/ k2x1 k GT .@x/k2x1 ˇˇ ˇ b ˇ ˇ L.@x; / D ˇ kL.3/ k1x2 L.4/ 0 ˇ ˇ ˇ ˇk G.@x/k1x2 0 < ˇ 74 M. Chumburidze
The matrices L.@x; / and b L.@x; / we use in technical point of view efficientlysolving the system of partial differential equations of CPTE in Green-Lindsayformulation. In next section we will solve the matrix equations (2).
2 Fundamental and Singular Solutions
In our investigations the fundamental [16] and singular solutions together with basicpotentials becomes an useful tool for the development of approximate methodssolution of stationary problems. Let us construct the matrix of fundamental solutions [4, 16] of the operator (2):
˚.x; / D j˚ij .x; /j4x4 D L.@x; /'.x; / (3)
where L is the associated with L.@x; / matrix:
L.@x; /L.@x; /'.x; / D L.@x; /'.x; /L.@x; / D IdetL.@x; / (4)
where I is the 4 4 dimensional unit matrix. According to (2) and (3) we have:
4 Y detL.@x; / D . C 2/. C ˛/. C ˇ/ . C k2 /'.x; / D 0 (5) kD1
From (5)we shall get:
4 X .1/ '.x; / D ak H0 .k jxj/ (6) kD1
.1/where H0 .k jxj/ is Hankel Function of the first kind (the zero order) [18], jxj Dq x21 C x22 ; k .k D 1; 4/ are parameters of thermo-elasticity [4, 16], '.x; / isunknown scalar, ak .k D 1; 4/ are constants, they are sought in such manner thatpartial derivatives of the eight order of function '.x; / has an isolated singularityof the kind lnjxj. If the matrix of fundamental solutions of the operator L.@x; / wedenote as b̊ .x; / by a direct check we can make sure that: b̊ .x; / D ˚ T .x; /: Let us construct the basic potentials: Z V.x; '/ D ˚.y x; /'.y/dy l l Z .1/ M .x; '/ D P.q/ .@y; n/ b̊ .y x; / T '.y/dy l Œb l Approximate Solution of Some Boundary Value Problems 75
Z M .2/ .x; '/ D Q.q/ .@y; n/ b̊ .y x; / T '.y/dy l Œb l Z U.x; '/ D ˚.y x; /'.y/dy ˝
where l 2 L2 .˛/ ; ˛ > 0; V.x; '/ is the single-layer potential, M .1/ .x; '/; M .2/ .x; '/are the mixed-type potentials, U.x; '/ is the volume potential. Conjugated operatorsbP.q/ ; b Q.q/ can be obtained from corresponding operators P.q/ ; Q.q/ replacing and .
3 Approximate Solutions
The aim of this section is to obtain approximate solutions of stationary problemsof CPTE in Green-Lindsay formulation. Similar methods of these techniques whichare extended to certain classes of problems are developed in the recent papers: [3, 10,16]. In particular, we formulate basic boundary value problems for two-dimensionalisotropic hom*ogeneous elastic materials with a center of symmetry. It is assumedthat surfaces are sufficiently smooth. Problem P.q/ .ı/. It is required to find regular solution U D .u; u3 ; u4 / – withthe following conditions:
8x 2 ˝ W L.@x; /U.x; / D H.x/ 8z 2 @˝ W P.q/ .@z; n.z//U.z/ D F .1/ .z/
Problem Q.q/ .ı/. It is required to find regular solution U D .u; u3 ; u4 / – withthe following conditions:
8x 2 ˝ W L.@x; /U.x; / D H.x/ 8z 2 @˝ W Q.q/ .@z; n.z//U.z/ D F .2/ .z/
The existence and uniqueness of this solution has been proved in [4]. Solution of the Problem P.q/ .ı/ will be found by the formula: Z Z 1 U.x/ D ˚.y x; /H.y/dy C Œb Q.q/ .@y; n/˚ T .y x; / T '.y/dy l (7) 2 ˝ l
where '.y/ is solution of the singular integral equations of normal type with zero(total) index: Z '.z/ C P.q/ .@z; n/Œb Q.q/ .@y; n/˚ T .y z; / T '.y/dy l D F .1/ .z/ l Z 1 P.q/ .@z; n/˚.y z; /H.y/dy (8) 2 ˝ 76 M. Chumburidze
Equation (8) is the singular integral equations with a Cauchy kernel [10] ofnormal type, which have an index equal to zero (total) and in this case the Fredholmtheorems hold [4, 5, 15]. Let us represent solutions in the following form: Z 1 U.x; / D GTP.q/ .x; yI ; ˝/H.y/dy C 2 ˝ Z C Œb Q.q/ .@y; n/GTP.q/ .x; yI ; ˝/ T F .1/ .y/dy l (9) l
where GP.q/ .x; yI ; ˝/ is the tensor of Green of Problem P.q/ .ı/ [4, 16]. Analogically we will get: Z 1 U.x; / D GT .x; yI ; ˝/H.y/dy C 2 ˝ Q.q/ Z C Œb P.q/ .@y; n/GTQ.q/ .x; yI ; ˝/ T F .2/ .y/dy l (10) l
where GQ.q/ .x; yI ; ˝/ is the tensor of Green of Problem Q.q/ .ı/ Formulas (9),(10) have an essential meaning to verify the generalized Fourierseries method [8, 16] for Problem P.q/ .ı/ and Problem Q.q/ .ı/. Let us construct auxiliary domains and surfaces. Let us consider b̋ ˝ finitedomain,bounded by the closed surfaces of Holder class :@ b̋ be sufficiently smoothsurface – boundary of b̋ , fxk g1kD1 @ b̋ be everywhere accounted set of points. The next theorems are proved there [3, 4, 6, 16]: ˚ 1Theorem 1 Accounted set of the vectors P.q/ .@y; n/˚ .j/ .y xk / kD1 ; j D1; 2; 3; 4; y 2 @˝, is linearly independent and full in the Hilbert space L2 .@˝/. ˚ 1Theorem 2 Accounted set of the vectors Q.q/ .@y; n/˚ .j/ .y xk / kD1 ; j D1; 2; 3; 4; y 2 @˝, is linearly independent and full in the Hilbert space [1] L2 .@˝/. Let us prove the following theorem:Theorem 3 For any " > 0 can be found the natural number N0 such, that whenN > N0 in any domain ˝ ˝, uniformly holds the inequality
jU.x/ U N .x/j < "; x 2 ˝;
where
X N X k Z .1/ j Œ jC3 1 U .x/ D N Xk ak ˚ ej .y x 4 ; i / ˚.y x; /H.y/dy kD1 jD1 2 ˝ Approximate Solution of Some Boundary Value Problems 77
Proof Introduce the following notations: kC3 k .y/ D P.q/ .@y; n/˚ ek .y xŒ 4 /; k D 1; 1
where k1 ek D k 4 4
Let us consider that ' k .y/ is orthonormal system of vectors j .y/ on @˝, then
X k j ' k .y/ D ak j .y/; k D 1; 1; y 2 @˝ (11) jD1
jwhere ak are coefficients of the orthonormalization. According to (3), we have:
X k jC3 ak P.q/ .@y; n/˚ ej .y xŒ 4 j ' k .y/ D /; k D 1; 1; y 2 @˝ (12) jD1
Let us consider that U.x/ be the direct solution of the Problem P.q/ .ı/(existences this solutions are proved [4]). Let us consider the vector: Z 1 V.x/ D U.x/ ˚.y x; /H.y/dy 2 ˝
Obviously V.x/ is the regular solution of the following boundary value problem:
8x 2 ˝ W L.@x; /V.x; / D 0; x 2 ˝ 8z 2 @˝ W P.q/ .@z; n.z//V.z/ D X .1/ .z/ (13)
where Z 1 X .1/ .z/ D F .1/ .z/ P.q/ .@z; n/˚.y z; /H.y/dy 2 ˝
According to the above X .1/ .z/ 2 C.0;˛/ .@˝/; ˛ > 0 we have:
X n .1/ X .1/.y/ Xk ' .k/ .y/; y 2 @˝ kD1 78 M. Chumburidze
where Z .1/ Xk D X .1/ .y/' .k/.y/dl @˝
Allow us introduce the vectors:
X N .1/ X N .1/ X k X N X k .1/ j jC3 Xk ' .k/ .y/ D Xk ak ˚ ej .y xŒ 4 jV N .x/ D Xk ak j .y/ D ; i / kD1 kD1 jD1 kD1 jD1
Let us show that it is approximation solution of Problem P.q/ .ı/: Obviously V N .x/ is the regular solution of the following boundary value problem:
8x 2 ˝ W L.@x; /V N .x; / D 0; x 2 ˝ P .1/ 8z 2 @˝ W P.q/ .@z; n.z//V N .z/ D NkD1 Xk ' .k/ .z/ (14)
But, according to (9) we have: Z X N 1 .1/ V N .x; / D Œb Q.q/ .@y; n/GTP.q/ .x; yI ; ˝/ T Xk ' .k/ .y/dy l (15) 2 l kD1
and also we have: Z 1 V.x; / D Œb Q.q/ .@y; n/GTP.q/ .x; yI ; ˝/ T X .1/ .y/dy l (16) 2 l
Hence, from the formulas (15) and (16) and according to the assumptions ofGreen tensor [4] and applying the Cauchy-Bunyakovski inequality [9], we have:
V.x/ D lim V N .x/ N!1
Analogically we can show that approximate solution of Problem Q.q/ .ı/ hasfollowing form:
X N X k Z .2/ j Œ jC3 1 U .x/ D N Xk bk ˚ ej .y x 4 ; i / ˚.y x; /H.y/dy kD1 jD1 2 ˝ Approximate Solution of Some Boundary Value Problems 79
where Z .2/ Xk D X .2/ .y/! .k/ .y/dl; @˝ Z 1 X .2/ .z/ D F .2/ .z/ Q.q/ .@z; n/˚.y z; /H.y/dy 2 ˝
X k ! .k/ .y/ D j bk j .y/; jD1 jC3 j .y/ D Q.q/ .@y; n/˚ ej .y xŒ 4 /
4 Conclusion
Thus, two-dimensional stationary problems of CPTE in the Green-Lindsay for-mulation have been investigated by applying the boundary integral method incombination with the harmonic potentials theory and generalized Fourier seriesanalysis. Approximate solutions of boundary value problems for a finite domainbounded by closed surface of Holder class have been constructed. As a result, it is shown that the approximate method is reliable for obtainingeffective solutions (explicitly) of boundary value problems for isotropic hom*oge-neous elastic materials with a center of symmetry.
Acknowledgements I would like to express my special gratitude to memories of my professorsTengis Burchuladze and Davit Gelashvili.
References
1. Bachman, G., Narici, L.: Fourier and Wavelet Analysis. Universitext. Springer, Berlin/New York (2000) 2. Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s Law: A Challenge to Theorists. Mathe- matical Physics. Imperial College Press, London (2000) 3. Burchuladze, T., Gegelia, T.: Development of the Potential Method in Elasticity Theory. Mecniereba, Tbilisi (1985) 4. Chumburidze, M.: Non-classical Models of the Some Theory of Boundary Value Problems. LAP LAMBERT Academic Publishing, Saarbrucken (2014) 5. Chumburidze, M., Lekveishvili, D.: Effective Solution of Boundary Value Problems of the Theory of Thermopiezoelasticity for a Half-Plane. American Institute of Physics (2013). doi:10.1063/1.485476 6. Chumburidze, M. Lekveishvili, D.: Approximate solution of some mixed boundary value problems of the generalized theory of couple-stress thermo-elasticity. Int. J. Math. Comput. Nat. Phys. Eng. (2014). WASET. http://waset.org/Publication/9998377 80 M. Chumburidze
7. Chumburidze, M., Lekveishvili, D., Khurcia, Z.: Solutions Of boundary-value problems of the generalized theory of couple-stress thermo-diffusion. J. Math. Syst. Sci. 3(7), 365–370 (2013). David Publishing, New York 8. Constanda, C.: Generalized Fourier Series. Mathematical Methods for Elastic Plates. Springer, London (2014) 9. Dragomir, S.: A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities. J. Inequal. Pure Appl. Math. (JIPAM) 4(3), 222–226 (2003)10. Eshkuvatov, Z., Long, N.: Approximate solution of singular integral equations of the first kind with Cauchy kernel. Appl. Math. Lett. (2009). doi:10.1016/j.aml.2008.08.00111. Ezzat, M., Zakaria, M.: Generalized thermoelasticity with temperature dependent modulus of elasticity under three theories. J. Appl. Math. Comput. 14, 193–212 (2004)12. Gelashvili, D.: To Ward the Theory of Dynamic Problems of Couple-Stress Thermodiffusion of Deformable Solid Micropolar Elastic Bodies. North-Holland, Amsterdam (1979)13. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)14. Green, E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)15. Hakl, R., Zamora, M.: Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations. Bound. Value Probl. 2014(1), 113 (2014). Springer16. Kupradze, V. et al.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermo Elasticity. North-Holland, Amsterdam-New York (1983)17. Lord, H., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)18. Orlando, F.: Hankel Functions. Mathematical Methods for Physicists, 3rd edn. Academic, Orlando (1985)19. Qing, H., Fanghua, L.: Elliptic Partial Differential Equations. Courant Institute of Mathemati- cal Science, New York (1997)20. Rezazadeh, G., Vahdat, A.: Thermoelastic damping in a micro-beam resonator using modified couple stress theory. Acta Mechanica 223(6), 1137–1152 (2012). Springer21. Sherief, H., Hamza, F., Saleh, H.: The theory of generalized thermoelastic diffusion. Int. J. Eng 5, 591–608 (2004). Elsevier22. Tripathi, J., Kedar, G., Deshmukh, K.: Dynamic problem of generalized thermoelasticity for a semi-infinite cylinder with heat sources. J. Thermoelast. 2(1), 01–08 (2014)23. Youssef, HM.: Theory of two-temperature-generalized thermoelasticity. J. Appl. Math. IMA 71(3), 383–390 (2006) Symmetry-Breaking Bifurcations in LaserSystems with All-to-All Coupling
Juancho A. Collera
Abstract We consider a system of n semiconductor lasers with all-to-all couplingthat is described using the Lang-Kobayashi rate equations. The lasers are coupledthrough their optical fields with delay arising from the finite propagation time ofthe light from one laser to another. As a consequence of the coupling structure, theresulting system of delay differential equations is equivariant under the symmetrygroup Sn S1 . Since symmetry gives rise to eigenvalues of higher multiplicity,implementing a numerical bifurcation analysis to our laser system is not straightfor-ward. Our results include the use of the equivariance property of the laser system tofind symmetric solutions, and to correctly locate steady-state and Hopf bifurcations.Additionally, this method identifies symmetry-breaking bifurcations where newbranches of solutions emerge.
1 Introduction
Semiconductor lasers are highly sensitive to optical feedback that even a smallamount of optical feedback is enough to produce chaotic instabilities [7, 8]. In 1980,Lang and Kobayashi [9] examined the influences of external optical feedback onsemiconductor laser properties. A single mode laser was examined where a portionof the laser output is reflected back to the laser cavity from an external mirror. Thedimensionless form of the Lang-Kobayashi (LK) rate equations derived in [2] aregiven by the following system of differential equations
P D .1 C i˛/N.t/E.t/ C ei˝ E.t /, E.t/ (1) P T N.t/ D P N.t/ .1 C 2N.t//jE.t/j2 ,
J.A. Collera ()Department of Mathematics and Computer Science, University of the Philippines Baguio, Gov.Pack Road, Baguio City 2600, Philippinese-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 81J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_8 82 J.A. Collera
where E.t/ is the complex electric field, N.t/ is the excess carrier number,and the fixed delay time represents the external cavity roundtrip time of thefeedback light. The parameters ˛, , ˝, T and P correspond to the linewidthenhancement factor, feedback strength, angular frequency of the solitary laser,electron decay rate, and pump parameter, respectively. System (1) was shown tocorrectly describe the dominant effects observed experimentally, see for example[10–12]. In 2006, Erzgraber et al. [4] studied a model of two mutually delay-coupledsemiconductor lasers in a face-to-face configuration. The two lasers are coupledthrough their optical field, and the finite propagation time of the light from one laserto the other constitute the time delay. A special case with zero detuning given by thefollowing LK-type rate equations
EP 1 .t/ D .1 C i˛/N1 .t/E1 .t/ C eiCp E2 .t /, EP 2 .t/ D .1 C i˛/N2 .t/E2 .t/ C eiCp E1 .t /, (2) T NP 1 .t/ D P N1 .t/ .1 C 2N1 .t//jE1 .t/j2 , T NP 2 .t/ D P N2 .t/ .1 C 2N2 .t//jE2 .t/j2 ,
where Cp is called the coupling phase parameter, was shown to have greatimportance as this case organizes the dynamics for small non-zero detuning. Noticethat (2) is in fact an extension of the LK equations in (1) into the case of twomutually delay-coupled semiconductor lasers. In this paper, we consider a system of n semiconductor lasers with all-to-all coupling. The lasers are coupled through their optical fields with time delayarising from the finite propagation time of the light from one laser to another. Thisgeneralizes the one-laser case in (1) and the zero-detuning two delay-coupled laserscase in (2). As a consequence of the coupling structure, the resulting system of delaydifferential equations (DDEs) is equivariant under a symmetry group. Symmetricsystems are known to give rise to eigenvalues with higher multiplicity [6] and thismakes numerical bifurcation analysis of such systems harder to implement. Ourresults include employing a group-theoretic approach from [6] to overcome thisproblem. This method not only locates steady-state and Hopf bifurcations correctlybut also identifies symmetry-breaking (SB) bifurcations where branches of newsolutions emerge. The paper is organized as follows. In Sect. 2, we introduce our model, itssymmetry properties and basic solutions. Then, in Sect. 3 we show how to findsymmetric solutions and use numerical continuation to find branches of solutionsand their stability. In Sect. 4, we give our main result which is a method of findingand classifying steady-state and Hopf bifurcations. We also look at the symmetrygroup of bifurcating branches of solutions from SB bifurcations. Then, lastly wegive our conclusions. Symmetry-Breaking Bifurcations in Laser Systems with All-to-All Coupling 83
2 Laser Systems with All-to-All Coupling
We now consider the n-laser system with all-to-all coupling as described by thefollowing LK rate equations, for j D 1; : : : ; n,
X n EP j .t/ D .1 C i˛/Nj .t/Ej .t/ C eiCp Ek .t /; kD1; k¤j (3) T NP j .t/ D P Nj .t/ .1 C 2Nj .t//jEj .t/j2 :
To determine the symmetry group of system (3), we first define the action of thepermutation group Sn and the circle group S1 to the state variables .Ej .t/; Nj .t//,j D 1; : : : ; n, as follows:
.Ej .t/; Nj .t// D .E1 . j/ ; N1 . j/ /; and # .Ej .t/; Nj .t// D .Ej ei# ; Nj /;
where 2 Sn and # 2 S1 . Notice that the action of Sn permutes the position of thelasers, while S1 acts only on the optical fields Ej .t/ for all j. Moreover, observe thatif .Ej .t/; Nj .t// is a solution to (3) for all j, then .Ej .t/; Nj .t// and # .Ej .t/; Nj .t//are also solutions to (3) for all j. Hence, system (3) is equivariant under the groupSn S1 . Basic solutions to (3), called compound laser modes (CLMs), are of the formEj .t/ D Rj ei!tCij and Nj .t/ D Nj for j D 1; : : : ; n, where !, Rj > 0, j , and Nj areall real-valued for all j, and with 1 D 0. We refer to CLMs with Rj D R, j D 0,and Nj D N for all j, as the fully symmetric CLMs and are given by
Ej .t/ D Rei!t ; and Nj .t/ D N; (4)
for j D 1; : : : ; n. These fully symmetric CLMs are fixed by elements .; 0/ of Sn S1for all 2 Sn . We use the symbol S0n to denote this symmetry group of CLMs in (4).Note that this subgroup S0n of Sn S1 is isomorphic to Sn and is generated by allelements of the form .; 0/ where 2 Sn . We call such symmetry group of CLMsas isotropy subgroup [5].
3 Symmetric CLMs
In this section, we provide a method of finding symmetric CLMs, that is, CLMsfixed by an isotropy subgroup of Sn S1 . In particular, we use the case when theisotropy subgroup is S0n to find the fully symmetric CLMs in (4). 84 J.A. Collera
Substituting the ansatz in (4) to system (3) and then following a similarcomputation in [4], we obtain the following transcendental equation in ! p ! C .n 1/ 1 C ˛ 2 sin.Cp C ! C tan1 ˛/ D 0: (5)
Once a value for ! is found by solving (5), we can thenp compute for correspondingvalues of N D !=.˛ C tan.Cp C ! // and R D .P N/=.1 C 2N/. Fullysymmetric CLMs in (4) are obtained using these values of R, !, and N. It isworth noting that the isotropy subgroup of the CLMs that we seek providesrelations amongst Rj , Nj , and j , and this to some extent simplifies the form of thetranscendental equation, such as in (5), which is key in finding symmetric CLMs. A branch of CLMs can be obtained through numerical continuation by varyinga single parameter. We use DDE-Biftool [3] to obtain such branch of solutions. Weemploy the same technique as in [4] to follow a CLM as an equilibrium in DDE-Biftool. We choose to vary the coupling phase parameter Cp for two reasons. First,because system (3) has a 2-translational symmetry in Cp , and secondly, becausethe coupling phase can be changed accurately in experiments [1].Example 1 Consider system (3) with n D 4, and parameters ˛ D 2:5, T D 392,P D 0:3, D 20, D 0:1, and Cp D 10. Solving for ! in (5) gives 11 CLMs whichare shown in the left panel of Fig. 1 as dots. By following any of these CLMs inDDE-Biftool, a branch of fully symmetric CLMs is obtained which is the ellipse inthe left panel of Fig. 1. The stability of this branch is also determined using DDE-Biftool. The right panel of Fig. 1 shows the stable and unstable parts of the branchin dashed line and solid line, respectively.
0.2 0.2
N 0 N 0
−0.2 −0.2
−0.8 −0.4 0 0.4 0.8 −0.8 −0.4 0 0.4 0.8
ω ω
Fig. 1 (Left) A branch of fully symmetric CLMs obtained in DDE-Biftool by varying the couplingphase parameter Cp . (Right) Stable part of the branch is shown in dashed line while unstable partof the branch is shown in solid line. Regular steady-state and Hopf bifurcations are marked with(˘) and (ı), respectively. For SB bifurcations, we use (4) and () to marked pitchfork and SBHopf bifurcations, respectively Symmetry-Breaking Bifurcations in Laser Systems with All-to-All Coupling 85
Symmetric systems, such as (3), are known to have eigenvalues of highermultiplicity. Hence, implementing a numerical bifurcation analysis on such systemis not straight-forward. In the section that follows, we develop a method to locatesteady-state and Hopf bifurcations along a branch of solutions and classify thesebifurcations into regular and symmetry-breaking.
4 Symmetry-Breaking Bifurcations
We now give our main result which is a method of finding and classifying steady-state and Hopf bifurcations. We first determine the linearized system corresponding to (3) around the fullysymmetric CLM (4). We follow a similar computation done in [13] for the one-lasermodel in (1). In polar form, the LK rate equations in (3) are given by X n RP j .t/ D Nj .t/Rj .t/ C Rk .t / cos Cp C 'k .t / 'j .t/ , kD1; k¤j X n Rk .t / 'Pj .t/ D ˛Nj .t/ C sin Cp C 'k .t / 'j .t/ , (6) Rj .t/ kD1; k¤j 1 h ˇ ˇ2 i NP j .t/ D P Nj .t/ .1 C 2Nj .t// ˇRj .t/ˇ ; T Tfor j D 1; : : : ; n. If we let Xj .t/ D Rj .t/; 'j .t/; Nj .t/ and Yj .t/ D Xj .t /, then (6)can be written in the form XP j .t/ D f .Xj .t/; Y1 .t/; : : : ; Yj1 .t/; YjC1 .t/; : : : ; Yn .t//; forj D 1; : : : ; n. Now, let X.t/ D ŒX1 .t/; : : : ; Xn .t/ T and Y.t/ D ŒY1 .t/; : : : ; Yn .t/ T sothat the full system can be written as X.t/ P D F.X.t/; Y.t//. The fully symmetricCLM in (4) written in polar form is given by X D ŒX1 .t/; : : : ; Xn .t/ T whereXj .t/ D ŒR; !t; N T for all j. To obtain the linear variational equation around X , wefirst compute for A WD dXj .t/ f .Xj / and B WD dYj .t/ f .Xj /. Now, let M1 be the block-diagonal matrix with the block A on the main diagonal and the block 0 elsewhere,and let M2 be the block matrix with the block 0 on the main diagonal and the blockB on all off main diagonal entries. Then, the Jacobian matrix evaluated at X isdF.X / D ŒM1 j M2 . The linear variational equation around the fully symmetricCLM is given by X.t/ P D M1 X.t/ C M2 X.t /, and its corresponding characteristicequation is det ./ D 0 where ./ D In M1 e M2 . Notice that if we letA WD I3 A and B WD e B, then 2 3 A B B 6B A B7 6 7 L WD ./ D 6 : : : 7 4 :: : : :: 5 B B A
where blocks A and B are 3 3 matrices. 86 J.A. Collera
2 We now use an method from [6] to examine the eigenvalues o of L. Let D e n i k 2k .n1/k T 3and define Vk D v; v; v; : : : ; v j v 2 R , for k D 0; 1; 2; : : : ; n 1. ˚ Observe that V0 D Œv; v; v; : : : ; v j v 2 R3 and the action of L on V0 is given T
by L Œv; : : : ; v T D Œ.A C .n 1/B/v; : : : ; .A C .n 1/B/v T : This means that theeigenvalues of LjV0 are those of A C .n 1/B. In general, the eigenvalues of LjVk , Pn1 jkfor k D 0; 1; 2; : : : ; n 1, are those of A C jD1 B: When n is odd andk ¤ 0, the complex numbers k ; 2k ; : : : ; .n1/k can be grouped into pairs ofconjugates. Observe that jk C .nj/k D 2 cos.2kj=n/ since jk and .nj/k are Pn1 jk P.n1/=2conjugates. Hence, jD1 D jD1 2 cos.2kj=n/ D 1 using the Dirichlet PNkernel identity 1 C jD1 2 cos.j/ D sin.N C 12 /= sin. 21 /: When n is even andk ¤ 0, we have n=2 D 1. Consequently,
2 1 n X n1 X sin.k k=n/ D .1/ C jk k 2 cos.2kj=n/ D .1/k C 1 jD1 jD1 sin.k=n/
by pairing up conjugates, and then using the Dirichlet kernel identity. Now, sincesin.k k=n/ D .1/kC1 sin.k=n/ and sin.k=n/ ¤ 0 for 1 k n 1, we getP n1 jk jD1 D 1. This is essentially the same for the case when n is odd. Therefore,the eigenvalues of LjVk , for k D 1; 2; : : : ; n1, are those of AB. This means that theproblem of solving the characteristic equation det ./ D 0, reduces to solving theequations det.A C .n 1/B/ D 0 and det.A B/ D 0. Furthermore, the eigenvaluesof L from A C .n 1/B are simple while those from A B are of multiplicity n 1.Also, notice that the symmetry group S0n acts trivially on V0 while its action on Vk ,for k D 1; 2; : : : ; n 1, is non-trivial. This means that SB bifurcations are obtainedfrom block AB while bifurcations from block AC.n1/B are regular bifurcations. We now illustrate the above technique in finding ordinary and SB bifurcations.Example 2 Continuing from Example 1, we first look for SB bifurcations. Pitchforkbifurcations along the branch of fully symmetric CLMs are found by looking at theintersections of the curves det.A B/jD0 D 0 and (5), while SB Hopf bifurcationsare found by finding the intersections of the curves det.A B/jDiˇ D 0 with ˇ >0, and (5). Two pitchfork bifurcations and two SB Hopf bifurcations were foundand are shown in the right panel of Fig. 1 with markers (4) and ( ), respectively.Similarly, we can get the regular bifurcations from the block A C 3B. Saddle-nodebifurcations are obtained from the intersections of the curves det.A C 3B/jD0 D0 and (5), while regular Hopf bifurcations were found by intersecting the curvesdet.A C 3B/jDiˇ D 0 with ˇ > 0, and (5). Two saddle-node bifurcations and sixregular Hopf bifurcations were obtained and are shown in the right panel of Fig. 1using markers (˘) and (ı), respectively. Identification of SB bifurcations is important because they give rise to newbranches of solutions. We now examine the symmetry group of branches ofsolutions that emerge from the SB bifurcations obtained in Example 2. To do this, Symmetry-Breaking Bifurcations in Laser Systems with All-to-All Coupling 87
0.67 Re(E1(t)) 0.66 0.2 0 0.25 0.5 0.75 1 0.67 Re(E2(t)) 0.66N 0 0 0.25 0.5 0.75 1 0.67 Re(E3(t)) 0.66 −0.2 1 0.25 0.5 0.75 1 0.67 Re(E4(t)) 0.66 −0.8 −0.4 0 0.4 0.8 ω 0 0.25 0.5 0.75 1 t
Fig. 2 (Left) A new branch of CLMs shown in dotted curve emanates from the pitchforkbifurcations (4) and has symmetry group S03 . (Right) Time-series plots showing that a branchof periodic solutions emerging from the SB Hopf bifurcations (ı) has spatio-temporal symmetryZ2 .; /
we use DDE-Biftool to perform numerical continuation, again varying the couplingphase parameter Cp . A new branch of symmetric CLMs shown as the dotted curvein the left panel of Fig. 2 is obtained. This new branch of CLMs emanates from thepitchfork bifurcations (4) and has a smaller symmetry group S03 compared to thesymmetry group S04 of the fully symmetric CLMs. Similarly, a branch of periodic solutions is obtained by following a SB Hopfbifurcation ( ) in DDE-Biftool. This branch of periodic solutions has isotropysubgroup Z2 .; /, which is a subgroup of S4 S1 that is isomorphic to Z2 andis generated by the order-two element .; /. Here, is an element of S4 whoseaction interchanges lasers 1 and 2, and lasers 3 and 4. Hence, the element .; /interchanges lasers 1 and 2 (respectively, lasers 3 and 4) and then shifts the phaseof each laser by half a period. The right panel of Fig. 2 shows a time series of thereal part of the Ej .t/ for j D 1; 2; 3; 4 where the interval for t is one period. Noticethat interchanging the curves for Re E1 .t/ and Re E2 .t/ (respectively, Re E3 .t/ andRe E4 .t/) and then shifting them by half a period preserves the plots. Indeed, thebifurcating branch of periodic solutions emanating from the SB Hopf bifurcationshas isotropy subgroup Z2 .; /.
5 Conclusion
The main contribution of this research is the identification of symmetry-breakingbifurcations in symmetric systems. The n-laser system (3) with all-to-all couplingis an example of a symmetric system. Symmetry gives rise to eigenvalues of highermultiplicity and this makes numerical bifurcation analysis of such system harderto implement. However, we showed that symmetry actually aids us in finding sym-metric solutions in Example 1, and in locating steady-state and Hopf bifurcations 88 J.A. Collera
along the branch of fully symmetric CLMs as illustrated in Example 2. The actionof the symmetry group decomposes the physical space into invariant subspaces.This, in turn, gives us a means to identify symmetry-breaking bifurcations wherenew branches of solutions emerge.
Acknowledgements This work was funded by the UP System Emerging InterdisciplinaryResearch Program (OVPAA-EIDR-C03-011). The author also acknowledged the support ofthe UP Baguio through RLCs during the A.Y. 2014–2015.
References
1. Agrawal, G.P., Dutta, N.K.: Long-Wavelength Semiconductor Lasers. Van Nostrand Reinhold, New York (1986) 2. Alsing, P.M., Kovanis, V., Gavrielides, A., Erneux, T.: Lang and Kobayashi phase equation. Phys. Rev. A 53, 4429–4434 (1996) 3. Engelborghs, K., Luzyanina, T., Samaey, G.: DDE-BIFTOOL v. 2.00: A MATLAB package for bifurcation analysis of delay differential equations. Technical report TW-330. Department of Computer Science, K.U. Leuven, Leuven (2001) 4. Erzgraber, H., Krauskopf, B., Lenstra, D.: Compound laser modes of mutually delay-coupled lasers. SIAM J. Appl. Dyn. Syst. 5, 30–65 (2006) 5. Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Birkhauser Verlag, Basel (2002) 6. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. II. Springer, New York (1988) 7. Gray, G.R., Ryan, A.T., Agrawal, G.P., Gage, E.C.: Control of optical-feedback-induced laser intensity noise in optical data recording. Opt. Eng. 32, 739–745 (1993) 8. Gray, G.R., Ryan, A.T., Agrawal, G.P., Gage, E.C.: Optical-feedback-induced chaos and its control in semiconductor lasers. In: SPIE’s 1993 International Symposium on Optics, Imaging, and Instrumentation, San Diego, pp. 45–57. International Society for Optics and Photonics (1993) 9. Lang, R., Kobayashi, K.: External optical feedback effects on semiconductor injection laser properties. IEEE J. Quantum Electron 16, 347–355 (1980)10. Mork, J., Tromborg, B., Mark, J.: Chaos in semiconductor lasers with optical feedback: theory and experiment. IEEE J. Quantum Electron 28, 93–108 (1992)11. Sano, T.: Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconduc- tor lasers with optical feedback. Phys. Rev. A 50, 2719–2726 (1994)12. van Tartwijk, G.H.M., Levine, A.M., Lenstra, D.: Sisyphus effect in semiconductor lasers with optical feedback. IEEE J. Select. Top. Quantum Electron. 1, 466–472 (1995)13. Verduyn Lunel, S.M., Krauskopf, B.: The mathematics of delay equations with an application to the Lang-Kobayashi equation. In: Krauskopf, B., Lenstra, D. (eds.) Fundamental Issues of Nonlinear Laser Dynamics. AIP Conference Proceedings, vol. 548. American Institute of Physics, New York (2000) Effect of Jet Impingement on Nano-aerosol SootFormation in a Paraffin-Oil Flame
Masoud Darbandi, Majid Ghafourizadeh, and Mahmud Ashrafizaadeh
Abstract In this paper, the effects of mico-jet impingement on the formation ofsoot nano-particles, CO, CO2 , and C6 H6 species in a turbulent paraffin-oil flameare investigated numerically. In this regard, we use a two-equation - turbulencemodel, a PAH-inception two-equation soot model, a detailed chemical kineticconsisting of 121 species and 2613 elementary reactions, and steady flameletcombustion model. We take into account the turbulence-chemistry interaction byusing presumed-shape probability density functions PDFs. We also take into accountthe radiation heat transfer of soot and gases assuming optically-thin flame. In thefirst place, we solve a documented experimental test case and compare the flamestructure to evaluate our numerical results. Then, we embed a micro-scale injector atthe burner wall, split the incoming air-flow between primary and secondary streams,inject the secondary air into the burner via the embeded injector, and comparethe results for different values of impinging-jet mass flow rate. Our results showthat the mico-jet impingement affects the reactive flow behavior within the burnerand results in a compact-flame near the fuel-injector nozzle. Our calculations alsoshow that the emission of CO2 is slightly affected by the mico-jet impingement.They show that increasing the mass flow rate of the micro-impinging-jet effectivelyreduces the formation and emission of the soot nano-aerosol and CO, C6 H6pollutants. It is also found that mico-jet impingement leads to a more uniformprofiles of temperature, soot volume fraction, and mass fractions of CO, CO2 , andC6 H6 at the burner outlet.
M. Darbandi ()Department of Aerospace Engineering, Center of Excellence in Aerospace Systems, Institutefor Nanoscience and Nanotechnology, Sharif University of Technology, Tehran, P. O. Box11365-8639, Irane-mail: [emailprotected]. GhafourizadehDepartment of Aerospace Engineering, Center of Excellence in Aerospace Systems, SharifUniversity of Technology, Tehran, P. O. Box 11365-8639, IranM. AshrafizaadehDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan, P. O. Box84156-83111, Irane-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 89J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_9 90 M. Darbandi et al.
1 Introduction
There are several types of flame holders to maintain the combustion process insidea burner. Combustion can be stabilized using (1) an external energy supply suchas heated plate, torch, highly reactive chemicals, lasers, electric arcs, chemicaladditives, (2) bluff bodies such as V-gutters, cylinders, spheres, flat plates, (3) stepssuch as rearward facing steps, forward facing steps, side dumps, cavities, (4) flowinstabilities such as swirls, cyclones, (5) jets such as reverse flow jets, opposed jets,and fuel jet blockages [1]. The choice of flame holding method depends on theburner size, weight, pressure loss, flammability limits, material temperature, andstructural limitations. Most of these methods stabilize flames through establishingrecirculation zones which carry high-temperature products upstream to separateflow region for igniting the incoming flows. Among the above mentioned methods,the opposed jet, i.e. an impinging jet, is an interesting flame-holding methodregarding the limitations. So, a deep understanding on the behavior of impingingjets would help to keep a stable combustion in combustors burning reactants andforming products. Many researchers have studied the formation of CO and CO2 pollutants incombustion processes while nano-particulate soot, incorporated with hazardousaromatics, i.e. C6 H6 , needs special attention and considerations. Even a lowemission of these pollutants can cause serious health problems, i.e. different organscancer. On the other hand, C6 H6 has been widely used as an important componentof gasoline to enhance the octane number of fuels. So, this aromatic compoundrequires to be understood very well due to vast applications in engines as well as itscarcinogenicity. Literature shows aerosol modeling of soot formation in laminar flames hasbeen studied in last decades [2, 3]. Few researchers also studied soot formation inturbulent flames fueled by simple hydrocarbons [4, 5]. There is a lack of resourcesstudying the formation of nano-particulate soot aerosol in turbulent flames burningcommon fuels such as paraffin-oil, gasoline, jet fuel, or diesel. Brooks and Moss[4] proposed soot acetylene-inception model and showed that soot formation insimple fuels is limited by inception rate of particles from single-ring aromaticsformed from acetylene. However, soot formation in practical fuels, which havehigher amounts of carbon atoms as well as aromatics compounds, is limited bygrowth rate of aromatics. In this regard, Hall et al. [6] proposed a soot model,i.e. soot PAH-inception model, with inception rate based on two and three-ringedaromatics formed from single-ring aromatic species, i.e. C6 H6 and C6 H5 radical. Back to our past publications, we have already simulated the nano-aerosol sootformation in turbulent non-premixed flames fed with methane [7, 8], ethylene[9, 10], and propane [11, 12]. In this paper, we use and extend our previous studiesand simulate the nano-aerosol soot formation in a turbulent non-premixed flameburning paraffin-oil as its fuel. In this regard, we employ a two-equation sootmodel to solve for the soot mass fraction and the number density considering Effect of Jet Impingement on Nano-aerosol Soot Formation in a Paraffin-Oil Flame 91
the soot formation and its oxidation based on PAHs and OH agents, respectively.We utilize a steady flamelet model as our combustion model considering a large-detailed kinetic reaction mechanism with 2613 reversible chemical reactions and121 chemical species. We use a two-equation - turbulence model with round-jetcorrections and take into account the turbulence-chemistry interaction using somepresumed-shape PDFs. We also take into account the radiation heat transfer of thesoot and gases assuming an optically-thin flame and calculate their radiations locallyonly by emissions. To evaluate our numerical solution, we simulate a benchmarkturbulent paraffin-oil non-premixed flame inside a burner and compare the obtainedresults with those of experiment as well as another numerical study. The obtainedresults indicate our numerical simulation can predict soot volume fraction, mixturefraction, and temperature distributions of the flame. Then, we embed a micro-scaleinjector at burner wall, split incoming air-flow between primary- and secondary-airstreams and inject the secondary-air into burner via the micro-scale injector andcompare the results. We also study mass-flow-rate effect of the micro-scale injectoron nano-aerosol soot formation, the emissions of CO, CO2 , and C6 H6 from turbulentnon-premixed paraffin-oil flame inside burner.
2 The Governing Equations
In the cylindrical coordinates, i.e. r, z, the fluid flow conservation laws consisting ofcontinuity, r-momentum, and z-momentum are given by u r .V/ C D0 (1) r
@p u e @u r .Vu/ D C r .e r u/ e 2 C (2) @r r r @r
@p e @v r .Vv/ D C r .e r v/ C g (3) @z r @r
The radial and axial components of the velocity vector are u and v, respectively. Themixture density, velocity vector, pressure, and effective viscosity are represented by, V, p, and e , respectively. The transport equations for turbulence quantities, i.e.turbulence kinetic energy and its dissipation rate , are given by e e @ r .V/ D r r C C G (4) r @r e e @ r .V/ D r r C C .c1 G c2 / (5) r @r 92 M. Darbandi et al.
In a confined jet, the turbulence model constants for Eqs. (4) and (5) are taken fromRef. [13]. To model combustion in a turbulent diffusion flame, we use the steadyflamelet model. In this study, we choose a detailed kinetic scheme, i.e., 121 chemicalspecies and 2613 chemical reactions, to perform our simulations. The transportequations for the first two moments of mixture fraction, i.e., f and f 002 , are given by e e @f r .Vf / D r rf C (6) f f r @r e 002 e @f 002 r Vf 002 D r rf C C cg e .r f /2 c f 002 (7) f f r @r
As is known, the results from pre-computed laminar flamelets and turbulentstatistics can be tabulated as a 3D lookup table in terms of mixture fraction,mixture fraction variance, and the scalar dissipation rate. If so, all thermo-chemicalquantities in the solution domain can be obtained from this 3D lookup table [14]. To simulate the aerosol dynamics, we need to solve two transport equations forthe soot mass fraction m and the soot number density n [7]. These equations aregiven by e e @m r .Vm / D r r m C C S m (8) soot soot r @r e e @n r .Vn / D r r n C C Sn (9) nuc nuc r @r
Assuming a unit Lewis number, the total enthalpy equation h is given by e e @h r .Vh/ D r rh C C qrad (10) h h r @r
The radiation source term in the energy conservation law, can be determined locallyonly by emission assuming an opticallyP thin flame. Finally, the density is obtainedfrom the equation of state as p D RT nmD1 Ym =Wm where m counts the numberof chemical species in the mixture from 1 to n total number of species. Thetemperature, gas constant, mass fraction, and molecular weight are represented byT, R, Y, and W, respectively.
3 Computational Method
Back to our past experiences [7–12], we choose a hybrid finite-volume-elementFVE discipline and break the solution domain into a large number of quadrilateralelements. We use our past experiences in treating non-staggered (collocated) grid Effect of Jet Impingement on Nano-aerosol Soot Formation in a Paraffin-Oil Flame 93
arrangement and implement finite element shape functions and physical influenceupwind scheme PIS for the diffusion and convection terms, respectively. Using PISscheme we also handle the pressure-velocity coupling effectively.
4 The Benchmark Test Case and Validation
A paraffin-oil turbulent non-premixed flame is chosen to verify our numericalsolutions. We employ the experimental conditions of Young et al. [15] to performour simulations. Figure 1 shows the configuration of the burner consuming paraffin-oil. Because of the symmetry of problem, we consider a rectangular solution domainapplying the symmetry boundary conditions at the center line. The computationaldomain has 0:0775 0:6 m dimensions, i.e. R0 D 0:0775 m and L D 0:6 m, seeFig. 1. The fuel nozzle diameter is 1.5 mm, i.e. R1 D 0:75 mm. This fuel nozzleinjects the gaseous paraffin-oil, which consists of 80 % n-decane, i.e. C10 H22 , and20 % toluene, i.e. C7 H8 , as fuel at a speed of 22.28 m/s into the burner. The oxidizer,i.e. co-flow air stream, which consists of 23.3 % oxygen and 76.7 % nitrogen, entersthe burner at a speed of 0.234 m/s. The initial temperatures of paraffin-oil and airare 598 K and 288 K, respectively. As understood, the paraffin-oil is evaporated anddelivered through a heated line and injected into the burner. The turbulence intensityand eddy length scale for both the paraffin-oil and air streams are 3 % and 0.02 m attheir nozzle exits, respectively. To evaluate the accuracy of our numerical solution in simulating turbulentreacting flow and aerosol modeling of soot nano-particles, we solve the test casegiven in Young et al. [15] and compare the predicted flame structure, i.e. thedistributions of mixture fraction, temperature, and species concentrations, with thedata collected by this reference as well as another available numerical solution [16].Wen et al. [16] numerically solved this test case considering a detailed chemicalkinetic mechanism with 141 species and 1015 elemental reactions to constructthe flamelet library. Figure 2 presents the axial distributions of mixture fraction,temperature, and soot volume fraction at centerline, i.e. r D 0. It also shows theradial distributions of mixture fraction, temperature, and soot volume fraction atz D 0:3 m above the burner. The figure shows that there are good agreements withthe data reported by Young et al. [15]. Considering the experimental data [15], ourresults are more satisfactory than those of obtained by Wen et al. [16] in the contextof same assumptions and models for turbulence, combustion, radiation, and soot.Our numerical results predict the flame length of 0.3 m, i.e. the stoichiometric valueof mixture fraction for n-decane, i.e. fst D 0:0615, which also corresponds to thepeak of axial temperature distribution. As seen in Fig. 2, in far downstream of burnerexit, the mixture fraction at centerline is lower than its stiochiometric value, i.e. thefuel-lean region, which indicates the fuel is fully depleted and that the burner is longenough to result in a full burn of paraffin-oil injected into the burner. As seen, thepeak of axial temperature distribution is lower than the adiabatic flame temperatureof paraffin-oil/air flame, i.e. 2366 K. There is a discrepancy between our results with 94 M. Darbandi et al.
Fig. 1 The configuration ofthe burner consumingparaffin-oil [15]
the measured data. It can be attributed to (1) the relatively simple models used in ourcalculations, e.g., the turbulence model, the radiation model, and the soot model, (2)the assumptions made to simplify the problem, e.g., the gas-phase nucleation andthe free-molecular-regime coagulation assumptions used in soot modeling, and theoptically-thin flame assumption used in calculating the radiative heat transfer rate,which took into account only the most radiating species. Evidently, the use of moresophisticated radiation and soot models would help to overcome such shortcomingsand predict the soot characteristics and flame structure more accurately, which isbeyond the scope of this paper. Effect of Jet Impingement on Nano-aerosol Soot Formation in a Paraffin-Oil Flame 95
(a) r =0 (b) r =0 (c) r =0
(d) z =0.3 m (e) z =0.3 m (f) z =0.3 m
Fig. 2 The current axial and radial distributions of mixture fraction, temperature, and soot volumefraction in the flame and comparison with the experimental data [15] and numerical solution [16];(a) r D 0, (b) r D 0, (c) r D 0, (d) z D 0:3 m, (e) z D 0:3 m, and (f) z D 0:3 m
5 The Results and Discussion
In this section, we study the effect of mico-jet impingement on the reactive flowbehavior and the resulting emissions of CO, CO2 , C6 H6 species and soot nano-aerosol using the current developed numerical method. We first study the effect ofmico-jet impingement on the above parameters comparing with the benchmark testcase. So, we embed a micro-scale injector at the burner wall, split the incoming air-flow between the primary- and secondary-air streams and inject the primary air intothe burner via the embedded micro-scale injector. The dimension of micro-scaleinjector is 100 m embedded at a height of z D 0:1 m above the fuel nozzle exit.We inject a mass flow rate of 2.7 g/s of the total incoming-air via the micro-scaleinjector while the rest of incoming air would enter the burner as primary-air. Theother parameters, i.e. the geometry and boundary conditions (BCs), are similar tothe benchmark test case. Figure 3 shows the effect of mico-jet impingement on the distributions of streamfunction, temperature, OH and C6 H6 mass fractions in the turbulent paraffin-oilflame. In each subfigure, the left part depicts the distributions for benchmarktest case, i.e., without any mico-jet impingement, while the right part depicts thedistribution for the burner incorporated with the mico-jet impingement, which 96 M. Darbandi et al.
(a) (b) (c) (d)
Fig. 3 The effect of mico-jet impingement on the distributions of (a) stream function (kg/s), (b)temperature (K), (c) OH mass fraction, and (d) C6 H6 mass fraction in the turbulent paraffin-oilflame
injects a mass flow rate of 2.7 g/s via the embedded injector inserted at theburner wall. As seen, the mico-jet impingement causes two recirculation zonesupstream and downstream of the micro-injector, which affect the flame envelop.The upstream recirculation, which is due to the injected air micro-jet, would causethe entrance of fresh air to the flame and this results in a well mixing of thatwith the gaseous high-temperature products. The downstream recirculation, whichis due to the injected air micro-jet, would cause the return of high-temperatureproducts at regions downstream of the burner, which is eventually mixed withthe low-temperature gaseous reactants, i.e. the exhaust gas recirculation EGR.Such enhancement in mixing performance due to the appeared recirculation zones,which are formed upstream and downstream of the micro-impinging jet, wouldresult in high-temperature combustion products at the burner outlet and a boostup in the combustion efficiency. The mixing enhancement also leads to a uniformtemperature of exhaust gases at the burner outlet. So, the mico-jet impingementimproves the combustion efficiency via enhancing the existing mixing. As seen, themico-jet impingement also results in a compact distribution of OH mass fractionnear the fuel injector exit. There are several definitions for the flame length. Inthe numerical simulations, this parameter is defined as the location of maximumOH concentration, the stoichiometric line, or the location of maximum temperaturegradient. Considering the aforementioned definition, it is observed that the mico-jetimpingement reduces the flame length and results in a compact flame. The compactflame together with the resulting EGR phenomenon would cause the by-products tobe fully-burnt at the regions near the fuel-injector exit. So, C6 H6 species would be Effect of Jet Impingement on Nano-aerosol Soot Formation in a Paraffin-Oil Flame 97
(a) (b) (c) (d)
Fig. 4 The effect of mico-jet impingement on the distributions of (a) CO mass fraction, (b) CO2mass fraction, (c) soot volume fraction, and (d) soot particles diameter (m) in the turbulent paraffin-oil flame
formed within the burner at regions near fuel-injector exit and so their emissionswould be reduced at the outlet, i.e., a very low concentrations release into thesurrounding ambient. As seen in Fig. 4, the EGR, due to the appeared two recirculation zones of micro-impinging-jet, would cause the rebrurn of combustion by-products, i.e. the CO, andthe combustion products, i.e. CO2 , at the regions near the fuel-injector exit. So, themico-jet impingement would reduce the emission of CO at the outlet. As seen, theappeared recirculation zones of micro-impinging-jet would result in a well mixingof combustion products with the unburnt reactants, which in turn results in a uniformdistribution of combustion products, i.e. the CO2 , within the burner and at the outlet.As seen also, the nano-aerosol soot formation within the burner is reduced as a resultof EGR, i.e. due to the appearance of two recirculation zones. The soot depletionwithin the burner would result in low concentrations of the soot in the exhaust gasesreleased into the surrounding atmosphere. Now, we summarize, compare, and study the effect of different values of micro-impinging-jet mass flow rate on the exhaust-gases temperature and the emissionsof CO, CO2 , C6 H6 , and the soot nano-aerosol released from the current turbulentparaffin-oil flame. In this regard, we consider different values of micro-impinging-jet mass flow rate, i.e. 1.8, 2.7, 3.2, and 3.6 g/s. The other parameters, i.e. thegeometry and BCs, are similar to the benchmark test case. As observed in Table 1,the emissions of CO, C6 H6 , and the soot nano-aerosol would decrease as the micro-impinging-jet mass flow rate increases. It is also observed that the exhaust-gasestemperature would increase as the micro-impinging-jet mass flow rate increases. 98 M. Darbandi et al.
Table 1 The mass-flow-rate effect of mico-jet impingement on temperature and emissions ofpollutants at the burner outletJet mass flow Temperature (K) CO mass CO2 mass Soot volume Soot particles C6 H6 massrate (g/s) fraction fraction fraction diameter (m) fraction– 767 3.60E06 0.0576 1.13E06 1.20E07 5.84E111.8 774 3.73E06 0.0573 9.33E07 1.10E07 9.04E112.7 747 0 0.0579 5.46E07 9.37E08 03.2 787 0 0.0579 3.38E07 7.63E08 03.6 807 0 0.0578 1.90E07 5.97E08 0
This indicates the role of mico-jet impingement in the amount of combustionefficiency enhancement.
6 Conclusion
Using a numerical study, we have investigated the effect of mico-jet impingementon the formation of soot nano-aerosol and the emissions of CO, CO2 , C6 H6 froma turbulent paraffin-oil flame. We have first evaluated our numerical simulation bysimulating a benchmark turbulent paraffin-oil non-premixed flame in a burner andcomparing the flame structure with those of measured and another available numer-ical solution. We have compared the distributions of mixture fraction, temperature,and soot volume fraction in the flame. The comparison showed a good agreementwith those of experiment. Then, we embedded a micro-scale injector in the burnerwall, split the incoming air-flow between primary and secondary streams, injectedthe secondary-air into the burner via the micro-scale injector, and compared theresults obtained for different values of micro-impinging-jet mass flow rates. Ournumerical study have shown that mico-jet impingement changes the reactive flowpattern within the burner and reduces the flame length. They have also shown thatas the mass flow rate of micro-impinging-jet increases, the emissions of soot nano-aerosol, CO, and C6 H6 reduce. The profiles of exhaust-gas temperature, soot volumefraction, mass fractions of CO, CO2 , and C6 H6 in exhaust gases also become moreuniform in the case of a mico-jet impingement.
Acknowledgements The authors would like to thank the financial support received from theDeputy of Research and Technology in Sharif University of Technology. Their financial supportand help are greatly acknowledged. Effect of Jet Impingement on Nano-aerosol Soot Formation in a Paraffin-Oil Flame 99
References
1. Civilian, D.L.D.: Numerical analysis of two and three dimensional recessed flame holders for scramjet applications. PhD thesis, Air University, Air Force Institute of Technology (1996) 2. Leung, K.M., Lindstedt, R.P., Jones, W.P.: A simplified reaction mechanism for soot formation in nonpremixed flames. Combust. Flame 87, 289–305 (1991) 3. Kennedy, I.M., Yam, C., Rapp, D.C.: Modeling and measurements of soot and species in a laminar diffusion flame. Combust. Flame 107, 368–382 (1996) 4. Brookes, S.J., Moss, J.B.: Predictions of soot and thermal radiation properties in confined turbulent jet diffusion flames. Combust. Flame 116, 486–503 (1999) 5. Kronenburg, A., Bilger, R.W., Kent, J.H.: Modeling soot formation in turbulent methane-air jet diffusion flames. Combust. Flame 121, 24–40 (2000) 6. Hall, R.J., Smooke, M.D., Colket, M.B.: Predictions of soot dynamics in opposed jet diffusion flames, chapter 8. In: Sawyer, R.F., Dryer, F.L. (eds.) Physical and Chemical Aspects of Combustion: A Tribute to Irvin Glassman. Combustion Science and Technology Book Series, vol. 189, pp. 189–230. Gordon and Breach, Amsterdam (1997) 7. Darbandi, M., Ghafourizadeh, M., Schneider, G.E.: Aerosol modeling of soot nanoparticles in a turbulent diffusion flame using an extended detailed kinetic scheme. In: 52nd Aerospace Sciences Meeting, National Harbor. AIAA 2014-0580 (2014) 8. Darbandi, M., Ghafourizadeh, M., Jafari, S.: Simulation of soot nanoparticles formation and oxidation in a turbulent nonpremixed methane-air flame at elevated pressure. In: 13th IEEE International Conference on Nanotechnology, Beijing, pp. 608–613. IEEE-NANO 2013-0382 (2013) 9. Darbandi, M., Ghafourizadeh, M., Schneider, G.E.: Extending a numerical procedure to simulate the micro/nanoscale soot formation in ethylene-air turbulent flame using acetylene- route nucleation. In: 11th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, Atlanta. AIAA 2014-2385 (2014)10. Darbandi, M., Ghafourizadeh, M., Schneider, G.E.: Numerical study on surface oxidation of carbonaceous nano- and micro-particles in a heavily sooting ethylene turbulent jet flame. In: 53rd AIAA Aerospace Sciences Meeting, Kissimmee. AIAA 2015-2084 (2015)11. Darbandi, M., Ghafourizadeh, M., Schneider, G.E.: Prediction of soot nano/micro particles emission (smoke pollution) in a turbulent propane nonpremixed flame. In: The CSME International Congress, Toronto (2014)12. Darbandi, M., Ghafourizadeh, M., Schneider, G.E.: Two-phase flow simulation of nano- and micro-particulate soot in an air-preheated nonpremixed turbulent propane flame. In: 22nd Annual Conference of the CFD Society of Canada, Toronto (2014)13. Kent, J.H., Honnery, D.: Soot and mixture fraction in turbulent diffusion flames. Combust. Sci. Technol. 54, 383–397 (1987)14. Sozer, E., Hassan, E.A., Yun, S., Thakur, S., Wright, J., Ihme, M., Shyy, W.: Turbulence- chemistry interaction and heat transfer modeling of H2 /O2 gaseous injector flows. In: 48th AIAA Aerospace Sciences Meeting, Orlando. AIAA 2010-1525 (2010)15. Young, K.J., Stewart, C.D., Moss, J.B.: Soot formation in turbulent nonpremixed kerosine-air flames burning at elevated pressure: experimental measurement. Proc. Combust. Inst. 25(1), 609–617 (1994)16. Wen, Z., Yun, S., Thomson, M.J., Lightstone, M.F.: Modeling soot formation in turbulent kerosene/air jet diffusion flames. Combust. Flame 135, 323–340 (2003) Normalization of Eigenvectors and CertainProperties of Parameter Matrices Associatedwith The Inverse Problem for Vibrating Systems
Mohamed El-Gebeily and Yehia Khulief
Abstract Solutions of the equation of motion of an n-dimensional vibrating systemM qR C DPq C Kq D 0 can be found by solving the quadratic eigenvalue problemL./x WD 2 Mx C Dx C Kx D 0. The inverse problem is to identify realdefinite matrices M > 0; K > 0 and D 0 from a specified pair .; Xc /of n-eigenvalues and their corresponding eigenvectors of the eigenvalue problem.We assume here that D U C iW, where U 0 and W > 0 are diagonalmatrices. The well posedness of the inverse problem requires that the matrix Xcbe specially normalized. It is known that for such specially normalized Xc , thereexist a nonsingular matrix XR and an orthogonal matrix , both real, such thatXc D XR .I i/. The identified matrices depend on a matrix polynomial Pr ./ DUr C Wr T C Wr Wr T ; r D 1; 0; 1, where Ur D <.r / and Wr D =.r /.In this work we give an explicit characterization of normalizers of Xc , introducesome new results on the class of admissible orthogonal matrices and characterizethe invertibility of the polynomials Pr ./ in terms of the invertibility of r . Forr D 1; 1 this is equivalent to identifying M > 0; K > 0. For r D 2 but Ur notstrictly negative, we give an example to show that Pr ./ is indefinite for all .
1 Introduction
In this article we are interested in the identification problem associated with afree vibrating beam or pipe. The discretized equation of motion that represents thetransverse vibration of an Euler-Bernoulli beam with no axial deformations has thegeneral form [11].
M qR C DPq C Kq D 0; (1)
where M; D; K are called the mass matrix, damping matrix and stiffness matrix,respectively. For a physical stably vibrating system, M and K are strictly positive
M. El-Gebeily () • Y. KhuliefKing Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabiae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 101J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_10 102 M. El-Gebeily and Y. Khulief
definite (M > 0; K > 0) while D is negative definite (D 0). All these matrices areassumed to be of dimension n. It is well known [9] that solving (1) is equivalent tosolving the eigenvalue problem
L./x D 0; (2)
where L./ is the second order matrix polynomial
L./ D 2 M C D C K: (3)
Solutions .; x/ of (2) appear in conjugate pairs and for vibrating beams or pipes, has a non-positive real part: <./ 0 and a nonzero imaginary part: =./ ¤ 0.Hence, it suffices to consider the case =./ > 0. Let 1 ; 2 ; ; n be eigenvaluesand x1 ; x2 ; ; xn be their corresponding eigenvectors. Following the notation in [8],we let
U D diag.< .k //nkD1 ; W D diag.= .k //nkD1 ;; (4) D U C iW; X c D x1 x2 : : : xn
The inverse problem for (1) is one of identifying the real matrices M > 0; K >0 and D 0 from a specified pair .; Xc /. There is a great deal of interest andvast literature devoted to this inverse problem for vibrating beams and other relatedvibrating systems including system monitoring and fault detection, inverse Sturm-Liouville problems, applied physics, and signal processing. The reader is referred to[1–3, 9, 10, 12] and the references therein. The well posedness of the inverse problem requires that the matrix Xc be speciallynormalized [8] in the sense that there is a complex diagonal matrix such that thenormalized matrix
XQ c WD Xc (5)
satisfies
XQ R XQ TR D XQ I XQ TI ; (6)
where XQ R D <.XQc / and XQ I D =.XQc /. To the best of our knowledge, no formula hasbeen given for finding a normalization matrix except through solving case by casea quadratic system of equations involving the elements of , see, e.g. [4–6, 8]. In thiswork we give an explicit formula for , which turns out to be also a characterizationof such normalization matrices. This is done in Sect. 3 in which we also give anillustrative example of an actual vibrating beam problem. Furthermore, it follows from the polar decomposition of XQ c that its real andimaginary parts are related by
XQ I D XQ R ; Eigenvector Normalization of Vibrating Systems 103
where is a real orthogonal matrix. Therefore,
XQ c D XQ R .I i/: (7)
We also show that the spectrum of satisfies
./ fz 2 C W arg.z/ < =2g: (8)
This result is then used to show that a certain matrix Q associated with the socalled Jordan triples is invertible, which eliminates the need to check this conditionfrom Algorithm 1 in [8]. Other properties of and a certain matrix polynomialdefined in terms of it are discussed in Sect. 4, where we characterize the invertibilityof this polynomial in terms of the given parameter . This has direct consequenceon the identification of positive definite M; K. We also provide an example to showthat the identification of a D 0, may not be possible to in general. In Sect. 2, we collect some facts about discretized vibrating systems which wewill need in this paper.
2 Preliminaries
Let U; W; be as defined in (4) and, for r 2 Z, let
r D r ; Ur D <.r /; and Wr D =.r /: (9)
Here, 0 D I. We will always assume that U 0; W > 0. It is well known that.L.// D [ N D .A B/, where DM K 0 AD ; BD : M 0 0 M
With Xc ; ; M defined as above, given matrices X 2 Cn2n ; J 2 C2n2n ; Y 2 2nnC , the triple .X; J; Y/ is called a Jordan triple for L./ if 0 0 X D Xc X c ; J D ; Y D Q1 ; (10) 0 M 1
where X QD (11) XJ 104 M. El-Gebeily and Y. Khulief
and N denotes complex conjugate. To determine real symmetric matrices K and Dfrom and Xc , it suffices that .X; J; X T / be a Jordan triple for L./. In this case, wemust have
X eRT D X eR X eIT ; eI X (12)
for some normalization X eR C iX ec D X eI of Xc . When (12) is satisfied, we have therepresentations eI D X X e eR ; ec D X X e eR I i (13)
efor some real orthogonal matrix .
3 Eigenvector Normalization
In this section we address the question: which normalizations (if any) of Xcallow (12) to hold? Define the complex matrix by
D Xc1 XN c (14)
Proposition 1 Assume Xc D XR C iXI is a given n n matrix whose columns areeigenvectors of a symmetric system and let be defined by (14). There exists acomplex diagonal n n normalization matrix such that X ec D Xc satisfies (12) ifand only if there exists an n n invertible matrix A such that1. A is diagonal,2. AT D A and3. 2 D A: ec D Xc satisfies (12). The latter can be written inProof Suppose exists. Then Xmatrix form as T R2 I2 2R I XR XR XI D 0: 2R I I2 R2 XIT I I I iIObserving that D 2I2n , we can rewrite the last equation as iI iI I iI
" # 0 2 T Xc Xc X c D0 2 0 XcT Eigenvector Normalization of Vibrating Systems 105
Multiplying on the left and right by Xc1 and XcT , respectively, and putting WDXc1 XN c we get 0 2 T I D0 (15) 2 0 I T 0 2 Clearly, the kernel of the operator 2 W Cn ! C2n is f0g. Furthermore, 0 I the operator I W C2n ! Cn has rank n, and therefore, its kernel also has rank n: T 0 2 For (15) to hold, the operator 2 must map onto the kernel of the 0 I operator I : The latter is spanned by the columns of : Therefore, there Iexists an n n change of base matrix A such that 0 2 T D A: 2 0 I I
Hence, we must have
2 D A; 2 D A T :
Since 1 D , N we may rewrite the above equations as
2 D A D A N T: (16)
Furthermore, since the matrices in the last two equalities of (16) must be diagonalmatrices and is invertible, we conclude that AN D AT . The only if part of the proof can be done by reversing the above steps. Proposition 1 suggests the following algorithm to find a normalization matrix . Thesteps parallel the three parts of the proposition.Algorithm 1 To compute a normalization diagonal matrix :1. Perform column reduction operations on to reduce it to a lower triangular form. (This fixes the upper triangular part of A and, by Proposition 1-2, the lower triangular part. At this point, the matrix A is actually diagonal.)2. Ensure that the diagonal part of A is purely imaginary so that Proposition 1-2 is satisfied.3. Calculate from Proposition 1-3. 106 M. El-Gebeily and Y. Khulief
Since column reduction operations are not unique, we expect that the normalizationmatrix is not unique. The following example shows that this is the case. It alsoexplores an alternative approach for finding that works in special situations.Example 1 The coefficients of a finite element discretization of a vibrating beam(see [7], Chapter 8) are given by 2 3 :0929 0 0161 :0967 6 0 1:4881 :0967 :558 7 MD6 4 :0161 :0967 :0464 :1637 5 ; 7
:0967 :558 :1637 :7448 2 3 :0052 0 :0026 :0326 6 0 1:088 :0326 :272 7 K D 104 6 7 4 :0026 :0326 :0026 :0326 5 :0326 :272 :0326 :544
and D D 0:2M C :005K: The eigenvalues and ‘un-normalized’ eigenvectors of thissystem are 2 3 2 3 128:86 0 0 0 186:81 0 0 0 6 0 15:46 0 0 7 6 0 7 D6 7 C i 6 0 76:84 0 7; 4 0 0 1:43 0 5 4 0 0 23:05 0 5 0 0 0 0:13 0 0 0 3:60 2 3 2 3 0:6 0:3 2:3 20:1 0:9 1:3 30:8 71:8 6 0:3 0:4 0 1:4 7 6 0:4 1:9 0:4 4:9 7 10 Xc D 6 3 7 6 7 4 2:5 2:5 3:2 59:5 5 C i 4 3:6 12:5 42:6 212:4 5 1:0 0:5 0:3 1:6 1:4 2:4 4:1 5:9
The real part of this matrix is invertible, therefore, we will put 2 3 1:4497 0 0 0 6 0 4:9605 0 0 7 D XR1 XI D 6 4 7; 0 0 13:187 0 5 0 0 0 3:5725
which happens to be a diagonal matrix. In this rather simple case, (12) simplifies to
R D I .I C / .I /1 : Eigenvector Normalization of Vibrating Systems 107
This equation shows the arbitrariness in assigning, say I : If we choose I D I, then 2 3 5:4473 0 0 0 6 0 1:5050 0 0 7 R D 6 4 7: 5 0 0 1:1641 0 0 0 0 0:5626
The normalized eigenvectors are 2 3 2 3 2:5 0:9 2:8 83:1 5:6 2:2 38:2 20:3 6 1:0 1:3 3:0 5:7 7 6 2:3 3:3 0:5 1:4 7 10 X c D 6 3e 4 10:0 8:7 38:8 7 C i6 7: 245:9 5 4 22:3 21:3 52:8 60:0 5 3:9 1:7 3:7 6:8 8:6 4:1 5:1 1:7
4 Consequences of Normalization
From now on, we will assume that Xc is properly normalized so that (12) is satisfied.It was shown in [8] that, to determine a mass matrix M > 0, it is necessary that
C T > 0; (17)
where is a real orthogonal matrix satisfying (13). In this section, we will discusssome consequences of this condition. Let 1 ; 2 ; ; n be an orthonormal systemof eigenvectors of T corresponding to its eigenvalues ei'1 ; ei'2 ; ; ei'n with '1 ; '2 ; ; 'n < . Define the matrices ˘ D diag ei'1 ; ei'2 ; ; ei'n and ˚ D Œ1 ; 2 ; ; n : (18)
Notice that T ˚ D ˚˘ and ˚ D ˘ ˚ . If matrix Q of (11) is nonsingular,then .X; J; X T / is a Jordan triple for L./ and we have
M D .XJX T /1 ; (19) 2 T 1 D D M.XJ X / M; (20) K D .XJ 1 X T /1 : (21)
Equations (19), (20) and (21) determine a symmetric system whose eigenvalues andeigenvectors are and Xc , respectively. We shall see later (see Lemma 2 below)that (17) ensures the invertibility of Q. This will require a characterization of theeigenvalues of the matrix T , which will be given in Lemma 1. 108 M. El-Gebeily and Y. Khulief
The set of orthogonal matrices can be parameterized using real skew-symmetricmatrices C as
D .I C/ .I C C/1 (22)
Condition (17) restricts the class of parameters as in the following lemma. Weshould observe here that skew-symmetric matrices have pure imaginary eigenvalues.Lemma 1 Let be a given orthogonal matrix and define C by (22). The followingare equivalent:1. Equation (17) holds p p2. .C/ i.1= 2; 1= 2/, that is, every eigenvalue iı of the parameterization matrix C satisfies p jıj < 1= 2 (23) 3. T CC WD fz 2 C W < .z/ > 0g :Proof 1 ” 2. Let be an orthogonal matrix and let C be its correspondingparameter matrix defined by (22). Then
C T D .I C/ .I C C/1 C .I C/1 .I C C/ :
If iı is an eigenvalue of C, then by the spectral mapping theorem,
1 iı 1 C iı C 1 C iı 1 iı
is a corresponding eigenvalue of C T . Then, by (17),
1 2ı 2 > 0; 1 C ı2
which is equivalent to (23). 1 ” 3. If (17) holds, then 0 < ˚ C T ˚ D ˘ ˚ ˚ C ˚ ˚˘ D 2< .˘ / ;
which implies 3. If 3. holds, then for any y 2 Cn , we may write y D ˚˛ for some˛ 2 Cn . Then y C T y D ˛ ˚ C T ˚˛ D 2˛ < .˘ / ˛ > 0;
which implies 1. Eigenvector Normalization of Vibrating Systems 109
Lemma 2 The matrix Q defined by (11) is invertible. Consequently, .X; J; X T / is aJordan triple for L./.Proof It follows from the invertibility of Xc and (13) that both XR and I i areinvertible. Furthermore, I C i is also invertible since its eigenvalues are conjugatesof those of I i: Now, X XR .I i/ XR .I C i/ QD D XJ XR .I i/ XR .I C i/ Xc 0 I .I i/1 .I C i/ D : 0 Xc .I i/1 .I C i/
Put
D i .I i/1 .I C i/ : (24)
A straightforward calculation shows that . / RC and hence, has a welldefined square root. Then iI 0 I .I i/1 .I C i/ iI
D : i iI .I i/1 .I C i/ 0 C
Therefore, the invertibility of Q is equivalent to the invertibility of C ,which, in turn, is equivalent to the invertibility of 1 . To show theinvertibility of the latter, we write 1 D 1=2 I 1=2 1 1=2 1=2 1=2 1=2 1 D 1=2 I 1=2 1=2 1=2 1=2 1=2 D 1=2 I B1 B 1=2 ;
where B D 1=2 1=2 and where we made use of the fact that D 1 . 1Therefore, we have to show that I B B is invertible. If not, then, invoking thespectral mapping theorem, we have, for some ı 2 .B/ D ./,
ı = .ı/ 0D1 D 2i ; ı ıwhich contradicts the assumption that W > 0.Next, we investigate the invertibility of the matrices M; D; K as identifiedfrom (21), (22) and (23). For r 2 Z, let r ; Ur ; Wr be as defined in (9) and let 110 M. El-Gebeily and Y. Khulief
be a real orthogonal matrix. Define the matrix function Pr ./ D < .I i/ r I i T D Ur C Wr T C Wr Ur T :
Then 0 r I i T Pr ./ D I C i I i r 0 I C i T
and we have the following proposition.Proposition 2 Pr ./ is invertible if and only if r is invertible.Proof Pr ./ is invertible if and only if ker Pr ./ D f0g : Note that Pr ./ D 0 r I .I i/ i I I i T ; where is given by (24). Therefore, r 0 i T 0 r Iker Pr ./ D ker i I : Arguing as in Proposition 1, the r 0 i T 0 r I operator W Cn ! C2n has rank n; and the operator i I W r 0 i T
C2n ! Cn has rank n and kernel space of dimension n: It follows that 0 r I ker i I D f0g (25) r 0 i T 0 r Iif and only if maps into the orthogonal complement of the kernel r 0 i T Iof i I : The kenel of i I is spanned by the columns of and it is i
straightforward T to see that its orthogonal complement is spanned by the columns of i
: Therefore, (25) holds if and only if there exists an invertible matrix B such Ithat T 0 r I i
D B: r 0 i T I
The second of the above equations gives
r D B: Eigenvector Normalization of Vibrating Systems 111
Under our assumptions on U and W, P1 ./; P1 ./ are always invertible. This, inturn is equivalent [8] to the positive definiteness of M; K. However,the followingexample shows that P2 ./ may be indefinite if U is not strictly negative definite.Example 2 Let 20 10 UD ;W D : 00 02
Then 2 3 0 4 0 D Ci : 0 4 0 0
All orthogonal 2 2 matrices have the form cos sin cos sin or : sin cos sin cos
The eigenvalues of the first type are e˙i and by Lemma 1, is admissible only if 2 .0; =2/. The eigenvalues of the second kind are 1; 1 and thus, they do notcorrespond to an admissible . For the first kind, we calculate 8 cos 7=2 cos 2 C 7=2 4 sin 7=2 sin 2 P2 ./ D 4 sin 7=2 sin 2 7=2 cos 2 7=2 pwhich has eigenvalues 1;2 D 4 cos 1=2 162 98 cos 2: It can be shownthat 1 < 0; 2 > 0 for all 2 .0; =2/. Therefore, P2 ./ is non-definite for anyadmissible :
Acknowledgements This work is funded by KACST-NSTIP Project No. 12-ADV3005-04. Theauthors acknowledge the support provided by King Abdulaziz City for Science & Technology andKing Fahd University of Petroleum & Minerals.
References
1. Bai, Z.J.: Constructing the physical parameters of a damped vibrating system from eigendata. Linear Algebra Appl. 428(2–3), 625–656 (2008) 2. Chu, M.T.: Inverse eigenvalue problems. SIAM Rev. 40, 1–39 (1998) 3. Chu, M.T., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Oxford University Press, Oxford (2005) 4. De Angelis, M., Imbimbo, M.: A procedure to identify the modal and physical parameters of a classically damped system under seismic motions. Advances in Acoustics and Vibration, Hin- dawi Publishing Corporation, Article ID 975125, 11 pages (2012). doi:10.1155/2012/975125 112 M. El-Gebeily and Y. Khulief
5. Friswell, M.I., Prells, U.: A measure of non-proportional damping. Mech. Syst. Sig. Process 14(2), 125–137 (2000) 6. Garvey, S.D., Penny, J.E.T.: The relationship between the real and imaginary parts of complex modes. J. Sound Vib. 212(1), 64–72 (1998) 7. Kwon, Y.W., Bang, H.: The Finite Element Method Using Matlab. CRC Press, Boca Raton (1997) 8. Lancaster, P., Prells, U.: Inverse problems for damped vibrating systems. J. Sound Vib. 283, 891–914 (2005) 9. Lancaster, P., Tismenetsky, M.: The Theory of Matrices. Academic, Orlando, Academic Press, New York (1985)10. Lancaster, P., Zaballa, I.: On the inverse symmetric quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 35(1), 254–278 (2014)11. Reddy, J.N., Wang, C.M.: Dynamics of fluid-conveying beams, CORE Report No. 2004-03, Centre for Offshore Research and Engineering, National University of Singapore (2004)12. Xu, S.F.: An Introduction to Inverse Algebraic Eigenvalue Problems. Peking University Press/Friedr. Vieweg & Sohn, Beijing/Braunschweig/Wiesbaden (1998) Computational Aspects of Solving InverseProblems for Elliptic PDEs on PerforatedDomains Using the Collage Method
H. Kunze and D. La Torre
Abstract The treatment of an inverse problem on a perforated domain is com-plicated heavily by the presence of the perforations or holes. We present severaltheoretical results that provide relationships between the problem on the perforateddomain and the same problem on the corresponding unperforated/solid domain. Theresults establish that we can approximate the solution of the inverse problem on theperforated domain by instead solving the inverse problem on the associated soliddomain. Examples are provided.
1 Introduction
In this paper, we are interested in the parameter estimation inverse problem forelliptic PDEs. One physical setting for such problem is the estimation of the thermaldiffusivity in a lamina based on (perhaps noisy) observational data of its equilibriumtemperature. Indeed, the examples in the final section of the paper include imagesthat can be thought of as isotherm plots in a lamina. In Sect. 2, we present theCollage Method approach for solving such inverse problems. In the past [5], we haveestablished that the Collage Method compares favourably with established methods,such a Tikhonov regularization [9]. The complication in the current work is that wewish to consider such an inverse problem on a perforated domain. A perforated domain (or porous medium) is a material characterized by apartitioning of the total volume into a solid portion often called the “matrix” anda pore space usually referred to as “holes.” Mathematically speaking, these holescan be either materials different from that of the matrix or real physical holes. When
H. Kunze ()Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]. La TorreDepartment of Economics, Management, and Quantitative Methods, University of Milan, Milan,ItalyDepartment of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi, UAEe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 113J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_11 114 H. Kunze and D. La Torre
formulating differential equations over porous media, the term “porous” implies thatthe state equation is written in the matrix only while boundary conditions should beimposed on the whole boundary of the matrix, including the boundary of the holes.Solving differential equations over a perforated domain is typically a complicatedtask because the size and distribution of the holes within the material play animportant role in its characterization. Simulations conducted over a perforateddomain that includes a large number of matrix-hole interfaces present numericalchallenges since a very fine discretization mesh and a large computation timeare required. The direct problem is of great interest in various areas of science,engineering, and industry (see, for example, [2] and [3] for discussion of real-world problems). One way to treat these (perhaps idealized) problems with rigorousmathematics is called hom*ogenization (see [3] and [8]), which takes advantage ofthe multiscale nature of the perforated domain. The inverse problem over a perforated domain inherits all of these challenges.In this paper, we seek to avoid the complications of working with the inverseproblem solution machinery on the perforated domain. In Sect. 3, we present severaltheoretical results that connect the problem on the perforated domain with the sameproblem on the corresponding unperforated (solid) domain. To frame the situation,we set up the two types of problems here. Let ˝ 2 R2 be compact and convex and ˝B be collection of circular m holes[iD1 B.xj ; "j /, where xj 2 ˝, the radii "j > 0, and the holes B.xj ; "j / are non- m
overlapping and strictly inside ˝. We let " D maxj "j , the maximum hole radius.Let ˝" denote the closure of ˝ n ˝B . Then, we consider the problem .P" / on theperforated domain r .K .x; y/ru.x; y// D f .x; y/ in ˝" (P" ) u.x; y/ D 0 on @˝"
and the associated problem .P/ on the corresponding unperforated domain r .K .x; y/ru.x; y// D f .x; y/ in ˝ ; (P) u.x; y/ D 0 on @˝
where 2 Rn is a parameter belonging to the compact set . The Dirichletboundary conditions above can be replaced by Neumann boundary conditions, inparticular on the boundaries of the holes ( @u @n D 0 on @˝B ). In each case, the inverse problem is to estimate the parameter (perhapscoefficients defining the diffusivity K and/or the source/sink function f ) givenobservational data values of the solution u.x; y/. Given observational data for the solution of problem .P" /, we solve the inverseproblem for .P/; the results we present in Sect. 3 show that the obtained parametervalues solve (approximately) the inverse problem for .P" /. Computational Aspects of Solving Inverse Problems on Perforated Domains 115
2 The Collage Theorem for Elliptic PDEs
The variational equation associated with an elliptic PDE can be written as
a.u; v/ D .v/; v 2 H; (1)
where .v/ and a.u; v/ are linear and bilinear maps, respectively, both defined ona Hilbert space H. We denote by h; i the inner product in H, kuk2 D hu; ui andd.u; v/ D ku vk, for all u; v 2 H. The inverse problem of interest may now be viewed as follows: Suppose that wehave an observed solution u and a given (restricted) family of bounded, coercivebilinear functionals a .u; v/, 2 , and a family of bounded linear functionals .Then, by the Lax-Milgram theorem, for each 2 there exists a unique u 2 Hsuch that .v/ D a .u ; v/ for all v 2 H. We would like to determine if thereexists a value of the parameter such that u D u or, more realistically, such thatku uk is small enough. The following theorem is useful for the solution of thisproblem.Theorem 1 (Generalized Collage Theorem [6]) For all 2 , suppose thata .u; v/ W H H ! R is a family of bilinear forms and W H ! R isa family of bounded linear functionals. Let u denote the solution of the equationa .u; v/ D .v/ for all v 2 H, as guaranteed by the Lax-Milgram theorem. Then,given a target element u 2 H,
1 ku u k F .u/; (2) mwhere ˇ ˇ F .u/ D sup ˇa .u; v/ .v/ˇ (3) v2H; kvkD1
and m > 0 is the coercivity constant of a . In order to ensure that the approximation u is close to a target element u 2 H,we can, by the Generalized Collage Theorem, try to make the term F .u/=m asclose to zero as possible. If inf2 m m > 0 then the inverse problem can bereduced to the minimization of the function F .u/ on the space , that is,
min F .u/: (4) 2
We refer to the minimization of the functional F .u/ as a “generalized collagemethod” because of philosophical connection with our earlier approach to inverseproblems for ODEs [7]; in that work, the word “collage” was used due toconnections to ideas in fractal imaging [1]. Such an optimization problem has a 116 H. Kunze and D. La Torre
solution that can be approximated with a suitable discrete and quadratic program,derived from the application of the Generalized Collage Theorem and the use of anorthonormal basis in the Hilbert space H [6].
3 Theoretical Results
We introduce the Sobolev spaces H D H01 .˝/ and H" D H01 .˝" /. Since anyfunction in H" can be extended to be zero over the holes, it is trivial to prove that H"can be embedded in H. We let ˘" u be the projection of u 2 H onto H" ; it followsthat
ku ˘" ukH ! 0 whenever " ! 0:
When Neumann boundary conditions are considered, it is still possible to extend afunction in H" to a function of H: these extension conditions are well studied (see[8]) and they typically hold when the domain ˝ has a particular structure. The variational formulations of (P" ) and (P) are
find u 2 H" such that a" .u; v/ D " .v/; 8v 2 H" (P" )
and
find u 2 H such that a .u; v/ D .v/; 8v 2 H: (P)
The generalized collage distance associated to .P/ is stated in (3) in Theorem 1,while the generalized collage distance associated to .P" / is ˇ ˇ F" .u/ D sup ˇa .u; v/ .v/ˇ : (5) " " v2H" ; kvkH" D1
In the results that follow, we assume that the continuous and bilinear forms a" anda are uniformly coercive and bounded with respect to and ", namely there existstwo positive constants m and M such that for all 2 8 ˆ ˆ a .u; u/ mkuk2 8u 2 H" ˆ " ˆ < a .u; v/ Mkukkvk 8u; v 2 H" " (H1) ˆ a .u; u/ mkuk2 ˆ 8u 2 H ˆ :̂ a .u; v/ Mkukkvk 8u; v 2 H Computational Aspects of Solving Inverse Problems on Perforated Domains 117
We also assume that the linear functionals " and are uniformly bounded withrespect to and ", namely there exists a positive constant such that ( " .u/ kuk 8u 2 H" (H2) .u/ kuk 8u 2 H
Under the hypotheses (H1) and (H2), (P" ) and (P) have unique solutions u" and u ,respectively, for each 2 and for each fixed choices of "j , j D 1; : : : ; m. We now state three results relating (P" ) and (P), first presented with proofs in [4].Proposition 1 The following estimate holds:
F .u/ M k˘" u u" kH" C ku ˘" ukH (6) m mProposition 2 There exists a constant C, that does not depend on ", such that thefollowing estimate holds:
F .˘" u/ F" .˘" u/ C C" (7)
for all 2 , " > 0.Proposition 3 Suppose that F .u/; F" .v/ W ! RC are continuous for all u 2 H,v 2 H" , and " > 0. Let " be a sequence of minimizers of F" .u/ over . Thenthere exists "n ! 0 and 2 such that "n ! , with a minimizer of F .u/over . The practical fallout of these results is that we have some justification inminimizing F .u/ (the generalized collage distance for the problem on the domainwith no holes) to obtain estimates of the parameter values for .P" / (the problemon the domain with holes). The quality of the estimate may be affected by the sizeof the largest hole; certainly, when all holes are sufficiently small, we expect theapproximation to be good.
4 Examples
Example 1 We place nine holes of assorted sizes inside ˝ D Œ0; 1 2 , as in Fig. 1 Wechoose K.x; y/ D Ktrue .x; y/ D 8 C 3x2 C y2 and consider the problem 8 < r .K.x; y/ru.x; y// D 2x2 C y2 ; in ˝" ; u.x; y/ D 0; on @˝; (8) : @u @n .x; y/ D 0; on @˝B ; 118 H. Kunze and D. La Torre
Fig. 1 The domain and level curves of solutions for Example 1
Table 1 Results for the inverse problem in Example 1. Due to the normalization of 0 , the truevalues are .1 ; 2 ; 3 / D .8; 3; 1/ı (%) 1 2 30 7.1935 2.9783 1.24311 7.1586 3.0466 1.25733 7.0737 3.1945 1.2934
where ˝B is the union of the nine holes. We solve the diffusion problem numericallyand sample the solution u" at 49 uniformly-distributed points strictly inside ˝. Thelevel curves of the solution are illustrated in Fig. 1. If a sample point lies inside ahole, we obtain no information at the point. Now, beginning with the observational data points, we consider the inverseproblem of estimating K and f . To this end, we define K .x; y/ D 0 C 1 x2 C 2 y2 .Using the data values and f .x; y/ D 4x2 C y2 , we seek to estimate the values of iin K .x; y/ by applying the generalized collage theorem to solve the related inverseproblem on ˝ with no holes. The results for various cases, with relative noise of ı% added, are presented inTable 1. The results worsen as noise is added, but remain reasonably good. 1Example 2 For " 2 f0:1; 0:025; 0:01g, define N" D 10" and
[ N" 1 1 ˝B D B" i "; j " ; i;jD1 2 2 Computational Aspects of Solving Inverse Problems on Perforated Domains 119
a domain with N"2 uniformly-distributed holes all of radius ". Choosing K.x; y/ DKtrue .x; y/ D 9 C 3x C 2y, we consider the steady-state diffusion problem 8 < r .K.x; y/ru.x; y// D 2x2 C y2 ; in ˝" ; u.x; y/ D 0; on @˝; (9) : @u @n .x; y/ D 0; on @˝ B :
For each fixed value of ", we solve the diffusion problem numerically and samplethe solution at 49 uniformly-distributed points strictly inside ˝, obtaining noinformation at the point if it lies in a hole. Using the data values, with relative noiseof ı% added, we use the generalized collage theorem to solve the related inverseproblem, knowing f .x; y/ D 2x2 C y2 and seeking a diffusivity function of the formK.x; y/ D 0 C 1 x C 2 y. The level curves of each solution are illustrated in Fig. 2. When the hole is toolarge, as in the N D 1 case, the estimates are very poor. In this case, the hole needsto be incorporated into the macroscopic-scale model, as it can’t be considered partof the smaller-scale model. In the other cases of the table, the estimates are good. We see that as the size of the holes decreases (even while the number increases),the solution to the inverse problem produces better estimates of the parameters. Asthe noise increases, the results worsen but remain good (Table 2).
Fig. 2 Level curves of solutions in Example 2, with " D 0:1, 0:025, and 0:01
Table 2 Results for the Recovered parametersinverse problem in " N" ı (%) 0 1 2Example 2. True values are.0 ; 1 ; 2 / D .10; 3; 2/ 0.1 1 0 14:4169 10:1594 1:8807 1 14:2991 10:2965 1:8130 3 14:0300 10:5812 1:6543 0.025 4 0 8:7683 4:1071 2:4944 1 8:7188 4:1853 2:5131 3 8:6024 4:3496 2:5587 0.01 10 0 8:8155 3:7757 2:3408 1 8:7645 3:8531 2:3661 3 8:6453 4:0158 2:4252 120 H. Kunze and D. La Torre
Acknowledgements H. Kunze thanks the Natural Sciences and Engineering Research Council forsupporting this research.
References
1. Barnsley, M.F.: Fractals Everywhere. Academic, New York (1989)2. Civan, F.: Porous Media Transport Phenomena. Wiley, Hoboken (2011)3. Espedal, M.S., Fasano, A., Mikelić, A.: Filtration in Porous Media and Industrial Application. Lecture Notes in Mathematics, vol. 1734. Springer, New York (1998)4. Kunze, H., La Torre, D., Collage-type approach to inverse problems for elliptic PDEs on perforated domains. Electron. J. Differ. Equ. 48, 1–11 (2015)5. Kunze, H., La Torre, D., Levere, K., Ruiz Galán, M.: Inverse problems via the “generalized collage theorem” for vector-valued Lax-Milgram-based variational problems. Math. Probl. Eng. (2015). http://dx.doi.org/10.1155/2015/7646436. Kunze, H., La Torre, D., Vrscay, E.R.: A generalized collage method based upon the Lax– Milgram functional for solving boundary value inverse problems. Nonlinear Anal. 71(12), 1337– 1343 (2009)7. Kunze, H., Vrscay, E.R.: Solving inverse problems for ordinary differential equations using the Picard contraction mapping. Inverse Probl. 15, 745–770 (1999)8. Marchenko, V.A., Khruslov, E.Y.: hom*ogenization of Partial Differential Equations. Birkhauser, Boston (2006)9. Tychonoff, A.N.: Solution of incorrectly formulated problems and the regularization method. Dokl. Akad. Nauk SSSR 151, 501–504 (1963) Dynamic Boundary Stabilizationof a Schrödinger Equation Througha Kelvin-Voigt Damped Wave Equation
Lu Lu and Jun-Min Wang
Abstract In this paper, we study an interconnected system of a Schrödinger and awave equation with Kelvin-Voigt (K-V) damping, where the K-V damped waveequation performs as a dynamic feedback controller. We show that the systemoperator generates a C0 -semigroup of contractions in the energy state space, and thesystem is well-posed. By detailed spectral analysis, we know that the spectral of thesystem operator composes of two parts, point spectrum and continuous spectrum.Moreover, the points in the spectra all have negative real parts. It follows that theC0 -semigroup generated by the system operator achieves asymptotic stability.
1 Introduction
There has been much interests in the problem of dynamic feedback controller,which is modeled by coupled partial differential equations (PDEs) or interconnectedPDEs. This kind of problems are quite challenging, yet have became attractive toresearchers these days. Control design and stability analysis for such systems havebecome active over the past 5 years, see [8, 9] and the references therein. In [9],an interconnected system of Schrödinger equation and heat equation is carefullystudied, which replaces the static feedback by dynamic feedback governed by aheat equation. It shows the exponential stability of the system and the Gevery classproperty of the semigroup. The boundary and internal stabilizations for Schrödingerequation are considered in [6], where the stability of the systems is achieved byusing multiplier techniques. Two kinds of boundary controllers for Schrödingerequation are concerned in [2], which shows that a simple proportional collocatedboundary controller can exponentially stabilize the system but the decay rate cannotbe prescribed, while the backstepping method can ensure it to have arbitrary decayrate. It is also known that viscoelastic materials have been widely used in engineeringand lots of researchers have put much efforts to analyze the dynamic behavior of
L. Lu () • J.-M. WangSchool of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R. Chinae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 121J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_12 122 L. Lu and J.-M. Wang
vibration for elastic structures with viscoelasticity over the past several decades.Kelvin-Voigt (K-V) damping is one of those most commonly used viscoelasticmodel, due to its easy and huge applications in modern technology. And there hasbeen an abundance of literature on the study of elastic system with viscoelasticdamping. In [3, 4], it is shown that local K-V damping can ensure the exponentialstability of both a string and an Euler-Bernoulli beam system when the materialparameter is smooth enough at the interface. However, if the smooth conditioncannot be satisfied, the string system is non-exponentially stable even if the materialparameter is a constant. Passive control of a wave equation with internal K-Vdamping is studied in [1]. Results there reveal that the spectrum of the systemoperator consists of point spectrum and continuous spectrum, due to the fact that theresolvent of the system operator for a viscoelastic system is not compact anymore. In this paper, we present a dynamic input/output feedback controller, which feedsthe K-V damped wave equation and Schrödinger equation into each other throughthe boundary. The coupled Schrödinger-wave system (as shown in Fig. 1) is writtenas follows: 8 ˆ ˆ yt .x; t/ C iyxx .x; t/ D 0; 0 < x < 1; t > 0; ˆ ˆ ˆ ˆ ztt .x; t/ zxx .x; t/ ˛zxxt .x; t/ D 0; 1 < x < 2; t > 0; ˆ ˆ ˆ < y.0; t/ D z.2; t/ D 0; t 0; y.1; t/ D kzt .1; t/; t 0; (1) ˆ ˆ ˆ ˆ ˛z .1; t/ C z .1; t/ D iky .1; t/; t 0; ˆ ˆ xt x x ˆ ˆ y.x; 0/ D y0 .x/; 0 < x < 1; :̂ z.x; 0/ D z0 .x/; zt .x; 0/ D z1 .x/; 1 < x < 2;
where ˛ > 0, k ¤ 0. The two equations are coupled at x D 1 with interconnectedconditions and fixed at each end.
y0 (x) y(x, t) Schrödinger equation yt (x, t) + iyxx (x, t) = 0 y(1, t) = kzt (1, t) kzt (1, t) −ikyx (1, t)
Wave equation with K-V damping ztt (x, t) − zxx (x, t) − αzxxt (x, t) = 0 z0 (x) αzxt (1, t) + zx (1, t) = −ikyx (1, t) z(x, t)
Dynamic feedback controller
Fig. 1 Block diagram for the dynamic boundary feedback of the coupled system Stabilization of Coupled Schrödinger-Wave System 123
By introducing the following transformation w.x; t/ D y.1 x; t/; 0 < x < 1; t > 0; (2) u.x; t/ D z.x C 1; t/; 0 < x < 1; t > 0;
then (1) becomes 8 ˆ ˆ wt .x; t/ C iwxx .x; t/ D 0; 0 < x < 1; t > 0; ˆ ˆ ˆ < tt u .x; t/ u xx .x; t/ ˛u xxt .x; t/ D 0; 0 < x < 1; t > 0; w.1; t/ D u.1; t/ D 0; t 0; (3) ˆ ˆ ˆ w.0; t/ D kut .0; t/; ˆ t 0; :̂ ˛uxt .0; t/ C ux .0; t/ D ikwx .0; t/; t 0;
Accordingly, the initial conditions for system (3) are w.x; 0/ D w0 .x/; u.x; 0/ Du0 .x/; ut .x; 0/ D u1 .x/; 0 < x < 1. The energy function for (3) is given by Z 1 1 E.t/ D jw.x; t/j2 C jux .x; t/j2 C jut .x; t/j2 dx: (4) 2 0
In this paper, we analyze the spectrum of (3) in which the system operator hasno compact resolvent. We first set up the system operator and show it generates aC0 -semigroup of contractions, and the system is well-posed. By detailed spectralanalysis, we obtain that the residual spectrum is empty and the continuous spectrumcontains only one negative point. Moreover, all the eigenvalues of the system lie inthe open left half plane. Therefore, this controller design moves the eigenvalues ofthe Schrödinger and wave equations into the second quadrant. It follows that theC0 -semigroup generated by the system operator achieves asymptotic stability.
2 Well-Posedness of System (3)
We consider system (3) in the energy space H D L2 .0; 1/ HE1 .0; 1/ L2 .0; 1/,where HE1 .0; 1/ D fg 2 H 1 .0; 1/jg.1/ D 0g. The norm in H is induced by the innerproduct Z 1 h i hX1 ; X2 i D f1 .x/f2 .x/ C g01 .x/g02 .x/ C h1 .x/h2 .x/ dx; (5) 0 124 L. Lu and J.-M. Wang
where Xs D . fs ; gs ; hs / 2 H , s D 1; 2. Define the system operator of (3) by 8 ˆ ˆ A . f ; g; h/ D .if 00 ; h; .g0 C ˛h0 /0 /; 8 . f ; g; h/ 2 D.A /; ˆ ˆ 8 ˇ 0 9 ˆ < ˇ g C ˛h0 2 H 1 .0; 1/; ˆ ˆ ˇ > > < ˇ = ˆ ˆ D.A / D . f ; g; h/ 2 H ; A . f ; g; h/ 2 H ; ˇ f .1/ D 0; f .0/ D kh.0/; : ˆ ˆ ˆ ˇ > :̂ :̂ ˇ > ˇ g0 .0/ C ˛h0 .0/ D ikf 0 .0/ ; (6)Then (3) can be written as an evolution equation in H : 8 ˆ < dX.t/ D A X.t/; t > 0; dt (7) :̂ X.0/ D X ; 0
where X.t/ D .w.; t/; u.; t/; ut .; t//.Theorem 1 Let A be given by (6). Then A 1 exists, and hence 0 2 .A /, theresolvent set of A . Moreover A is dissipative in H and A generates a C0 -semigroup eA t of contractions in H .Proof For any given . f1 ; g1 ; h1 / 2 H by,
A . f ; g; h/ D .if 00 ; h; .g0 C ˛h0 /0 / D . f1 ; g1 ; h1 /; (8)
we have ( 00 f .x/ D if1 .x/; h.x/ D g1 .x/; .g0 .x/ C ˛h0 .x//0 D h1 .x/; f .1/ D g.1/ D 0; f .0/ D kh.0/; g0 .0/ C ˛h0 .0/ D ikf 0 .0/; g1 .1/ D 0; (9) and the solution of (9) is given by 8 Z x Z 1 ˆ ˆ 0 ˆ ˆ f .x/ D f .0/.x 1/ i .1 x/f1 ./d i .1 /f1 ./d; ˆ ˆ 0 ˆ ˆ Z 1 x Z x ˆ ˆ ˆ < g.x/ D ik.x 1/f 0 .0/ ˛g1 .x/ .1 /h1 ./d C .x 1/ h1 ./d; x 0 ˆ ˆ ˆ ˆ h.x/ D g1 .x/ ˆ ˆ ˆ ˆ Z 1 ˆ ˆ 0 :̂ f .0/ D i .1 /f1 ./d kg1 .0/: 0 (10) Stabilization of Coupled Schrödinger-Wave System 125
Hence, we get the unique solution . f ; g; h/ 2 D.A / to equation (8), thus A 1exists. Now we show that A is dissipative in H . Let X D . f ; g; h/ 2 D.A /. Thenwe have ˝ ˛hA X; Xi D .if 00 ; h; .g0 C ˛h0 /0 /; . f ; g; h/ Z 1 Z 1 Z 1 D .if 00 /f dx C h0 g0 dx C .g0 C ˛h0 /0 hdx 0 0 0 ˇ1 Z 1 Z 1 ˇ1 Z 1 ˇ ˇ D if 0 f ˇ C i jf 0 j2 dx C h0 g0 dx C .g0 C ˛h0 /hˇ .g0 C ˛h0 /h0 dx 0 0 0 0 0 Z 1 Z 1 D if 0 .1/f .1/ C if 0 .0/f .0/ C i jf 0 j2 dx C h0 g0 dx C .g0 .1/ C ˛h0 .1//h.1/ 0 0 Z 1 Z 1 .g0 .0/ C ˛h0 .0//h.0/ h0 g0 dx ˛ jh0 j2 dx 0 0 Z 1 Z 1 Z 1 0 0 2 D if .0/f .0/ C i jf j dx C h0 g0 dx 0 ikf .0/h.0/ ˛ jh0 j2 dx 0 0 0 Z 1 h0 g0 dx 0 Z 1 Z 1 Z 1 Z 1 D ˛ jh0 j2 dx C i jf 0 j2 dx C h0 g0 dx h0 g0 dx (11) 0 0 0 0
and Z 1 RehA X; Xi D ˛ jh0 j2 dx 0: (12) 0
Hence A is dissipative and A generates a C0 -semigroup eA t of contractions in Hby the Lumer-Phillips theorem [7]. The proof is complete.
3 Spectral Analysis
In this section, we consider the eigenvalue problem of (3). Let A X D X, where0 ¤ X D . f ; g; h/ 2 D.A /, then f ; g; h satisfy: 8 00 ˆ ˆ f .x/ if .x/ D 0; ˆ ˆ ˆ ˆ ˆ ˆ h.x/ D g.x/; < .1 C ˛/g00 .x/ 2 g.x/ D 0; (13) ˆ ˆ ˆ ˆ ˆ ˆ f .1/ D g.1/ D 0; f .0/ D kh.0/; ˆ :̂ 0 ˛h .0/ C g0 .0/ D ikf 0 .0/: 126 L. Lu and J.-M. Wang
1Let p./ D 1 C ˛, when p./ ¤ 0 i.e. ¤ , (13) changes to ˛ 8 00 ˆ ˆ f .x/ D if .x/; ˆ ˆ ˆ ˆ 2 2 ˆ 00 < g .x/ D g.x/ D g.x/; 1 C ˛ p./ (14) ˆ ˆ ˆ ˆ f .1/ D g.1/ D 0; f .0/ D kg.0/; ˆ ˆ :̂ .1 C ˛/g0 .0/ D p./g0 .0/ D ikf 0 .0/:
We can get p p q q 2 2 ix p./ x p./ x f .x/ D a1 e ix C b1 e ; g.x/ D c1 e C d1 e ; (15)
where a1 ; b1 , c1 and d1 are constants. Substituting these into the boundary conditionsof (14), we have 8 p p ˆ ˆ a1 e i C b1 e i D 0; ˆ ˆ ˆ ˆ q q ˆ < 2 2 c1 e p./ C d1 e p./ D 0; (16) ˆ ˆ a1 C b1 D k.c1 C d1 /; ˆ ˆ ˆ ˆ q :̂ p./ 2 .c d / D ikpi.a b /: p./ 1 1 1 1
Then (14) has the nontrivial solution if and only if the characteristic equationdet ./ D 0, where 2 p p 3 e i e i 0 0 6 q q 7 6 2 2 7 6 0 0 e e p./ 7 6 7 p./
./ D 6 7: (17) 6 1 1 k k 7 6 7 4 p p q q 5 2 2 ik i ik i p./ p./ p./ p./
Lemma 1 Let A be defined by (6). Then for each 2 p .A /, we have Re < 0.Proof By Theorem 1, since A is dissipative, we have for each 2 .A /, Re 0.So we only need to show there is not any eigenvalue on the imaginary axis. Let D ˙i2 2 p .A / with 2 RC and X D . f ; g; h/ 2 D.A / be its associatedeigenfunction of A . Then by (12), we have Z 1 RehA X; Xi D ˛ jh0 j2 dx D 0: 0 Stabilization of Coupled Schrödinger-Wave System 127
Hence h0 .x/ D 0. By the second and third equations of (13), we haveh.x/ D g.x/ D 0. Then by the first equation of (13) and its boundary conditionswe have: ( 00 f .x/ D if .x/; f .0/ D f 0 .0/ D f .1/ D 0:
A direct computation yields f .x/ D 0. Hence, X D . f ; g; h/ D 0. Therefore, thereis no eigenvalue on the imaginary axis. This completes the proof. 1Proposition 1 Let A be defined by (6). Then D … p .A /. ˛ 1Proof When D , then p./ D 0. From the third and second equation of (13), ˛we can easily get g.x/ D h.x/ D 0. Following the same manner as the proof of 1Lemma 1, we have f .x/ D 0. Hence, X D . f ; g; h/ D 0. So that D … ˛p .A /. Theorem 2 Let A be defined by (6). The eigenvalues of A have the followingasymptotic expressions: 1(i) When ! , the asymptotic eigenvalue is given by: ˛ p 1 1 ˛A 2n D 2 2 3 C 4 4 5 C O.n5 /; (18) ˛ n ˛ n ˛ where p p 2 1 B 1 C B 2i i k2 i AD p Œ p p e ˛ ; B D : (19) 2i i B.1 C e ˛ / ˛ ˛ ˛2
(ii) When jj ! 1, there are two families of eigenvalues given by: " # 2 ln2 r 1n D .n C /j ln rj C .n C / i C O.n1 /; (20) 2 2 4
' 2 ˛ 2 1 h ' i 3n D ˛.n C / C ln r C ˛.n C / ln r i C O.n1 /: 2 4 ˛ 2 (21) 128 L. Lu and J.-M. Wang
Here r; and ' are three constants given by p ˛ 2 C k8 0<rD p < 1; ln r < 0; ln r1 D ln r > 0; (22) ˛ C 2˛k2 C k4
8 p ˆ 2˛k2 < arctan ˛k4 ; ˛ k4 > 0; D 2; p ˛ D k4 ; (23) :̂ 2˛k2 arctan k4 ˛ ; ˛ k4 < 0;
and 8 p ˆ 2˛k2 < arctan k4 ˛ ; k4 ˛ > 0; ' D 2; p ˛ D k4 ; (24) :̂ 2˛k 2 4 arctan ˛k 4 ; k ˛ < 0:
Moreover,
Re1n ; Re3n ! 1; when n ! 1: (25)
Proof Due to q space limitation, we give the outline of the proof here. From (17), and 2let s./ WD p./ 2 C, a direct computation gives, h p p ih p i det ./ D e i es./ e i es./ p./s./ C ik2 i h p p ih p i C e i es./ e i es./ p./s./ ik2 i : (26)
Let det ./ D 0, asymptotic expressions of the eigenvalues can be achieved. (I) When ! ˛1 , let " D C ˛1 , then " ! 0. We get 2n as shown in (18) and (19).(II) When ! 1, we consider WD i2 ; 0 arg 2 . We then get 1n and 3n (given by (20) and (21)), when belongs to 0 arg 8 and 8 < arg 3 8 , respectively. There is no high frequency solution in 3 8 < arg 2 . Stabilization of Coupled Schrödinger-Wave System 129
Proposition 2 Let A be defined by (6). Then its adjoint operator A has thefollowing form: 8 ˆ ˆ A . f ; g; h/ D .if 00 ; h; .g0 ˛h0 /0 /; 8 . f ; g; h/ 2 D.A /; ˆ ˆ 8 ˇ 0 9 ˆ < ˇ g ˛h0 2 H 1 .0; 1/; ˆ ˆ ˇ > > < ˇ = ˆ ˆ D.A / D . f ; g; h/ 2 H ; A . f ; g; h/ 2 H ˇ f .1/ D 0; f .0/ D kh.0/; : ˆ ˆ ˆ ˇ > :̂ :̂ ˇ > ˇ g0 .0/ ˛h0 .0/ D ikf 0 .0/ ; (27) ˚ 1 SProposition 3 Let A is defined by (6). Then .A / D ˛ p .A /. ˚ Proposition 4 Let A be defined by (6). Then r .A / D ;, and c .A / D ˛1 . ˚ SProof From Propositions 1 and 3, we have ˛1 … p .A / and ˛1 p .A / D.A /. The desired results will be got if ˛1 … r .A /. Now we suppose ˛1 2r .A /, then ˛1 2 p .A /. By A X D ˛1 X, where X D . f ; g; h/ 2 D.A /, weget 8 ˆ 00 i ˆ ˆ f .x/ ˛ f .x/ D 0 ˆ ˆ ˆ ˆ ˆ ˆ 1 ˆ ˆ h.x/ D g.x/ ˆ ˆ ˛ < 1 (28) ˆ ˆ .g0 .x/ ˛h0 .x//0 D h.x/ ˆ ˆ ˛ ˆ ˆ ˆ ˆ f .1/ D g.1/ D 0; f .0/ D kh.0/; ˆ ˆ ˆ ˆ 0 :̂ ˛h .0/ g0 .0/ D ikf 0 .0/:
From the second and third equation of (28), we get g.x/ D h.x/ D 0. Then f .x/satisfy 8 < f 00 .x/ i f .x/ D 0; ˛ (29) : f .0/ D f 0 .0/ D f .1/ D 0:
Simple computation shows that f .x/ D 0. This implies that X D . f ; g; h/ D 0,which is a contradiction. So ˛1 … r .A /. The proof is complete. 130 L. Lu and J.-M. Wang
4 Asymptotic Stability of System (3)
Definition 1 A C0 -semigroup T.t/ is called asymptotically (strongly) stable, if
lim jjT.t/xjj D 0: t!1
Theorem 3 ([5]) Let T.t/ be a uniformly bounded C0 -semigroup on a Banachspace X and let A be its generator. If \ .A/ iR c .A/;
and c .A/ is countable, then T.t/ is asymptotically stable.Theorem 4 Let A be defined by (6). Then the system (3) achieves asymptoticstability.Proof Since from Lemma 1 and Proposition 3, we know that when 2 .A /,Re < 0. So we have \ 1 .A / iR D ; c .A / D ; ˛
and c .A / is obviously countable. Then system (3) achieves asymptotic stability byTheorem 3.
5 Conclusions
In this paper, we use a wave equation with K-V damping to be a dynamic feedbackcontroller for a Schrödinger equation. We give the asymptotic expression of theeigenvalues, and also the exact composition of the spectrum of the system operator.At last, the asymptotic stability of the system was achieved.
Acknowledgements This work was supported by the National Natural Science Foundation ofChina.
References
1. Guo, B.Z., Wang, J.M., Zhang, G.D.: Spectral analysis of a wave equation with Kelvin-Voigt damping. Z. Angew. Math. Mech. 90(4), 323–342 (2010)2. Krstic, M., Guo, B.Z., Smyshlyaev, A.: Boundary controllers and observers for the linearized Schrödinger equation. SIAM J. Control Optim. 49(4), 1479–1497 (2011) Stabilization of Coupled Schrödinger-Wave System 131
3. Liu, K., Liu, Z.: Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J. Control Optim. 36, 1086–1098 (1998)4. Liu, K., Liu, Z.: Exponential decay of energy of vibrating strings with local viscoelasticity. Zeitschrift für angewandte Mathematik und Physik ZAMP 53, 265–280 (2002)5. Luo, Z.H., Guo, B.Z., Morgul, O.: Stability and Stabilization of Infinite Dimensional Systems with Applications. Communications and Control Engineering Series. Springer, London (1999)6. Machtyngier, E.: Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32, 24–34 (1994)7. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)8. Wang, J.M., Krstic, M.: Stability of an interconnected system of Euler-Bernoulli beam and heat equation with boundary coupling. ESAIM: Control, Optim. Calc. Var. 21(4), 1029–1052 (2015)9. Wang, J.M., Ren, B., Krstic, M.: Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans. Autom. Control 57, 179–185 (2012) Molecular-Dynamics Simulations Using SpatialDecomposition and Task-Based Parallelism
Chris M. Mangiardi and R. Meyer
Abstract This article discusses the implementation of a hybrid algorithm forthe parallelization of molecular-dynamics simulations. The hybrid algorithm com-bines the spatial decomposition method using message passing with a task-based,threaded approach for the parallelization of the workload. Benchmark simulationson a multi-core system and an Intel Xeon Phi co-processor show that the hybridalgorithm provides better performances than the message-passing or threadedapproaches alone.
1 Introduction
Molecular-dynamics (MD) is a computer simulation method that is widely used incomputational physics, chemistry and materials science. The method is described indetail in Refs. [1, 3]. Since MD is frequently used to perform large-scale simulationson high-performance computers, it is important to develop MD algorithms that makethe best use of modern computing architectures. This article discusses a hybrid approach that uses a two-level approach forthe parallelization of MD simulations. The first level is based on the spatialdecomposition method [10] and is implemented with the Message-Passing Interface(MPI) [6]. The second parallelization level employs the cell-task method for theparallelization of the work-load within the spatial domains and is implemented withIntel’s Threading Building Blocks Library [11]. If a MD simulations uses a short-range force model and the simulated systemis sufficiently hom*ogeneous, the well-known spatial decomposition method [10]provides an effective means for the parallelization of the simulation. In this method,the system being simulated is divided into equally sized and shaped sub-volumes,wherein each of these sub-volumes are processed by a processor, in order toimprove the performance and reduce the time required to run the simulation. Thesesub-volumes of the system, however, require utilizing message passing in orderto communicate their particles’ data to neighbouring sub-volumes. The spatial
C.M. Mangiardi () • R. MeyerLaurentian University, Sudbury, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 133J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_13 134 C.M. Mangiardi and R. Meyer
decomposition method works best when the particles are evenly distributed betweensub-volumes, as in hom*ogeneous systems. If the system is inhom*ogeneous, loadbalancing problems occur and the parallel efficiency is reduced. The cell-task method [7–9] is made to work well for inhom*ogeneous systems,wherein the work load would not be evenly balanced with the spatial decompositionmethod. This method works in a similar manner to the spatial decompositionmethod, in that it splits the system into sub-volumes; although for this methodthese sub-volumes are the width of the interaction range (Verlet-radius), and there istypically thousands of these sub-volumes. Each of these sub-volumes are scheduledto run on a processor core in an order wherein no two of the sub-volumes currentlybeing simulated interact with the same particles, in order to remove the requirementof cache blocking. This method, unlike spatial decomposition, does not require theuse of message passing, since the design is for a shared-memory system. The hybrid method discussed in this article utilizes both of these methods, byfirst dividing the simulation into large sub-volumes for distribution to separateprocessors, as in the spatial decomposition method. After this point, the cell-taskmethod is used to create many smaller sub-volumes of the system for dynamicscheduling across the processor cores. Communication between the larger sub-volumes is still required for particles near the border with the other subsections. The primary rationale for the implementation of the hybrid method is that itextends the range of the cell-task method to more than one compute node. However,even on a single node, the hybrid approach can be advantageous. While the cell-taskmethod is more efficient for inhom*ogeneous systems, the situations is less clearfor hom*ogeneous systems where spatial decomposition works well. In this case,the situation depends on system details since the overhead of the task managementin the cell-task method competes with the communication overhead of the spatialdecomposition method. Further, the spatial decomposition approach may have aslight advantage through a more localized memory access pattern. In this situation,a hybrid approach can lead to performance enhancements as it allows to interpolatebetween the cell-task method and spatial decomposition. This work focuses on the performance of the hybrid model on a single computenode. In Sect. 3, results from a series of benchmark simulations on two test machinesare presented. The results show that the hybrid method enables performance gainscompared to the pure spatial decomposition or cell-task approaches.
2 Benchmark Procedures
The parallelization methods are tested using a system with multi-core processorsand a separate system with an Intel Xeon Phi co-processor. The multi-core computenodes contain two Intel Xeon E5-2680 processors, each with eight cores using theAVX instruction set architecture. The Intel Xeon Phi co-processors utilized are the5110P model, with 60 physical cores, using hardware based threading for a total MD Simulations Using Spatial Decomposition and Task-Based Parallelism 135
of 240 threads (details on the Xeon Phi many-core architecture can be found inRef. [4]). For the purposes of this work, the measurement being utilized is the parallelspeedup factor. This represents the number of times faster than the baseline run thecurrent run is, and is measure as the baseline time divided by the current run time. On both systems, a variety of combinations of threads and MPI ranks are used, inorder to gauge and compare performances. Speedups are measured against baselineruns using a single MPI rank and a single thread. In addition to the simulationsusing the hybrid method, all MPI runs were carried out using a single thread perMPI rank, and all threaded runs on the multi-core system using a single MPI rank.On the Xeon Phi, all threaded runs were based upon a recompiled version of thecode without the MPI overhead, including its baseline run, as the speedups on thissystem are affected by the MPI overhead; conversely, this overhead does not effectthe speedups on the multi-core system. On the multi-core systems, tests were thendone using one, two, four, eight and sixteen MPI ranks, and varying numbers ofthreads to the total number of cores available per system. On the Xeon Phi co-processor, tests employed one, two, four, eight, sixteen, thirty, sixty, one-hundredtwenty, and two-hundred forty MPI ranks, with varying numbers of threads. All simulation runs, including the baseline, used vectorized implementations ofthe potentials [9] for better performance. All tests were run for 1000 time steps (2femtoseconds per step), with the Verlet-lists (neighbour-lists) regenerated every 10time steps. Each set of tests were run five times, taking the average time of each, inorder to more accurately measure the time required to perform the simulation on agiven system. A number of simulation systems are used to ensure results extend beyond a singlesystem. A bulk copper system with approximately 4.5-million particles, Cu (bulk),is used for its hom*ogeneity, which works well with spatial decomposition. A porouscopper system with two-million particles, Cu (porous), is used as it highlightsthe advantages of the cell-task method. Both these systems use the tight-bindingpotential [2], which has moderate force calculations. Additionally, an iron systemwith four-million particles, Fe (bulk), using the Mendelev potential [5] is included,as it has more complex force calculations and is less sensitive to memory access.Lastly, a liquid silver system with four-million particles, Ag (liquid), using theLennard-Jones potential [1] is used due to its fast force calculations which makes ithighly sensitive to memory access speed.
3 Results
As an example for all model systems, Fig. 1 shows the Cu (bulk) system’s speedupsusing different numbers of processor cores on the multi-core system. For the hybridmethod, only the data obtained with two MPI ranks is shown. The speedups withfour and eight ranks are nearly identical to those obtained with two ranks and havebeen omitted to avoid crowding of the figure. 136 C.M. Mangiardi and R. Meyer
16 Threaded MPI Hybrid (2 MPI ranks)
12 Speedup S(p)
0 0 4 8 12 16 Processing Cores p
Fig. 1 Parallel speedups S in simulations of the Cu (bulk) system on the multi-core system as afunction of the number of processing cores used p. The dashed line indicates the ideal speedupS.p/ D p
Table 1 Best parallel MPI Threaded Hybridspeedup factors in speedup speedup speedupsimulations involving System (ranks) (threads) (ranks threads)different systems on the Cu (bulk) 13.1 (16) 11.2 (16) 13.5 (2 8)multi-core processor Cu (porous) 7.9 (16) 12.6 (16) 13.6 (2 8) Fe (bulk) 13.1 (16) 10.4 (16) 13.2 (2 8) Ag (liquid) 7.6 (16) 7.4 (16) 4.7 (8 2)
The best speedup factors, together with the number of ranks or threads where theywere obtained, are shown in Table 1 for all test systems on the multi-core machine.Data is shown for spatial decomposition method (MPI), the cell-task approach(threaded) and the hybrid method utilizing the optimal combination of ranks andthreads. Figure 2 shows the Cu (bulk) system’s speedups on the Intel Xeon Phi system. Toavoid cluttering, only data for the cases of eight and thirty MPI ranks are shown forthe hybrid method. The best results for the hybrid method were attained using thirtyranks, however, as seen with the eight ranks, the behaviour of the speedups are notmonotonous, but instead are dependent upon the combination of ranks and threads.The best speedup factors with the corresponding numbers of ranks and threads aresummarized for all test systems in Table 2. From Fig. 1 and Table 1 it can be seen that when utilizing the tight-bindingpotentials on the multi-core system, the hybrid method produces better speedupsthan either the spatial decomposition or cell-task methods alone. For the hom*o- MD Simulations Using Spatial Decomposition and Task-Based Parallelism 137
120 Threaded MPI Hybrid (8 MPI ranks) Hybrid (30 MPI ranks)
90 Speedup S(p)
60
30
0 0 60 120 180 240 Processing Cores p
Fig. 2 Parallel speedups S in simulations of the Cu (bulk) system on the Intel Xeon Phi system asa function of the number of processing cores used p
Table 2 Best parallel MPI Threaded Hybridspeedup factors in speedup speedup speedupsimulations involving System (ranks) (threads) (ranks threads)different systems on the Intel Cu (bulk) 98.8 (240) 117.7 (230) 116.1 (30 8)Xeon Phi co-processor Cu (porous) 22.8 (240) 86.6 (185) 71.6 (4 40) Fe (bulk) 100.6 (240) 124.1 (240) 118.5 (16 15) Ag (liquid) 81.3 (240) 79.2 (240) 88.4 (120 2)
geneous system, the spatial decomposition method works well, allowing particlesto be evenly divided amongst the different cores. The task-based method similarlyworks well, but shows a slower performance then the other two methods for thistest system. This is in contrast to previous results [7, 8] where the performance ofspatial decomposition and the cell-task method was very similar for hom*ogeneoussystems. A likely reason for this discrepancy is the larger size of the test systems inthis work. Since the generation of the task schedule for the cell-task method does notparallelize well, and the number of tasks grows with the system size, larger systemsmight shift the balance towards spatial decomposition. As shown by Fig. 2 and Table 2, the behaviour is different on the Xeon Phi co-processor where the cell-task method and the hybrid approach outperform spatialdecomposition for the Cu (bulk) system. This is in agreement with the design of thisarchitecture which favours threaded programs and is not optimal for large numbersof MPI ranks [4]. When utilizing the Cu (porous) system, the speedups of the hybrid method aremore noticeable as compared to the spatial decomposition method. Due to the 138 C.M. Mangiardi and R. Meyer
system being inhom*ogeneous, the spatial decomposition method does not evenlydivide the system’s particles amongst the processor cores. This allows for boththe threaded and hybrid approaches to significantly outperform this method. Whenutilizing the Intel Xeon Phi, the system is divided into 240 areas with the spatialdecomposition method, resulting in a large portion of the co-processor cores beingidle, waiting on other cores to finish their work. This is significantly alleviated bythe hybrid method, which divides the system into less areas, and uses the task-based approach to reduce the number of idle cores. The cell-task method, however,outperforms the hybrid method on the Xeon Phi, as it does not split the systeminto any areas, which allows it to better allocate cores to reduce the effects of theinhom*ogeneity. For the Fe (bulk) system both the hybrid and cell-task methods outperform thespatial decomposition method. Despite the spatial decomposition method having asmaller portion the system to work on for each area, as compared to the hybridmethod, the overhead associated with MPI and the inter-process communicationcounteracts the speed improvements attained calculating the forces. This effect isreduced significantly in the hybrid method, as it is uses less MPI ranks thereforereducing the overhead of the spatial decomposition method. Further, the cell-taskmethod has larger speedups compared to the other methods for this system. This ismost likely due to the higher complexity of the force calculations for the Mendelevpotential. With more time spent per force calculation, the impact of limiting factorslike memory access time is reduced. The Ag (liquid) system performs significantly slower using the hybrid methodcompared to the other methods on the multi-core system. Due to this being aliquid system, particles are able to more freely move about the system, and as aresult, there is a high amount of inter-process communication required. Further,the overhead of creating the schedule utilizing less threads further degrades theperformance of the hybrid method. The Lennard-Jones potential is also simpler thanother potentials, reducing the amount of time spent in the calculations of forces, andinstead emphasizes other sections of the program, such as the Verlet-list generation,scheduler, and inter-process communication. On the Xeon Phi system, however, thehybrid method performs significantly better than the other two methods on the Ag(liquid) system. The different behaviour of this test system on the two machines isnot fully understood and requires more studying.
4 Conclusions
The results of the benchmark simulations shown in the previous section demonstratethat the hybrid method utilizing both spatial decomposition and task-based paral-lelism, is able to provide better performance on molecular-dynamics simulationsthan either method alone. The hybrid method takes advantage of each method’sbenefits, whilst minimizing their drawbacks. MD Simulations Using Spatial Decomposition and Task-Based Parallelism 139
For hom*ogeneous systems spatial decomposition works very well and oftendelivers, in particular on multi-core systems, a better performance than the cell-task method. The Intel Xeon Phi hardware and system software, however, favoursthreaded programming and is not designed to work optimally with 240 MPIranks. For simulations of hom*ogeneous systems, the hybrid model provides themeans to optimize the performance by reducing the number of threads and MPIcommunications. By its design, the cell-task method outperforms spatial decomposition forinhom*ogeneous systems since it avoids the load balancing problems that occurwith the spatial decomposition method when the number of particles varies betweendomains. In principle, the hybrid method suffers from the same issue, however sincethe hybrid methods operates with larger domains, inhom*ogeneities may averageout. By reducing the task management overhead of the cell-task method, the hybridapproach can therefore in certain cases still improve the total performance. An advantage of the hybrid parallelization method described in this work is thatit makes the simulation program adaptable. By choosing the combination of MPIranks and threads, the speed of the simulation can be optimized for the specific typeof the simulated system and the computer system on which the simulation is run. Overall, the performance gains of the hybrid method are much more pronouncedon the Xeon Phi co-processor than they are on the multi-core system. Thisemphasizes that novel many-core architectures require the development of newalgorithms to maximize the performance.
Acknowledgements This work has been made possible by generous allocation of computer timeon computer systems managed by Calcul Québec, the Shared Hierarchical Academic ResearchNetwork (SHARCNET) and Compute/Calcul Canada. Financial support by Laurentian Universityand the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefullyacknowledged.
References
1. Allen, M.P., Tildesley, D.J.: Computer Simulations of Liquids. Clarendon, Oxford (1987) 2. Cleri, F., Rosato, V.: Tight-binding potentials for transition metals and alloys. Phys. Rev. B 48(1), 22–33 (1993) 3. Frenkel, D., Smit, B.: Understanding Molecular Simulation. Academic, San Diego (2002) 4. Jeffers, J., Reinders, J.: Intel Xeon Phi Coprocessor High Performance Programming. Morgan Kaufman, New York (2013) 5. Mendelev, M.I., Han, S., Srolovitz, D.J., Ackland, G.J., Sun, D.Y., Asta, M.: Development of new interatomic potentials appropriate for crystalline and liquid iron. Philos. Mag. 83(35), 3977–3994 (2003) 6. Message Passing Interface Forum: http://www.mpi-forum.org/ (2016) 7. Meyer, R.: Efficient parallelization of short-range molecular dynamics simulations on many- core systems. Phys. Rev. E 88(5), 053,309 (2013) 8. Meyer, R.: Efficient parallelization of molecular dynamics simulations with short-ranged forces. J. Phys.: Conf. Ser. 540(1), 012,006 (2014) 140 C.M. Mangiardi and R. Meyer
9. Meyer, R., Mangiardi, C.M.: Parallelization of molecular-dynamics simulations using tasks. MRS Proc. 1753, mrsf14–1753–nn10–09 (2015)10. Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117(1), 1–19 (1995)11. TBB official website: http://threadingbuildingblocks.org/ (2016) Modelling of Local Length-Scale Dynamicsand Isotropizing Deformations: Formulationin Natural Coordinate System
O. Pannekoucke, E. Emili, and O. Thual
Abstract We propose an algorithm to model anisotropic correlation functions usingan approach based on the deformation of locally isotropic ones. In a previous work,a set of equations that allow to calculate the desired deformation was derived for aflat coordinates system. However this strategy is not adapted for curved geometryas the sphere (regional and global atmospheric models), where it is suitable to statethe local isotropy in terms of the local Riemannian metric. This paper introduces the theoretical background to deal explicitly with naturalcoordinate systems leading to a formulation adapted with the Riemannian metric. Itresults that the isotropizing deformation is obtained from the resolution of couplednon-linear equations depending on the geometry. This procedure is illustrated withina 2D setting.
1 Introduction
In variational data assimilation, the analysis state minimizes the cost function
J.X / D .X X b /T B1 .X X b / C .Y o HX /R1 .Y o HX /; (1) Twhere X 2 Rn is the state vector, Y o 2 Rp the observational vector, B D EŒ"b "b denotes the covariance matrix of the background error "b D X b X t , X b (X t )
O. Pannekoucke ()CERFACS/CNRS URA 1875, Toulouse, FranceCNRM/GAME, Météo-France/CNRS UMR 3589, Toulouse, FranceINPT-ENM, Toulouse, Francee-mail: [emailprotected]. EmiliCERFACS/CNRS URA 1875, Toulouse, FranceO. ThualCERFACS/CNRS URA 1875, Toulouse, FranceUniversité de Toulouse; INPT, CNRS; IMFT; F-31400 Toulouse, France
© Springer International Publishing Switzerland 2016 141J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_14 142 O. Pannekoucke et al.
is the background state (the true state), R D EŒ"o "oT denotes the covariance matrixof the observational error "o D Y o HX t , and H is the observation operator(assumed linear here) that maps the model state to observation locations. For largedimensional problems, like meteorological applications, the covariance matrix Bcannot be explicitly represented and is often modelled. Preliminary works have shown that it is possible to estimate and to usethe correlation length-scale, to produce a coordinate change that facilitates themodelling of the anisotropic covariance matrix B. The approach, developed in [1],relies on the local metric tensor gx defined from the Taylor expansion of a correlationfunction .x; x C ıx/ D 1 12 jjıxjj2gx C O.jıxj3 /, with jjxjj2E D xT Ex (thisformalism can be extended in the particular case where the gradient of is notzero at x D y). This local definition for gx serves to define a metric field g over adomain (thereafter the two notations Gx and G.x/ are used to denote the value of aquantity G at point x). In particular, isotropic correlation functions in Rn takes theform iso .u1 ; u2 / D iso .jju1 u2 jjRn /, where jj jjRn denotes the Euclidian normand where .u1 ; u2 / are two points in Rn . Considering a Riemannian manifold Membedded in Rn and denoting by ! the metric induced by the Euclidian metric ofRn , then the restriction of iso to the manifold defines correlation functions on M[2]. The local behavior of these functions is given by .x; x C ıx/ D iso jjıxjj!x .Then, the local metric tensor gx of at point x is often expressed in terms of theRiemannian metric !x as gx D L12 !x , where Lh is the constant length-scale, as h r2 2L2hencountered for, e.g. iso .r/ D e . By analogy with correlation modelling on thesphere, we say that the correlation functions on M are isotropic. As an example of Riemann manifold, we can consider the sphere of radiusa, parametrized thanks to the longitude/co-latitude coordinate system x D .; /and equipped with the natural metric ! defined from the square arc-length ds2 D!ij dxi dxj (in Einstein’s summation convention) with ds2 D a2 .sin /2 d 2 C a2 d 2 . The local metric tensor is a very attractive quantity. It can be diagnosed fromensemble estimation [3], associated with a filtering step to damp spurious samplingnoise [4]. This local metric is often associated with the local diffusion tensor x [5],defined by ij D 12 gij where gij denotes the inverse tensor of g, i.e. gij gjl D ıil whereıij denotes the Kronecker symbol. In data assimilation it is usual to find anisotropic correlation functions whichrequire sophisticated covariance models, e.g. the wavelets formulation [6], therecursive filter formulation [7] the diffusion equation formulation [8, 9] or thecoordinate change of isotropic correlations [10]. This coordinate change is a trickyway to reduce the numerical cost or facilitate the parallelization of the algorithm[1]: ones the computation of the isotropizing coordinate change has been done,anisotropic correlations simply result from low cost interpolations between the twocoordinate systems where efficient algorithms exist. For example a locally isotropiccorrelation field can be efficiently obtained by alternate applications of diffusionequation method, parallelized in the space dimension. Thereafter the application ofthe coordinate change consists in introducing a coordinate change, able to transform Modelling of Local Length-Scale Dynamics and Isotropizing Deformations:. . . 143
isotropic correlation into the desired anisotropic correlation. The coordinate changecan be estimated from ensemble method, e.g. [11] has proposed a procedure thatrelies on the wavelet estimation of the gradient of the deformation [12]. Legrandand Michel [13] illustrated the feasibility of the isotropization procedure for realdata, and the potential for data assimilation. Until now, the general framework thatleads to a local isotropy for the curved spaces has not been addressed. In [1] the coordinate change is obtained from the local metric tensor g.x/: itconsists in finding a differential map x.Qx/ that transforms a coordinate system xQ ,where the local metric tensor gQ .Qx/ is isotropic, into a coordinate system, where thelocal metric g.x/ is the anisotropic diagnosed one, and respecting the chain rule
gQ ˛ˇ D @xQ˛ xi @xQˇ xj gij : (2)
In particular, it has been shown possible to construct a coordinate change leadingto a local isotropic tensor gQ ˛ˇ .Qx/ D L21.Qx/ ı˛ˇ .Qx/. This coordinate change is found intwo steps:(a) first one has to find x.Qx/, solution of the coupled non-linear Poisson’s like equations
xi C jki @xQ˛ xj @xQ˛ xk D 0; (3)
where is the Laplacian operator, and jki D gil jkl denotes the Christoffel symbols of the second kind associated with the metric g defined from the Christoffel symbols of the first kind ijk D .@i gkj C @j gkj @k gij /=2;(b) in a second step, the appropriate coordinate change xQ .x/ is obtained as the inverse of the differential map x.Qx/. For general manifold, the metric ıij is not the one found in natural coordinates.As mentioned above, on a part of the sphere, the metric ıij must be replaced by thenatural metric !ij and a global isotropic correlation function is locally expressed interms of !ij so that gQ ij .Qx/ D L12 !ij .Qx/ where Lh is a constant length-scale. The local hisotropic version is thus gQ ij .Qx/ D L21.Qx/ !ij .Qx/, where L.Qx/ is a length-scale field. Itfollows that the metric !ij has to be taken into account in the isotropizing process. The aim of the present contribution is to extend the results from [1], by takinginto account natural coordinate systems. We first introduce a geometrical frameworkfor the isotropization issue in Sect. 2, where the isotropizing deformation appears asan harmonic map, solution of a system of non-linear partial differential equations.The isotropization procedure proposed is then tested into a simplified 2D setting inSect. 3. The conclusions are reported in Sect. 4. 144 O. Pannekoucke et al.
2 General Formalism for the Isotropization Procedure
In this section we describe the theoretical background for constructing the isotropiz-ing coordinate transform. This relies on Riemannian geometry [14, 15] and inparticular on the properties of harmonic maps [16].
2.1 Coordinate Change Equation Set
The geographical domain considered here is assumed to be represented by a firstRiemannian manifold M of dimension d and equipped with the metric !, it isdenoted by .M; !/. The diagnostic of the local metric tensor field g endows Mwith a second Riemannian structure, .M; g/. A third Riemannian structure .M; gQ /is considered, where the metric gQ takes the form gQ D L12 ! with L is an unknownlength-scale field. The isotropizing coordinate change takes the form of a differentialmap x.Qx/ from .M; gQ / to .M; g/ so that gQ and g are related according to Eq. (2). Froma theoretical point of view, x.Qx/ can be considered as an isometric diffeomorphismfrom .M; Q gQ / to .M; g/, where M Q is isomorphic to M. Note that when considering adifferential map f from M Q to .M; g/, it is possible to define a third metric f g on MQby . f g/˛ˇ D @xQ˛ f i @xQˇ f j gij . Then, Eq. (2) also reads gQ D x g. It can be shown from transformation rules of the Christoffel symbols of the second kind [15] Q˛ˇ D @xQ˛ xj @xQˇ xk @xi xQ jki C @2xQ˛ Qxˇ xi @xi xQ , that the coordinatefunctions xi are solutions the system of coupled non-linear equations
gQ xi C jki @xQ˛ xj @xQˇ xk gQ ˛ˇ D 0; (4)
where gQ denotes the Beltrami’s Laplacian with jki (respectively Qjki ) are theChristoffel symbols of the second kind associated with the metric field g (respec-tively gQ ). Hence, Eqs. (4) provides a necessary condition verified by the isotropizingcoordinate change, and can be used to find the unknown coordinate change.However multiple solutions can exist, which means that several length-scale fieldscould exist for a given metric g. A major difficulty is that these equations directlymake use of the unknown metric gQ . To better clarify these two statements we introduce a functional which is relatedto the energy of the deformation x.Qx/. A differential map, solution of Eq. (4), is an harmonic map [14, 16], that is astationary point (critical point), for variations R of the differential mappx.Qx/, of theenergy functional [17], EŒx.Qx/; gQ ; g D MQ 21 jjdxjj2 d M, Q where dM Q D gQ dQx1 dQxdis the invariant volume element of M Q and jj jj denotes the Hilbert-Schmidt norm.This implies that the coordinate notation for the energy density exQ Œx.Qx/; gQ ; g D1 2 x/; gQ ; g D 12 gQ ˛ˇ @xQ˛ xi @xQˇ xj gij . In particular, Eq. (4) are the Euler-2 jjdxjj reads exQ Œx.Q Modelling of Local Length-Scale Dynamics and Isotropizing Deformations:. . . 145
Lagrange of the energy E for variation of the differential map x.Qx/ [14]. Note thatEŒx.Qx/; gQ ; g is a quadratic functional but since it is not convex in x it may havemultiple critical points.
2.2 Conformal Equivalence Classes
For covariance modelling in data assimilation, a simplified decomposition of linearoperators can be introduced [6, 8, 18, 19]. For instance the background errorcovariance matrix B as specified in the non-separable spectral decomposition 1=2takes the form B D ˙ S1 Dh Ehv Dh ST ˙ T where linear operators are as T=2
follows: Ehv encodes the horizontal-vertical correlations, Dh encodes the horizontalcorrelations, S is the grid point to spectral transform operator and ˙ encodesthe grid point standard-deviation [18]. The separable formulation based on thediffusion equation [8, 9, 20] takes the form B D ˙ S1 Lh Lv W1 LTv LTh ˙ T wherethe linear operator W is the metric tensor, Lh and Lv encode the horizontal and thevertical diffusion time integration with the diffusion tensor deduced from objectiveestimation [3, 9, 20]. Note that, [21] has proposed a non-separable formulation of thediffusion formulation that relies on the wavelets. It results, from the linear operatordecomposition described above, that the horizontal correlation can be treated with alinear operator that is 2D for a given level. A particular property of the 2D case is that the energy functional is conformalyinvariant in gQ , i.e. it remains constant when replacing gQ by L.Q1x/2 !, i.e. EŒx; gQ ; g DEŒx; !; g (Weyl transformation and invariance, see [15]). Thanks to this invariance,the unknown length-scale field L can be eliminated by solving the problem withinthe conformal equivalent class of !. It results that, for two dimensional problems,one can replace Eq. (4) by equations
! xi C jki @xQ˛ xj @xQˇ xk ! ˛ˇ D 0; (5)
this can also be directly obtained from Eq. (4) when replacing gQ by L2 ! in 2D. Inthe particular case where !ij D ıij , Eq. (5) leads to Eq. (3) as previously found in[1]. The unknown metric gQ can be explicitly deduced from the solution x.Qx/ of Eq. (5)thanks to the metric change Eq. (2) that reads gQ D x g, and then provides the length- 1 1 1=2scale field L.Qx/ D 2 Trace x gxQ !xQ . We are now able to describe the isotropizing procedure. 146 O. Pannekoucke et al.
2.3 Isotropizing Procedure
The two step isotropizing procedure, detailed in Algorithm 1, is similar to the onedescribed in [1], using a finite difference scheme for spatial derivative, a forwardEuler time scheme, and a spline bi-cubic interpolation. First it consists in computingthe inverse isotropization transformation, obtained by solving Eq. (5). For thatpurpose the pseudo-time diffusion scheme (the heat flow) [16]
@ xi D ! xi C jki @xQ˛ xj @xQˇ xk ! ˛ˇ ; (6)
can be employed to find the stationary state, solution of Eq. (5), where the initialcondition is the identity map id.Qx/ D xQ . This defines a family of differentialmap x indexed by 2 Œ0; 1/, continuously dependent of . For diffeomorphiccompact manifold without boundary, if the algorithm converges toward a stationarydifferential solution, then this solution is an harmonic map, and this harmonic mapis continuously obtained from the initial condition as the limit lim !1 x [22].Note that the dynamics Eq. (6) is the gradient flow associated with the energy. As /a consequence, the tendency must be non-positive, dE.x d 0, so that the energyEŒx .Qx/; !; g is minimized along the path x .
Algorithm 1 Two-steps isotropizing procedure in curved spaceRequire: The metric fieldg is assumed to be known (e.g. estimated from an ensemble) 1: Compute ijk .x/ D 12 @j gik .x/ C @k gji .x/ @i gjk .x/ 2: Compute jki .x/ D gil .x/ljk .x/ 3: Set q1 D 500 and k1 D 2 4: Compute jki .x/ D gil .x/ljk .x/
5: # Step 1: Pseudo-diffusion 6: xi1 .Qx/ D xQi 7: for q from 1 to q1 do 8: for i from 1 to d do 9: jki .Qx/ D jki Œx1 .Qx/ (Spline interpolation) xi2 .Qx/ D xi1 .Qx/ C d ! xi1 C jki @Qx˛ x1 @Qxˇ xk1 ! ˛ˇ .Qx/ j10:11: end for12: xi1 .Qx/ D xi2 .Qx/13: end for14: x.Qx/ D x1 .Qx/
15: # Step 2: Inverse transform (fixed-point’s iterations)16: ı1 .Qx/ D 017: for k from 1 to k1 do18: ı2 .Qx/ D 12 fxŒQx ı1 .Qx/ xQg19: ı1 .Qx/ D ı2 .Qx/20: end for21: xQ .x/ D x 2ı1 .Qx/ Modelling of Local Length-Scale Dynamics and Isotropizing Deformations:. . . 147
The isotropizing coordinate change xQ .x/ is obtained as the inverse of the ıdifferential map x.Qx/. The differential map xQ .x/ is denoted D. The isotropization procedure is illustrated within a 2D setting in the next section.
3 Numerical Experiments
Anisotropic correlations are numerically constructed in 2D from globally isotropicones with a deformation derived from an arbitrary stream function. This construc-tion will be used to test the skill of local isotropization methods. The 2D settingis specified in order to mimic a portion of the sphere, which is often the case ofhorizontal coordinates employed in regional models.
3.1 Experimental Set-Up
To validate the isotropizing process, the following route is considered: a Riemannianmanifold .M; !/ is first introduced, that mimics the situation encountered inatmospheric modelling where the local metric ! is not flat ; a local isotropic metricgQ , with respect to the metric !, is then produced from a given length-scale field L ;a deformation D is constructed and used to generate an anisotropic metric g D D gQ(the pushforward metric of gQ by D). Using the isotropizing process, the challengeis to find from g and ! – the only information available in real applications – acoordinate change that transforms g into a local isotropic metric with respect to !. As a first step,a bi-periodic domain M is considered, parametrized with a singlebi-periodic chart U D Œ0; 1 Œ0; 1 ; x D .x; y/ discretized in n D 81 points alongthe two directions x and y. This coordinate system is similar to an angular chart onthe sphere, and is equipped with the metric !ij defined from the arc-length square ds D R ˛ C .1 ˛/.sin y/ dx2 C R2 dy2 ; 2 2 2
with ˛ D 0:2 and R D 1000 km, such that the resolution at x D .0:5; 0:5/ isisotropic with ds 12 km, the length of the domain along x for y Dp0:5 is equal toR D 1000 km, while the length for y D 0 or for y D 1 is equal to R ˛ 447 km.The length along y does not depends on the position x and is equal to R. A deformation D.x/ D x C d.x/ is constructed as follows. First, a wind fieldu0 .x/ D k r is introduced, where is a stream function, and it is normalizedso that u D .u; v/ D .u0 =max.ju0 j/; v0 =max.jv0 j//. Then, for each position x, thegeodesic curve x .t/ is computed, starting at x with the velocity Px .0/ D u.x/dtwhere dt D 0:05 is a magnitude factor. Denoting by V.t/ the velocity Px .t/, 148 O. Pannekoucke et al.
Fig. 1 Stream function (shading) and displacement field resulting from the geodesic timeintegration with initial velocity deduced from the stream current. The arrows indicate the directionand the intensity of the associated displacement
the geodesic curve is obtained as the time integration of the geodesic system n i i o dx dt D V ; dt D ˝jk V V , with initial condition x .0/ D x; V.0/ D u.x/dt i dV i j k
and where ˝jki are the Christoffel symbols of the second kind associated with themetric field !. The displacement field is then defined as d.x/ D x .1/ x. The stream function used for numerical experiment is shown in Fig. 1, with thedisplacement field deduced from the velocity field.
3.2 Anisotropic Correlations and Isotropizing Procedure
In a data assimilation framework, the metric g is computed from ensemble esti-mation, and it can be used within the isotropizing procedure we propose. In thiswork, a simulated experiment is designed to validate the isotropizing procedure anda synthetic anisotropic metric field is generated from the deformation of a locallyisotropic metric field. For this purpose, locally isotropic correlation functions are considered, specifiedby their local metrics gQ .Qx/ D L.Q1x/2 !.Qx/ where the varying length-scale field Lhas been arbitrarily fixed, with average value Lh D 50 km. The diffusion tensor Modelling of Local Length-Scale Dynamics and Isotropizing Deformations:. . . 149
Fig. 2 The initial locally isotropic metric, represented on a regular coordinate system (a), isdeformed under the action of the deformation D leading to anisotropic metric field (b). Then,the anisotropic metric field represented within the natural coordinate system (c) is diagnosed and ıemployed in the isotropizing process which provides the isotropizing deformation D (correspond- ıing theoretically to D1 ). The action of D on the anisotropic metric field is the nearly locallyisotropic metric field (d). The coordinate systems are reproduced in gray
associated with gQ is illustrated in Fig. 2a The coordinate change D is then appliedon these functions to obtain anisotropic correlations functions diagnosed by theirdiffusion tensor, Fig. 2b. The isotropization procedure is considered on the metric g (diagnosed within anatural coordinate system) as illustrated in Fig. 2c. In a first step, the Christoffelsymbols of second kind jki are computed from g. Then time integration of Eq. (6),with initial condition x D xQ and time step ı D ıx2 =4 (see [1] for the details), isachieved to obtain a solution of Eq. (5). In a second step, the inverse transformationis constructed following the fixed-point procedure ı D dŒx Q ı (valid in curvedgeometry), computed similarily to a second-order semi-Lagrangian scheme: the 150 O. Pannekoucke et al.
fixed-point problem ı D 12 dŒxı Q is iteratively first solved, then the displacement field is deduced as as ı D 2ı (see Algorithm 1). ı The diffusion tensor of the isotropization of the metric g by D is given in Fig. 2d,and appears close to the theoretical one in Fig. 2a. Hence, in this experiment, wehave shown possible to find a coordinate system in which the isotropizing metric isconformal to the natural metric !.
4 Conclusion
We have extended the preliminary work of [1] within a geometrical backgroundthat offers a new insight on the isotropizing issue. Here, we have shown how theformalism of the isotropization should be stated to incorporate the natural metric ona geographical domain, as those encountered in atmospheric data assimilation. In particular, we have shown that, in 2D, the diagnosis of the length-scale (i.e.the local metric tensor) can be used to find a coordinate change, solution of apseudo-diffusion scheme, that provides isotropic local metric. The pseudo-diffusionscheme used to solve the non-linear system of coupled Poisson’s equations, can beconsidered for real application since it relies on classical numerical componentsof Numerical Weather Prediction Systems or Chemical Transport Model (semi-Lagrangian scheme, first-order derivative and Beltrami’s Laplacian). Further work is to propose and to test more efficient algorithms than the pseudo-diffusion scheme.
Acknowledgements OP would like thanks Joseph Tapia, Jean-Pierre Otal and Marina Ville, JohnHarlim, Tyrus Berry and Dimitrios Giannakis for interesting discussions ; This work was supportedby the French LEFE INSU program and the MACC2 project within the FP7 E.U. reasearchprogram.
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7. Purser, R., Wu, W.S., Parrish, D., Roberts, N.: Numerical aspects of the application of recursive filters to variational statistical analysis. Part I: Spatially hom*ogeneous and isotropic Gaussian covariances. Mon. Weather Rev. 131, 1524 (2003) 8. Weaver, A., Courtier, P.: Correlation modelling on the sphere using a generalized diffusion equation (Tech. Memo. ECMWF, num. 306). Q. J. R. Meteorol. Soc. 127, 1815 (2001) 9. Weaver, A.T., Mirouze, I.: On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation. Q. J. R. Meteorol. Soc. 139(670), 242 (2013)10. Desroziers, G.: A coordinate change for data assimilation in spherical geometry of frontal structures. Mon. Weather Rev. 125, 3030 (1997)11. Michel, Y.: Estimating deformations of random processes for correlation modelling in a limited area model. Q. J. R. Meteorol. Soc. 139, 534 (2013)12. Clerc, M., Mallat, S.: The texture gradient equation for recovering shape from texture. IEEE Trans. Pattern Anal. Mach. Intell. 24, 536 (2002)13. Legrand, R., Michel, Y.: Modelling background error correlations with spatial deformations: a case study. Tellus 66, 23984 (2014)14. Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2005)15. Nakahara, M.: Geometry, Topology and Physics, 2nd edn. Taylor & Francis, New York (2003)16. Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109 (1964)17. Eells, J., Lemaire, L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1 (1978)18. Courtier, P., Andersson, E., Heckley, W., Pailleux, J., Vasiljević, D., Hamrud, M., Hollingsworth, A., Rabier, F., Fisher, M.: The ECMWF implementation of three-dimensional variational assimilation (3D-Var). I: formulation. Q. J. R. Meteorol. Soc. 124, 1783 (1998)19. Derber, J., Bouttier, F.: A reformulation of the background error covariance in the ECMWF global data assimilation system. Tellus A 51, 195 (1999)20. Massart, S., Piacentini, A., Pannekoucke, O.: Importance of using ensemble estimated back- ground error covariances for the quality of atmospheric ozone analyses. Q. J. R. Meteorol. Soc. 138, 889 (2012)21. Pannekoucke, O.: Heterogeneous correlation modeling based on the wavelet diagonal assump- tion and on the diffusion operator. Mon. Weather Rev. 137, 2995 (2009)22. Jost, J.: Harmonic mapping between Riemannian manifolds. In: Proceedings of the Centre for Mathematical Analysis, Australian National University (1983) Post-Newtonian Gravitation
Erik I. Verriest
Abstract Einstein’s field equations relate space-time geometry to matter andenergy distribution. These tensorial equations are so unwieldy that solutions are onlyknown in some very specific cases. A semi-relativistic approximation is desirable:One where space-time may still be considered as flat, but where Newton’s equations(where gravity acts instantaneously) are replaced by a post-Newtonian theory,involving propagation of gravity at the speed of light. As this retardation dependson the geometry of the point masses, a dynamical system with state dependentdelay results, where delay and state are implicitly related. We investigate severalcases with Lagrange’s inversion technique and perturbation expansions. Interestingphenomena (entrainment, dynamic friction, fission and orbital speeds) emerge.
1 Introduction
A post-Newtonian (pN) gravitation (see [7]) is developed, with the main assumptionthat gravity cannot act instantaneously at a distance, but has an interaction speedthat is limited by the speed of light, otherwise one would violate relativity [4].Section 2 establishes the formulas for the retarded potentials in analogy to theLiénard-Wiechert potentials in electromagnetic theory. From it, the retarded field,which determines the motion, is obtained by the gradient of the potential in Sect. 3.This results in a functional differential equation (FDE) with state dependent delay(See [3]). In particular, interesting new phenomena (entrainment, dynamic friction,orbital speeds) emerge from this pN theory. In Sect. 4 we discuss the dynamics fortwo masses flying apart in a rectilinear way, a (1-D) toy model for a supernova. Themain problem is the implicit relation between state and delay occurring in the FDEdescribing the motion. We discuss several cases where the Lagrange’s inversiontechnique [5] can be applied to render such a relation explicit [6].
E.I. Verriest ()Georgia Institute of Technology, Atlanta, GA 30306, USAe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 153J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_15 154 E.I. Verriest
2 Retarded Gravitational Potential
Let at time t0 the mass distribution be represented by the scalar function .r0 ; t0 /,the density field. This assumption requires already a notion of simultaneity andtherefore cannot be consistent with the theory of relativity. We postulate that thisdensity field generates a gravitational field, which assumes at a fixed position,r, and time t the value due to the superposition of the fields of all infinitesimalcontributions. However the contribution from r0 must have traveled a (straightline) distance jr r0 j, which takes a propagation time .r0 / D jr r0 j=c. Henceonly the past value, .r0 ; t .r0 //, contributes to the field at r at time t. Again,general relativity implies the warping of space and time, and hence geodesics (alongwhich signals propagate) are not straight lines in general. The following will beapproximately valid for weak fields and small speeds relative to c, the speed oflight.Assumption Given the time varying mass density .r0 ; t0 /, the gravitational poten-tial at a fixed point r and time t is given by the linear superposition Z .r0 ; t .r0 // .r; t/ D G dV.r0 /; (1) R3 jr r0 j
where G is the gravitational constant, and V.r0 / denotes the volume element atposition r0 . This equation is coupled with an implicit delay-state relation
c .r0 / D jr r0 j: (2)
If is continuous in its arguments, the field at a point r where .r; / ¤ 0 can beobtained by taking the integral over an -small ball around r away, since that partwill not contribute to the field by virtue of the asymptotic isotropy. Let us now assume that the mass distribution is due to a unit point mass in motion,prescribed by its trajectory, rp .t/. This means that the mass density is given by athree-dimensional Dirac delta .r0 ; t0 / D ı.r0 rp .t0 //. The gravitational potentialat the fixed point and time .r; t/ is then Z ı.r0 rp .tb // .r; t/ D G dV.r0 /; (3) R3 jr r0 j
where tb D t .r0 / is the retarded time satisfying (2). We evaluate this integral byfirst principles. First, a Dirac delta peaks where its argument equals zero. However,this information is not sufficient to describe this behavior in its entirety. The rate atwhich the zero is approached is also required. To figure this out, let first r00 be the Post-Newtonian Gravitation 155
solution to r00 D rp .t .r00 // and set r0 D r00 C rQ 0 : Substituting these expressionsin the Dirac delta, we get (I stands for the identity matrix)
1 ı.r00 C r0 rp .t .r00 C rQ 0 /// D ı.Qr0 /: (4) j det ŒI C rP p .t .r00 //rr0 .r00 / j
Using the identity detŒI C pq D detŒI C qp , and substituting (4) in (3), yields: Z ı.Qr0 / .r; t/ D G dV.r0 /: R3 jr r0 j j det.1 C rr .r00 /Prp .t .r00 ///j
Taking the gradient w.r.t. r0 of (2) and evaluating at r0 D rp gives
.r rp .tb //> crr0 D ŒI ; jr rp .tb /j
so that finally
G .r; t/ D ˇ ˇ: (5) ˇ .rrp .tb //> rP p .tb / ˇ jr rp .tb /j ˇ1 cjrr p .tb /j ˇ
This result could also have been obtained using the standard trick to derivethe Liénard-Wiechert potentials in some physics texts, by reducing the threedimensional Dirac to a one-dimensional time Dirac (see [5]).
3 Gravitational Field Due to a Moving Point Mass
The gravitational field is the gradient of the gravitational potential, F.r; t/ Drr .r; t/, obtained in Sect. 2. Thus
G 1 F.r; t/ D ˇ ˇ rr jr rp .tb /j C ˇ .rrp .tb //> rP p .tb / ˇ jr rp .tb /j2 ˇ1 cjrrp .tb /j ˇ ˇ ˇ G 1 ˇ .r rp .tb //> rP p .tb / ˇˇ ˇ ˇ rˇ r ˇ 1 : (6) jr rp .tb /j ˇ .rrp .tb //> rPp .tb / ˇ2 cjr rp .tb /j ˇ ˇ1 cjrrp .tb /j ˇ
From rr jr r0 j2 D 2jr r0 j rr jr r0 j and noting that the left hand side is equal to
rr .r r0 /> .r r0 / D 2.r r0 /> rr .r r0 / D 2.r r0 /> ; „ ƒ‚ … DI 156 E.I. Verriest
it follows that
.r r0 /> rr jr r0 j D : jr r0 j
Note that the gradient with respect to a column vector is represented by a row vector.Similarly, if jv0 j < c ˇ ˇ ˇ .r r0 /> v0 ˇˇ .r r0 /> v0 v> r r0 ˇ rr ˇ1 D rr 1 D 0 rr : cjr r0 j ˇ cjr r0 j c jr r0 j
Also, r r0 1 .r r0 /.r r0 /> rr D I : jr r0 j jr r0 j jr r0 j2
The matrix between the square brackets is a projection operator (projection onto theplane perpendicular to the vector r r0 ). Putting this all together, the gravitationalfield due to a moving mass is
.r rp .tb //> G F.r; t/ D ˇ ˇC (7) jr rp .tb /j3 ˇˇ1 .rrp .tb //> rP p .tb / ˇ ˇ cjrrp .tb /j G 1 rP p .tb /> .r rp .tb //.r rp .tb //> C ˇ ˇ I : jr rp .tb /j2 ˇ .rrp .tb //> rPp .tb / ˇ2 c jr rp .tb /j2 ˇ1 cjrrp .tb /j ˇ
If the gravitating mass moves in such a way that rP p is parallel to .r rp .tb //, thenthe second term in (7) is zero and the field is directed along r rp .tb /. Aligning thex-axis with this direction, force, position and velocities can be expressed in the x-coordinate. In particular, the x-component of the force acting on a unit mass particleat x is
sgn.x xp .tb //G Fx D ˇ ˇ; (8) ˇ xP .t / ˇ jx xp .tb /j2 ˇ1 sgn.x xp .tb // p c b ˇ
where as always, tb denotes the retarded time. This form displays two deviations from Newton’s law: First the delayed positionof the gravitating particle appears, and second, there is an aberration which dependson the (delayed) velocity. Post-Newtonian Gravitation 157
It should also be observed that a naive generalization to Newton’s law in the form
sgn.x xp .tb //G Fx D ; jx xp .tb /j2
instead of (8) cannot be correct.
3.1 Expression the Field in Terms of the Predicted Position
First, reorganize (7) in the form
G 1 rPp .tb /> F.r; t/ D ˇ ˇ C jr rp .tb /j2 ˇ .rrp .tb //> rPp .tb / ˇ2 c ˇ1 jrrp .tb /j c ˇ
G.r rp .tb //> ˇ ˇ ˇ .rr .t //> rP .t / ˇ2 jr rp .tb /j3 ˇ1 jrrpp .tbb /j p c b ˇ ˇ ˇ ˇ .r rp .tb //> rPp .tb / ˇˇ .r rp .tb //> rPp .tb / ˇ ˇ1 C : (9) jr rp .tb /j c ˇ jr rp .tb /j c
Introducing rbp D rp .tb /C .tb /Prp .tb /, which is the expected position of the particle inmotion at time t, if the velocity of the mass were fixed at the constant vp .s/ D rP p .tb /for tb .t/ D t .tb / s t, it follows that ˇ ˇ ˇ > ˇ ˇ1 .r rp .tb // rP p .tb / ˇ D ˇ 1 ˇ j1> .r rbp /j; ˇ jr rp .tb /j c ˇ ˇr rp .tb /ˇ rrp
.rr .t //>where 1rrp D jrrpp .tb b /j denotes the unit vector in the direction of rrp . Relativityimposes jPrp j < c, which with the Cauchy-Schwarz inequality reduces (9) to
G > rr .r; t/ D ˇ ˇ2 .r rbp / : (10) ˇ > ˇ jr rp .tb /j ˇ1rrp .r rbp /ˇ
Thus the gravitational force exerted by a particle in motion is directed towards thepredicted position based on a uniform motion given the delayed information (i.e.delayed position and velocity). 158 E.I. Verriest
v0 x(t − τ ) x(t)
d cτ x
Fig. 1 Uniform motion
3.2 Field Due to Particle with Uniform Velocity
For a particle moving with uniform velocity, rbp .t/ is the position of the particle attime t, rbp .t/ D rp .t/. Without loss of generality, we fix r, the observation point, atthe origin, and let the particle move along a line parallel to the x-axis, at a distance dwith uniform velocity v0 . For notational simplicity, the x-coordinate at t D 0 is takento be zero. From the geometry of the problem (See Fig. 1): a quadratic equation for results: .c2 v02 / 2 C 2v02 t .v02 t2 C d 2 / D 0: The field magnitude follows
G..d2 C v02 t.t //2 C d2 v02 2 / FD q : (11) .d2 C v02 t.t //2 .d2 C v02 .t /2 /.d2 C v02 t2 /
In Fig. 2 the magnitude is shown as function of time for d D 1, v0 D 1 and c D1:1; 2 and the Newtonian case (c D 1). As the mass closes in on the origin fort < 0, the delayed effect is quite pronounced as the speed c is increased. Whend D 0, the delayed gravitational force at time t exceeds Newton’s law by a factor1 C vc0 if the particle is moving towards the origin, and a factor 1 vc0 when it ismoving away from the origin. The same holds for all d in the limits respectively fort ! 1 and t ! C1.
3.3 Gravitational Current and Its Entrainment
Let the origin coincide with the observation point, r, and consider an infinitely longstraight line mass moving along its axis, parallel to the x-axis, at a distance d withuniform velocity v0 . Thus the field at the origin due to a gravitational current isstudied. Consider first a mass-element at p position .x0 ; d/ at time t D 0. From thegeometryp of the problem (Fig. 1): c .t/ D .x0 C v0 .t .t///2 C d2 : Letting D 2 2 .x0 C v0 tb / C d ; the x-component of the field at the origin at time 0 due this Post-Newtonian Gravitation 159
Fig. 2 Field at the origin due to particle with uniform motion. The symmetric curve correspondsto the Newtonian case (c D 1)
elementary current element is .x0 Cv0 tb /2 x0 C v0 tb v0 1 2 Fx .x0 ; v0 / D 2 ; (12) 3 v0 .x0 Cv0 tb / c 1C c 2 1C v0 0 Cv0 tb / .x c
where tb is the backward time for t D 0, i.e., tb D .0/: Likewise, the y-component
d v0 d 1 Fy .x0 ; v0 / D 2 : (13) v0 .x0 Cv0 tb / c 3 1C c 4 1 C v0 .x0 Cv0 tb / c
Integrating over x0 from 1 to 1 gives the total gravitational force of the masscurrent. The Figs. 3 and 4 show for d D 1 respectively the x and y components ofthe field at the origin as function of ˇ D v=c. In the static case (v0 D 0) this is Z 1 Gd 2G Fx .0/ D 0; Fy .0/ D dx D 2 : 1 .x2 C d2 /3=2 d
This static field is directed perpendicular to the current direction, and points towardsthe linear mass. With v0 ¤ 0 it is augmented by a velocity dependent term. For the 160 E.I. Verriest
Fig. 3 Parallel field component as function of v=c for G D d D 1
Fig. 4 Orthogonal field component as function of v=c for G D d D 1
parallel (to the current) component, this additive term behaves linearly near v D 0,and quadratically for the component directed towards the current line. The effectof the current is to entrain the surrounding mass. The large scale structure of theuniverse is made up of large filaments of galaxy superclusters that may be modelledas gravitational currents.Remark One may be tempted to approximate r.t / by its truncated Taylorexpansion in terms of the instantaneous delay. It was found that such a naive useof Taylor expansions can not yield solutions that are consistent in the Newtonianlimit. Post-Newtonian Gravitation 161
3.4 Dynamic Friction
The results of the previous section imply a phenomenon analogous to dynamicfriction which was analyzed by Chandrasekhar in a classical setting [1]. A lonestar colliding with a galaxy (moving through a uniformly distributed star field) willslow down. As only relative motion is significant, we may consider the problem asthat of a stationary mass point (say located at the origin) embedded in a steady flowof mass in the direction of the x-axis. Locally, at O, the problem is isotropic in the(y,z) plane orthogonal to the motion. Hence Fy and Fz experienced by the star arezero. There is however a resulting force in the x-direction. As seen in Sect. 3.3, thestar will be entrained by the moving field of stars and follows a law of the formxR D k.v xP /; for some k which is computable by a more detailed analysis and mustdepend on the density of the star field. As before, v is the initial relative speed ofthe star field with respect to the star. But this means that the galactic medium actsas a viscous medium on the star. At equilibrium, the star moves in unison with themedium.
4 Escape from Gravity
Consider the motion of two equal masses, m, moving symmetrically in onedimension, hence having the same speed with opposite directions (gravitationalfission, supernova). Let r.t/ and l.t/ respectively be the position of the massesmoving towards the right and the left. The symmetry imposes l.t/ D r.t/. SeeFig. 5.Assumption Newton’s law Fr .t/ D mRr.t/ holds, where Fr .t/ is force acting on themass moving towards the right at time t. The gravitational force felt by the particle
cτ m m O
(t) (t − τ ) r(t)Fig. 5 Gravitational Fission 162 E.I. Verriest
at r.t/ due to the particle moving towards the left leads to the the coupled set
Gm rR .t/ D (14) rP.t .t// 1C c c2 2 .t/
c .t/ D r.t/ C r.t .t//: (15)
We present two solution methods: Lagrange inversion and perturbation expansion.
4.1 Lagrange Inversion
The explicit form of .t/ can be obtained from c .t/ D r.t/ C r.t .t// about apoint r.t0 / D r0 , .t0 / D 0 , by Lagrange inversion [2] provided that r is analytic,and jPrj < c at time t D t0 , where it holds that c 0 D r0 C r.t0 0 /; 1 X .c.t t0 / C r0 r.t//i .t/ D .t t0 C 0 / i : (16) i1 iŠ
With D D d=ds, the coefficients of the series expansion are expressed as 8 9 ˆ < > = 1 i D Di1 i > : (17) :̂ cC r.t0 0 Cs/c 0 Cr0 ; s sD0
4.2 Solution via Perturbation Expansion
Postulate a solution to equations (14) and (15) as a series in D 1=c.
r.t/ D r0 .t/ C r1 .t/ C 2 r2 .t/ C : : : (18) 2 .t/ D 1 .t/ C 2 .t/ C : : : : (19) r.tb /The velocity factor 1 C c expands to 1 C Pr0 C 2 ŒPr1 rR0 1 C 3 ŒPr2 rR0 2 C1 .3/ 22 r0 1 rR1 1 C Matching the expansions in for the state dependent delaydifferential equation and eliminating k , k D 1; 2; : : : gives rR0 D (20) 4r02 Post-Newtonian Gravitation 163
Fig. 6 Position r.t/ as function of t for sub-escape with D 1 (r0 D 1; v0 D 1). The curvescorrespond to the Newtonian case (marginal escape), and first and second order perturbations(which return eventually to r0 )
rR1 D r1 2 rP0 (21) 2r03 4r0 g rR2 D 3 4 r12 C 2 3 .r1 rP0 C r2 / 2 rP1 (22) 4r0 r0 4r0 :: :
The solutions of order 0 (Newtonian case), and uppto order 1 and 2 are shown inFig. 6 for D 4, r0 D 1 and v0 D vesc jNewton D 2=r0 , the escape velocity inthe Newtonian case. In pN theory v0 D vesc jNewton is insufficient for escape. For D 1, r0 D 1 and v0 D 1, the post-Newtonian escape velocity was computed p found that for small 1=c the(up to second order for several values of c and it wasadjusted quantity r vesc2.c/ is close to vesc .1/ D 2. This led to the conjecture: vesc .1/ 1C c2
vesc .c/ D r vesc .1/ . 2 .1/ vesc 1 c2
5 Conclusions and Beyond
The physical aspect of the paper established post-Newtonian gravitation as a systemwith state dependent delay. The mathematics centers around an implicit relationof delay and state, which is resolved by Lagrange inversion and perturbationexpansions. The field of mass in uniform motion was described in detail. Its 164 E.I. Verriest
integrated form gives rise to the consideration of gravitational currents and theirensuing entrainment. This led to the emergence of dynamic friction. Finally wediscussed the escape velocity for splitting masses (fission). Lack of space did notallow the discussion of the Kepler problem in pN theory. As expected, stable orbitsexist, which is not the case had one considered the naive generalization (delayedNewton’s law) of the gravitational field mentioned in Sect. 3. It should also bepointed out that naive Taylor expansions lead to results that are inconsistent in theNewtonian limit.
References
1. Chandrasekhar, S.: Dynamical friction. I. General considerations: the coefficient of dynamical friction. Astrophys. J. 97, 255–262 (1943)2. Good, I.J.: Generalization to several variables of Lagrange’s expansion, with applications to stochastic processes. Proc. Camb. Philos. Soc. 56, 367–380 (1960)3. Hartung, F., Krisztin, T., Walther, H.O., Wu, J.: Functional differential equations with state- dependent delays: theory and applications. In: Canada, A., Drábek, P., Fonda, A. (eds.) Handbook of Differential Equations, vol. 3, pp. 435–545. Elsevier, Amsterdam (2006)4. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973)5. Smith, G.: Classical Electromagnetic Radiation. Cambridge University Press, Cambridge/ New York/Melbourne (1997)6. Verriest, E.I.: Inversion of state-dependent delay. In: Karafyllis, I., Malisoff, M., Mazenc, F., Pepe, P. (eds.) Recent Results on Nonlinear Delay Control Systems. Springer International Publishing Switzerland, 327–346 (2016)7. Will, C.: The renaissance of general relativity. In: Davies, P. (ed.) The New Physics, pp. 7–33. Cambridge University Press, Cambridge (1989) Part IIMathematical and Computational Methods in Life Sciences and Medicine A Quantitative Model of Cutaneous MelanomaDiagnosis Using Thermography
Ephraim Agyingi, Tamas Wiandt, and Sophia Maggelakis
Abstract Cutaneous melanoma is the most commonly diagnosed cancer and itsincidence is on the rise worldwide. Early detection and differentiation of a malignantmelanoma from benign cutaneous lesions provides an excellent chance for treatingthe disease. Thermography is a non-invasive tool that can be used to detectand monitor skin lesions. We model heat transfer in a skin region containing alesion. The model which is governed by the Pennes equation uses the steady statetemperature at the skin surface to determine whether there is an underlying lesion.Numerical simulations from the model ascertain whether the lesion is malignant orbenign.
1 Introduction
The skin is the largest organ in the body and has the most exposure to the externalenvironment. Thus, it is prone to lesions attributed to both internal and externalfactors. Skin lesions begin when alterations in cellular metabolism allow cells togrow without restriction. Skin lesions do constitute the majority of all cancersand their incidence is on the rise [1]. The majority of skin lesions are benign andharmless. Melanoma is a malignant skin cancer that can easily undergo metastasisand consequently lead to death if not detected early. Melanoma accounts for themost cancer deaths in the United States compared to other cancers [2]. One of the many tools for diagnosing skin cancers is thermography [3]. Theprocedure is based on established observations that the temperature of the skindirectly above a tumor is significantly higher than the one in the absence of a tumor.Thermography uses an infrared camera to map the temperature distribution over thedesired skin surface. The FDA in the United States approved thermography as anadjunct tool for diagnosing breast cancers in 1982 [4]. Technological advances in thepast decades have improved thermographic imaging so that temperature differencesof about 0:025 ıC can be detected [5].
E. Agyingi () • T. Wiandt • S. MaggelakisRochester Institute of Technology, Rochester, NY 14623, USAe-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 167J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_16 168 E. Agyingi et al.
The published literature contains mathematical models and simulations illustrat-ing thermography. The majority of the studies are for breast cancer [6–12] and onlya few are dedicated to skin cancer [13–16]. Most of these papers model tumors thatare vascularized and governed by some version of the Pennes bio-heat equation.The metabolic heat generating rate of tumors are known to range from 0.205–1.645 W/cm3 , which is about 20–200 times the metabolic heat generation rates ofnormal tissue [6]. Lin et al. [8] considered a tumor metabolic rate of 24:156 W/cm3 ,which is about 2500–5000 times that of normal tissue. In a more recent paper [17],the authors studied a prevascular breast tumor, and related the temperature of theskin surface to the tumor size, tumor depth and the tumor metabolic heat generatingrate. The most effective treatment for melanoma is surgical incision before metastasis.Almost all tumors start up as prevascular tumors. The tumor has a single location,and has no vasculature connecting it to the host capillary network. The aim of thispaper is to investigate whether melanoma can be distinguished from a benign skinlesion at the prevascular level. To achieve this, we model the tumor as a spheroid,consisting of a necrotic core and surrounded by a viable region of proliferatingcells. The work reported here builds on the one-dimensional model reported by[18], which was recently extended in [17]. In [17], it was assumed that the heatdiffusion was the dominant effect and the net effect of perfusion was zero in thehealthy region. This paper differs from [17] in that the boundary conditions takeinto account the effect of blood perfusion in the healthy region.
2 Mathematical Model
We present a model of heat transfer in prevascular skin tumors. We assume that thetumors are spheroids and in the interest of simplicity, we shall present the modelas a spatially two-dimensional model. We consider a two-dimensional cross sectionof the skin tissue, containing a circular shaped tumor as depicted in Fig. 1. Thetemperature of the cross-sectional domain is given by T.x; y/, where the x coordinateis the horizontal direction and y is the depth. The two dimensional cut is placed inthe .x; y/-plane so that the y D 0 level corresponds to the bottom layer of the skin,and y D d corresponds to the surface. The origin is located so that a x a,with the center of the tumor at .0; d=2/. The radius of the tumor is R. The equations governing heat flow in each portion of the entire region are derivedfrom the Pennes equation [19]:
@T Nc D r .KrT/ C mb cb .TA T/ C S; (1) @twhere is the tissue’s density, cN is the tissue’s specific heat, K is the tissue’s thermalconductivity, mb is the mass flow rate of blood, cb is the blood’s specific heat, TA isthe arterial blood temperature, and S is the metabolic heat generation rate. A Quantitative Model of Cutaneous Melanoma Diagnosis Using Thermography 169
y=d
healthy region viable region
necrotic x = −a x=a core
y=0
Fig. 1 Cross-sectional sketch of the skin containing a prevascular tumor
We note that at steady state, the time derivative is zero. Therefore the equationdescribing the temperature T.x; y/ in the healthy region is given by
Sh mb cb .TA Th / Th D ; Kh Kh
where Sh is the metabolic heat generation rate of the healthy tissue and Kh is itsthermal conductivity. For the viable tumor region, we remark that the tumor has no vasculature andtherefore the perfusion term in equation (1) vanishes. Consequently, at steady state,the equation describing the temperature Tt .x; y/ of the viable region of the tumor isgiven by
St Tt D ; Kt
where St is the metabolic heat generation rate of the tumor and Kt is its thermalconductivity. Finally, we note that the necrotic core is comprised of dead cells and generatesno heat. This portion is simply described by the Laplace equation Tc D 0. The boundary conditions of the various parts of the entire domain are listedbelow. The outer boundary conditions of the healthy region are provided by items(i)(iii). Items (iv)(v) provide inner boundary conditions for the healthy regionand outer boundary conditions for the viable tumor region, and item (vi) providesthe boundary conditions between the viable region and the necrotic core. (i) At the bottom layer, y D 0, Th .x; 0/ D Tb isˇ the temperature of the body. hˇ(ii) At the skin surface, y D d. Hence, K @T @y ˇ D .Th Ta /, where Ta is the yDd ambient temperature and is the surface heat transfer coefficient. 170 E. Agyingi et al.
(iii) At the boundaries x D ˙a, Th .a; y/ D Th .a; y/ is the temperature distribution of healthy tissue. To compute this value, we assume hom*ogeneity of the tissue and therefore reduce the problem into a one-dimensional equation of the form
Sh mb cb .TA Th / Th D Th00 .y/ D Kh Kh
and therefore,
mb cb Th Sh mb cb TA Th00 .y/ D ; Kh Kh Kh
where y is the tissue depth, Th .0/ D Tb , and Th0 .d/ D .Th Ta /=Kh . The solution of this equation is given by
Th .y/ D Tb C cosh.! y/ .Ta Tb / K! sinh.! d/ cosh.! d/ C sinh.! y/ sinh.! d/ C Kh ! cosh.! d/ q where D Tb mbScb TA and ! D mKbhcb .(iv) We assume that the temperature is continuous across the interface of the healthy tissue and the viable region of the tumor, i.e. Th .x; y/ D Tt .x; y/ when x2 C .y d=2/2 D R2 . (v) The heat flux is continuous across the interface of the healthy tissue and the viable region of the tumor, i.e. Kt rTt D Kh rTh when x2 C .y d=2/2 D R2 .(vi) On the interface between the viable region of the tumor and the necrotic core, we assume continuous temperature and flux as well.
3 Results and Discussion
In this section, we present numerical simulations of the model presented above. Theresults are for a single tumor and multiple tumors for a given skin cross section. Thecalculations were performed using a MATLAB finite element solver. All thermo-physical parameter values used were chosen within the range of published data. Weset Sh D 0:009 W/cm3 , Kh D Kt D 0:0042 W/((cm)ı C), D 0:0005 W/((cm2 )ı C),mb D 0:0005 g/(ml s) and cb D 4:2 J/gı C. We also set the arterial blood temperatureTA D 37 ı C and the body temperature, at the level y D 0, to be Tb D 37 ıC. Otherparameters used were chosen to investigate the behavior of the model. The first results presented in Fig. 2 are for a single tumor of radius 1 mm andcenter located at .0; 0:25/. Figure 2a provides a contour map and temperaturedistribution over the entire domain. As expected, we observe that heat diffuses awayfrom the tumor towards the cooler surrounding region. In Fig. 2b we examine the A Quantitative Model of Cutaneous Melanoma Diagnosis Using Thermography 171
a b 36.85 m = 0.001 b m = 0.0005 b 36.8 mb= 0.00025 mb= 0.0
36.75
36.7
Temperature(T) 36.65
36.6
36.55
36.5
36.45 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Axial Distance (x)
Fig. 2 Numerical simulations of one tumor with radius R D 1 mm and St D 20Sh . The plot in(a) is the temperature distribution over the cross section and (b) is the skin surface temperature fordifferent perfusion rates (mb )
steady state temperature at the skin surface for different perfusion rates. We note thatthe results in Fig. 2b are almost identical even when the perfusion rate was double(mb D 0:001) or neglected (mb D 0:0). This suggests that the effect of perfusion isnegligible for prevascular tumors, that is, very small tumors that are close to the skinsurface. This may be attributed to the fact that there are no large blood vessels inthe skin region. The average temperature increase at the skin surface, caused by thepresence of the tumor, was observed to be about 0:33 ı C, which is a very significantnumber that can be easily detected using an infrared camera. Next, we consider two tumors of the same radii 1 mm, with centers located at.0:4; 0:25/ and .0:0; 0:25/. We investigate two cases; firstly, the tumors have thesame St values, and secondly, one of the values is altered. The results are given inFig. 3. Figure 3a represents a 3D steady state temperature profile for tumors withthe same metabolic heat generating rates St D 20Sh . In Fig. 3b the metabolic heatgenerating rate of the right tumor was reduced to St D 10Sh. Figure 3c gives thesteady state temperature at the skin surface for the two cases. The results affirm thata tumor with a higher metabolic heat generating rate will produce more heat andconsequently a better temperature profile at the skin surface. The next results as presented in Fig. 4 are for two tumors with the same metabolicheat generating rates (St D 20Sh ) and different radii. The tumor on the left has aradius of 1 mm with center located at .0:4; 0:25/, while the tumor on the right hasa radius of 0:75 mm with center located at .0:4; 0:25/. A 3D steady state temperatureprofile for tumors is given in Fig. 4a and the steady state temperature at the skinsurface is given in Fig. 4b. Here we see that a bigger tumor will produce more heatcompared to a smaller tumor having the same metabolic heat generating rate. Finally, Fig. 5 illustrates two tumors with different metabolic heat generatingrates and different radii. The bigger tumor (i.e. left tumor) with radius 1 mm andcenter located at .0:4; 0:25/ was given a smaller metabolic heat generating rateSt D 5Sh . The smaller tumor (i.e. right tumor) with radius 0:75 mm and centerlocated at .0:4; 0:25/ was given a higher metabolic heat generating rate St D 20Sh .Figure 5a shows a 3D steady state temperature profile for tumors over the entire 172 E. Agyingi et al.
37.2 37.2
37.1 37.2 37.1 37.2
37 37 37 37
Temperature(T)Temperature(T)
36.8 36.8 36.9 36.9
36.6 36.6 36.8 36.8
36.4 36.4 36.7 36.7 0.5 0.5 0.4 1 0.4 1 36.6 36.6 0.3 0.5 0.3 0.5 0.2 0 y 0.2 0 y 0.1 −0.5 x 36.5 0.1 −0.5 x 36.5 0 −1 0 −1
(a) Case I (b) Case II 36.85 Same S h Different Sh 36.8
36.75
36.7 Temperature(T)
36.65
36.6
36.55
36.5
36.45 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Axial distance (x)
(c)
Fig. 3 Numerical simulation of two tumors of the same radii. The results in (a) for the same St ,(b) for different St and (c) the temperature profile at the skin surface for cases I and II
b 36.85 37.2 36.8 a Temperature(T)
37.2 37.1 36.75 Temperature(T)
37 37 36.7 36.8 36.9 36.65 36.6 36.8 36.6 36.4 36.7 36.55 0.5 0.4 1 36.6 36.5 0.3 0.5 y 0.2 0 0.1 −0.5 36.5 36.45 −1 x −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 Axial Distance (x)
Fig. 4 Numerical simulations of two tumors with same St D 20Sh and different radii. The plot in(a) is a 3D temperature distribution over the cross section and (b) is the skin surface temperature
domain, while Fig. 5b provides the steady state temperature at the skin surface. Theresults show that a smaller tumor with a high metabolic heat generating rate will A Quantitative Model of Cutaneous Melanoma Diagnosis Using Thermography 173
36.7a 37.1 b37.2 37 36.65
37 36.936.8 36.6
Temperature(T)36.6 36.8
36.4 36.55
36.2 36.7 0.5
0.4 36.5 36.6 0.3 1
0.2 0.5 0 0.1 36.5 −0.5 36.45 y −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 −1 Axial Distance (x) x
Fig. 5 Numerical simulations of two tumors with different St and different radii. The plot in (a) isa 3D temperature distribution over the cross section and (b) is the skin surface temperature
produce more heat compared to a bigger tumor with a low metabolic heat generatingrate. An analysis of how the results of a variant of the model for breast tumors dependon the underlying parameters was carried out in [17]. Among others, differentvalues of the depth of the tumor, the ambient temperature, and the internal bodytemperature were investigated. A similar analysis yields analogous results for thecurrent model for melanomas.
3.1 Future Work
There are several directions in which the model presented in this paper can beextended. The case of multiple tumors can be investigated by considering differentsize tumors at multiple, randomly chosen locations. This would give a betterunderstanding of possible heat signatures. The case of non-spherical tumors canbe explored as well. However, this requires a slight modification of the currentnumerical methods. Another aspect of interest is how the sizes of the blood vesselsin the neighborhood of the tumor modify the temperature profile of the tumor at theskin surface. The extension of the model from the 2-dimensional domain to 3 dimensionsis under way. The numerical methods have to be adjusted to deal with the addeddimension, but the extended model might discover some new phenomena notpresent in the 2-dimensional investigations. The inverse problem requires a different approach to the model. One possibleextension of our investigation is to set up an optimization problem to identify thelocation and size of the tumors for given heat signatures. 174 E. Agyingi et al.
4 Conclusion
We have presented a mathematical model for cutaneous melanoma at the prevascularlevel based on the Pennes bio-heat transfer equation. We studied the effect of bloodperfusion rate, tumor size and tumor metabolic heat generating rate. The modelsuggests that the perfusion rate does not influence the steady state temperatureat the skin surface for prevascular tumors. The size and tumor metabolic heatgenerating rate are shown to be very important factors in determining the steadystate temperature distribution of the region under consideration. These two factorsare therefore useful bio-markers that can be used to diagnose whether a skin lesionis dormant and harmless or malignant. The results of the model associate melanomawith a tumor that has a high metabolic heat generating rate and to one that isincreasing in size.
References
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16. Bhowmik, A., Repaka, R., Mishra, S.C.: Thermographic evaluation of early melanoma within the vascularized skin using combined non-Newtonian blood flow and bioheat models. Comput. Biol. Med. 53, 206–219 (2014)17. Agyingi, E., Wiandt, T., Maggelakis, S.: Thermal detection of a prevascular tumor embedded in breast tissue. Math. Biosci. Eng. 12, 907–915 (2015)18. Maggelakis, S.A., Savakis, A.E.: Heat transfer in tissue containing a prevascular tumor. Appl. Math. Lett. 8, 7–10 (1995)19. Pennes, H.H.: Analysis of tissue and arterial blood temperatures in the resting forearm. J. Appl. Physiol. 1, 93–122 (1948) Time-Dependent Casual Encounters Gamesand HIV Spread
Safia Athar and Monica Gabriela Cojocaru
Abstract In Tully et al. (Math Biosci Eng AIMS, 2015, to submitted) the authorsmodel and investigate casual sexual encounters between two members of a pop-ulation with two possible HIV states: positive and negative, using a Nash gameframework in which players try to maximize their expected payoff resulting out ofa possible encounter. Each player knows their own HIV status, but do not know theHIV status of a potential partner. They do however have a personal assessment of therisk that the potential partner may be HIV positive. Last but not least, each playerhas a ranked list of preferences of potential types of sexual outcomes: unprotected,protected, or no sexual outcome. In Tully et al. (Math Biosci Eng AIMS, 2015, tosubmitted), the game model is studied via 1- and 2-dimensional sensitivity analyseson parameters such as the utility values of unprotected sex of an HIV negativeindividual with an HIV positive, and values of personal risk (of encountering anHIV positive partner) perception. In this work, we introduce time as a variable which affects players’ riskperceptions, and thus their strategies. Given that HIV transmission happens whenan HIV positive player has a non-zero probability (strategy) of having unprotectedsex with a HIV negative player, we are also able to keep track of the time evolutionof the overall fraction of HIV positive individuals in the population, as reflectedas an outcome of repeated casual encounters. We model a continuous time dynamicgame (as in Cojocaru et al. (J Optim Theory Appl 127(3):549–563, 2005)) where wecompute the stable strategies of each player based on a dynamical system definedon a set of functions. We observe that with change in choices the HIV prevalence inthe population increases.
1 Introduction
HIV/AIDS (human immunodeficiency virus/acquired immunodeficiency syndrome)was first observed in California, 1980 [6]. Patients, at that time, were treated by thelocal physicians for intense fever, diarrhoea, weight loss and swollen lymph nodes.
S. Athar () • M.G. CojocaruUniversity of Guelph, 50 Stone Rd E, Guelph, ON N1G 2W1, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 177J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_17 178 S. Athar and M.G. Cojocaru
In 1982, 600 cases were reported by CDC, out of which 75 % were identified ashom*osexual or bisexual males. Therefore, the early name given to these symptomswas ‘GRID’ (gay related immune deficiency). In late 1982, it was changed to AIDSby CDC after determining the fact that this disease is not exclusive only to the gaypopulation. About 1500 lives were claimed by this disease in 1984 [5]. A poll in1985 indicated that nearly half of Canadians were concerned about getting infectedby this disease [9]. HIV is a unique virus among others as it incorporates its own DNA into thehost’s cellular DNA. The virus also contains a protein that takes over the host cell’sreproduction ability; is then using it as an aid for self replication. This virus impactsthe immune system resulting in life threatening infections. It can be transferredthrough blood, sem*n,vagin*l fluid or breast milk [12]. Nearly 15,000 cases of HIV were discovered in United States in 1985, whereas inAfrica the estimated number was approximately half a million. The latest researchproved that HIV first appeared in humans in central Africa. First evidence of theevolution of HIV emerged in 1985, when scientists found a virus in Macaquemonkeys that was closely related to HIV virus [11, 12]. HIV was classified as a pandemic by the World Health Organization (WHO).According to estimates by WHO and UNAIDS, 35 million people were living withHIV globally at the end of 2013. In the same year, 2.1 million people became newlyinfected, and 1.5 million died of AIDS-related causes. Although medical treatmentshave reduced the annual rate of HIV, the drop in new HIV infections is still notsignificant. The primary mode of transmission, for this disease, is sexual encountersin many countries. For instance, 80 % of the cases in the United States are due tounprotected sexual encounters [13]. According to [11], the spread of HIV infection is largely influenced by people’sway of thinking about their sexual encounters. Transmission of HIV in a populationmay increase if the individuals have unprotected sex. To better understand the spreadof HIV, population models have been used. These models are helpful in observingthe change in infected populations caused by different parameters. Usually, theconcept of probability is used to understand the process of decision-making inrelation to unprotected sex. Game theory is an important mathematical tool thathas been extensively used to describe the decision-making of individuals in certainnon cooperative situations. Different classes for the games are being used to modelindividual’s decision depending upon for example: linear payoff with zero or non-zero sum game as in a 2-player game; or non-linear payoff as in a multi-playergame [11]. In this paper, we are interested in studying a model of two players engagedin finding a casual sexual partner. The game here is dynamic, i.e., we considertime-dependency of equilibrium(Nash) strategies under time evolution of utilitiesand player’s risk assessment. The main purpose is to investigate the influenceof different parameters involved in the game upon the infected population overa certain period of time. This paper has following structure: Sect. 2 describesbriefly the one-shot 2-player game; Sect. 3 presents the dynamic game using theframe work of evolutionary variational inequalities; Sect. 4 discusses the results Time-Dependent Casual Encounters Games and HIV Spread 179
of the computational work along with the analysis of parameters. We close withconclusions and a few ideas for future work.
2 2-Player Game: A Brief Introduction
Tully et al. [11] described a casual sexual encounter between two individuals as agame. The status of the two players are known only to themselves, while playersare aged 15 years and above. Players are denoted by P1 and P2 whose HIV status ispositive and negative respectively. We denote by C the proportion of HIV positives(HIVC ) in total population, and by the proportion of HIV negatives (HIV ) inthe total population with the condition that C C D 1. The authors [11] used game models to find Nash equilibrium probabilities ofhaving unprotected sex (US) in casual encounters. The probabilities of unprotectedsex US for the two players are denoted by:
xi 2 Œ0; 1 2 I xi D .xi ; xiC /; i 2 f1; 2g
where xi represents the probability of Pi having US with an HIV individual and xiCrepresents the probability of Pi having US with an HIVC individual. The expectedutility for Pi ; i 2 f1; 2g, when interacting with HIV and HIVC individuals, is givenby: i E D Œxi U.US; C; / C .1 xi /U.notUS; C; / i EC D ŒxiC U.US; C; C/ C .1 xiC /U.notUS; C; C/
where i 2 f1; 2g By the term notUS means either the players have protected sex or no sex at all. Therefore, the overall expected utility for Pi ; i 2 f1; 2g is given by:
Ei .xi ; xiC / D .1 bC /E i i C b C EC
where represents the activity parameter for player Pi ; i 2 f1; 2g and bC , brepresent the risk assessments of P1 and P2 respectively, regarding the individualthey are engaging in a casual encounter with. The risk parameters bC and b aredefined as follows:
b D ˇ bself C .1 ˇ /C self bC D ˇC bC C .1 ˇC /C : (1) 180 S. Athar and M.G. Cojocaru
selfThe parameters bC and bself represent the personal belief of player P1 , respectivelyP2 about the level of HIV infection in the population. Their values are between 0and 1, with 1 representing the belief that everyone else in the population is infected. The parameters ˇ and ˇC represent the weight a player places upon personalassumptions of HIV prevalences. Their values range again between 0 and 1. Withthese in mind, C is defined as:
C D C .0/ C Œx1 C .0/ C x2C .0/ ;
where C .0/ is the initial fraction of HIVC in the population. By initial we meanhere: before the encounter, where as .0/ represents the initial fraction of HIV inthe population. We let D 0:02 be the transmission rate of HIV in the population[2]. Both players want to optimize their expected payoffs subject to constraintsdefined as follows: for each i 2 f1; 2g,Pi solves the optimization problem, ( max Ei WD Ei .x1 ; x2 / s.t xi 2 Ki WD Œ0; 1 2 \ fxi C xiC D 1g
Definition 1 Assume each player is rational and wants to maximize their payoff.Then the Nash equilibrium is a vector x 2 K WD K1 K2 which satisfies the inequalities: For all i fi .xi ; xi / fi .xi ; xi /, 8xi 2 Ki where xi D .x1 ; : : :; xi1 ; xiC1 ; : : :; x2 /.The authors used variational analysis to find the Nash equilibria for the above gameas in Definition 1.Definition 2 Given a set K Rn , closed, convex, non-empty and given F W K !Rn is a function, the variational inequality(VI) problem is to find a vector x 2 Ksuch that
hF.x /; y x i 0; 8y 2 K (2)
Specifically, the Nash game is reformulated into a variational inequality problem asfollows [3, 10]:Theorem 1 Provided the utility functions ui are of class C1 and concave (meaningui is convex) with respect to the variables xi , then x 2 K is a Nash equilibrium ifand only if it satisfies the VI:
hF.x /; x x i 0 8x 2 K (3)
where F.x/ D .rx1 u1 .x/; : : ::: rxn un .x// and rxi ui .x/ D . @u@x1 .x/ 1 ; @u@x2 .x/ 2 : : :: @un .x/ @xN / Time-Dependent Casual Encounters Games and HIV Spread 181
The variational inequality problem has at least one solution if the constraint set Kis closed, bounded and convex and F W K ! Rn is continuous. In other words, Kis a compact and convex set then solutions for variational inequality problems exist.Further, solution may be unique if the function F is strictly monotone on K [8]. In our case we have: F.x1 ; x2 / D . 5 Ei .x1 ; x2 // D . 5x1 E1 ; 5x2 E2 /.Here, K is closed, bounded and convex and F.x1 ; x2 /, being linear, is continuous;therefore, the existence of solution for the problem is proved. However, F.x1 ; x2 / isnot strictly monotone, therefore the strategy applied here, by the authors, is to lookat the Nash points .x1 ; x2 / 2 K as critical points of a set of differential equationsdriven by the vector field F and constraint set K:
dx D PTK .x. // .F.x. ///I x.0/ D .x1 .0/; x1C .0/; x2 .0/; x2C .0// 2 K D K1 K2d (4)
2.1 One-Shot Game: Example
We develop a base case for our further investigations. The parameters of the modelare set in such a way that we can compute the Nash equilibrium vector for an idealsituation. Ideal situation is developed to restrict the players to their own groups, i.e.,P1 interacts with HIV and P2 interacts with HIVC player. We solve VI problemattached to the game, using the following values in the Table 1 for the parametersinvolved. The utilities for US and notUS are also given in the following Table 1. We start with uniformly distributed initial conditions and find the unique Nashequilibrium for the 2-player game.The value for this equilibrium is (0,1,1,0). Thisshows that the two players are careful regarding the choice of their partner for casualencounters. The strategies of the two players are plotted to see the evolution towards theequilibrium point when different initial values have been considered. The plot inFig. 1 shows the strategy of P1 and P2 , when selecting their partners for a casual sexencounter.
Table 1 Parameters used in ˇ ˇC bself self bC base case 0.5 0.5 0.6 0.3 .0/ C .0/ 0.95 0.05 0.02 1 US.; / US.; C/ US.C; / US.C; C/ 1 0.5 0.5 1 notUS.; / notUS.; C/ notUS.C; / notUS.C; C/ 0.5 0.5 0.5 0.5 182 S. Athar and M.G. Cojocaru
Fig. 1 Phase portrait for the strategy selections of P1 and P2 ; when P1 being HIVC interact withHIV , whereas P2 interact with HIVC . We started with 150 uniformly distributed initial conditionsto solve (4) using Matlab
3 Time-Dependent 2-Player Encounter Game
We convert the one-shot game, as described in Sect. 2, into a time-dependent game.The purpose behind this extension is to see the effect of time on the decision-making process of the two players. Moreover, an additional purpose is to seehow this decision-making can affect the change in HIV prevalence (C ), if the2-player game is played repeatedly. We use the combined theories of projecteddynamical system (PDS) and evolutionary variational inequality (EVI), in order toinvestigate the dynamics of the 2-player game [1, 4]. These two theories have beenapplied to a number of other problems in different disciplines, such as operationalresearch, economics and finance etc. [1, 4]. For this purpose, we reformulatethe variational inequality problem of our 2-player game into a time-dependentvariational inequality, (EVI). We then measure the effects of time dependency onthe HIV prevalence and on the strategies used by the players. First, we start by defining constraint set K, as in Sect. 2, and rewrite it for theEVI. Our feasible vector u.t/ D .x1 .t/; x2 .t// D .x1 .t/; x1C .t/; x2 .t/; x2C .t// has tosatisfy the time-dependent constraints on xi ; i 2 f1; 2g, and it belongs to the set offunctions given by: K D fxi .t/ 2 L2 .Œ0; T ; R4 / W 0 xiC .t/; xi .t/ 1; xiC .t/ C xi .t/ D 1; i 2f1; 2g a:e 2 Œ0; T g Time-Dependent Casual Encounters Games and HIV Spread 183
Since F.x1 ; x2 / D .rE.x1 ; x2 //, we write F.u.t// D .rE.u.t/// where F WK ! L2 .Œ0; T ; R4 /. Then EVI is defined as:Definition 3 Find u 2 K such that
hF.u.t//; v.t/ u.t/i 0 8 v.t/ 2 K.t/ for a.a t
where
K D fxi .t/ 2 L2 .Œ0; T ; R4 / W 0 xiC .t/; xi .t/ 1; xiC .t/ C xi .t/ D 1; i 2 f1; 2g a.e in Œ0; T g
This is called point-wise form for EVI.Theorem 2 Let H be Hilbert space and let K H be non-empty, closed andconvex subset. Let F W K ! H be the Lipschitz continuous vector field withLipschitz constant b. Then the solutions of the time dependent variational inequalityDefinition 3 are the same as the critical points of the projected differential equation:
du.t; / D PTK.x/ .u.t; /; F.u.t; /// d
That is the point x 2 K such that PTK.x/ .u.t/; F.u.t/// 0, and the converse alsoholds.If we choose H D L2 .Œ0; T ; R4 / then the solution of EVI in Definition 3 and thecritical points of the equation in (2) are the same. The important point in (2) is the difference between the two times, t; . The timeŒ0; T represents the time interval the game is considered to be played over. As tvaries over the interval [0,T], we can obtain one or more curves representing howNash strategies of both players may change. The time is the simulated time ofevolving from an initial point of the differential equation (2) towards one of the Nashequilibria on this curve(s). For the computational purpose, a sequence of partitionftn0 ; tn1 ; : : ::; tnN g 2 Œ0; T is defined such that
0 D tn0 < tn1 <; : : :: < tnN D T:
Then for each tnj , we solve the time dependent variational inequality:
hF.u.tnj //; v u.tnj /i 0 for all v 2 K.tnj /;
where K.tnj / D fxi .tnj / 2 L2 .Œ0; T ; R4 / W 0 xiC .tnj /; xi .tnj / 1; xiC .tnj / C xi .tnj / D1; i 2 f1; 2g a.e in Œ0; T g. 184 S. Athar and M.G. Cojocaru
We compute all solutions of this finite dimensional variational inequality, byfinding the critical point of the projected dynamical system [4]:
PTK.x/ .u.tnj ; /; F.u.tnj ; /// D 0
4 Computation of Time-Dependent Nash Strategies and Their Effects on the HIV Prevalence
4.1 Base Case
We use the same values for the parameters in the base case as in the one-shot game.The only difference in this base case is that both the strategies and C are timedependent. The parameters used for the base case are described in Table 1. We runthis case for T D 10 periods, and reach the same equilibrium point as in the one-shotgame. No change in the value of C .t/ is observed (Fig. 2).
Fig. 2 Increase in HIV prevalence over a period of 10 periods Time-Dependent Casual Encounters Games and HIV Spread 185
4.1.1 Influence of US.; C/ and US.C; / on HIV Prevalence
We vary US.; C/ & US.C; / as a function of time, whereas the other parametersare kept constant. The change in choices about having US over the given periodof time can be considered the result of emotional decision-making and/or thelevel of information (Fig. 3). Gutnik et al. in [7] discussed the role of emotionsin decision-making. The role was considered as negative hindrance in the rationaldecision-making. According to new research, people use analytic and experimentalsystems to understand and assess the factors of risk. However, the emotions arethe most common factor that works in the experimental system. These emotionsdepend on past experiences or the perceived risks [7]. The players can change theirpreferences over the given period under the influence of emotion. These decisionsunder emotional circ*mstances perhaps occur due to lack of adequate knowledge ordue to the “heat of moment” (Table 2).
Fig. 3 3-D plots shows the change in HIV prevalence, when the US.; C/ and US.C; / aremade time dependent. Upper left shows the change in US.; C/ and US.C; / are linear functionof time; upper right shows US.; C/ and US.C; / are quadratic function of time. Lower leftshows US.; C/ and US.C; / are quadratic function as well but in different formation and lowerright shows US.; C/ and US.C; / are cubic function of time. The increase is much higher inlinear case, US.; C/ and US.C; / are linear function of time, as compared to the case whenthese choices are quadratic or cubic function of time 186 S. Athar and M.G. Cojocaru
Table 2 Effect of US.; C/ and US.C; / on HIV prevalence C .t/Cases US.; C/ US.C; / .t/Case 1 0:5 C 0:5 .t=10/ 0:5 C 0:5 .t=10/ 5 % to 32 %Case 2 0:5 C 0:5 .t=10/2 0:5.t/ C 0:5 .t=10/2 5 % to 24 %Case 3 0:5 C 0:5 .t=10/3 0:5 C 0:5 .t=10/3 5 % to 21 %
The time is taken as 10 periods and Matlab is used for simulations. We observe anincrease in HIV prevalence from 5 % to 32 % when US.C; / is linear function oftime; whereas the increase is 24 % when US.C; / in quadratic in time. The impactof US.C; / on increase in HIV prevalence is perhaps due to change in preferencesof player 1(HIVC) over the time.
5 Conclusion and Future Work
In this work we converted a discrete time 2-player game into a time-dependentgame. We investigated the effects of changes in utilities for having unprotected sexbetween HIVC and HIV players on the HIV prevalence rate over a given period oftime. Also, we examined the influence of individual’s personal risk perception abouttheir partners in casual encounters. In the near future we are interested in formulating a multi-player time-dependentgame, when two different types of HIV viruses, i.e., HIV 1 and HIV 2 areinvolved. The players with these viruses interact with HIV players from sameage groups or different age groups. Further, we want to investigate the impact ofprophylactic vaccination against HIV in younger population groups in multi-playerstime-dependent games.
References
1. Barbagallo, A., Cojocaru, M.-G.: Dynamic vaccination games and variational inequalities on time-dependent sets. J. Biol. Dyn. 4(6), 539–558 (2010) 2. Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory, vol. 200. SIAM, Philadelphia, (1995) 3. Cojocaru, M.-G., Greenhalgh, S.: Dynamic games and hybrid dynamical systems. Optim. Eng. Appl. Var. Inequal. Issue 13(3), 505–517 (2011) 4. Cojocaru, M., Daniele, P., Nagurney, A.: Projected dynamical systems, evolutionary variational inequalities, applications, and a computational procedure. In: Pareto Optimality, Game Theory and Equilibria, pp. 387–406. Springer, New York (2008) 5. Engel, J.: The Epidemic: A History of AIDS. HarperCollins, New York (2009) 6. Grmek, M.D.: History of AIDS: Emergence and Origin of a Modern Pandemic. Princeton University Press, Princeton, NJ (1993) Time-Dependent Casual Encounters Games and HIV Spread 187
7. Gutnik, L.A., et al.: The role of emotion in decision-making: a cognitive neuroeconomic approach towards understanding sexual risk behavior. J. Biomed. Inform. 39(6), 720–736 (2006) 8. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic, New York (1980) 9. Levon, B.: Facing a fatal disease. Maclean’s 1(6), 48 (1986)10. Moulin, H.: On the Uniqueness and Stability of Nash Equilibrium in Non-cooperative Games, vol. 130. North-Holland Publishing Company, Amsterdam (1980)11. Tully, S., Cojocaru, M.-G., Bauch, C.: Multiplayer games and HIV transmission via casual encounters. Math. Biosci. Eng. AIMS (2015, to submitted)12. World Health Organization: Health topics: HIV/AIDS. http://www.who.int/hiv/pub/guidelines/ en. Accessed 6 Mar 201313. World Health Organization: Health topics: HIV/AIDS. http://www.who.int/features/qa/71/en/ http://www.who.int/bulletin/volumes/85/11/06-033779/en/index.html. Accessed 6 Mar 2013 Modelling an Aquaponic Ecosystem UsingOrdinary Differential Equations
C. Bobak and H. Kunze
Abstract Aquaponic agriculture is a sustainable system which uses interdependentprocesses and has been growing in popularity. However, relatively little mathemat-ical and other academic research has been conducted in the practice. In this paper,we develop a system of ordinary differential equations to model the populationand concentration dynamics of the environment. Our model has an asymptoticallystable non-trivial equilibrium, representing the inherent symbiotic relationship ofthe variables. Values of the nine parameters in the system are estimated from theresearch literature. We provide simulated results simulated results to illustrate thenature of solutions to the system, and we present and discuss a sensitivity analysis.
1 Introduction
Aquaponics is a closed-loop agricultural system which uses a symbiotic relationshipbetween aquatic organisms and aquatic macrophytes. The system recirculates waterthrough an aquaculture environment (fish in a designated body of water) anda hydroponic structure (aquatic plants in soilless water) to create a sustainableenvironment which fully conserves water and nutrients. The key motivation behindaquaponics is using waste produced by fish in the system as a nutrient sourcefor the plants. This process not only allows the fish waste to act as a naturalfertilizer for the plants but also inherently cleans the water to be returned to thefish [6]. Thus the environment is in a natural state of stability, which we seek tomodel using a system of differential equations. This paper discusses backgroundtheory in aquaponics, develops a compatible aquaponic model in the form of asystem of differential equations, discusses equilibria of the model and their stability,presents specific solution simulations for parameter values derived from the researchliterature, performs a sensitivity analysis, and suggests future research direction.
C. Bobak () • H. KunzeDepartment of Mathematics and Statistics, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 189J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_18 190 C. Bobak and H. Kunze
2 Background: Research of Aquaponic Environments
Aquaponic agriculture systems have been growing in popularity due to their robusteconomic and ecological benefits. Both hydroponics and aquaculture have provento have a detrimental effect on the environment. Some studies have suggested thataquaponics has a 75 % smaller carbon footprint when compared to traditional farm-ing methods [7] and is able to satisfy human demand long-term [3]. Importantly,aquaponics achieves all of these benefits while maintaining economic viability forfarmers. The lower cost of resources and lower spatial requirements has led tosuggestions that aquaponic agriculture may be a viable solution in densely populatedurban regions such as Pakistan and India [3]. The basic aquaponic system consists of two main components: one for fishand one for plants. The system relies heavily on food input as many studies havedemonstrated that factors like protein content of food and feeding frequency havethe largest effect on the efficiency of the system. These factors highly contributeto fish growth and are also directly related to the amount of fish waste in theenvironment [6]. Organic nitrogen in fish waste naturally converts to ammonia through biologicaldegradation [4]. Ammonia is highly toxic to fish, and is an inefficient nutrient sourcefor plants [6]. In order for ammonia to be used as fertilizer for the plants, it mustgo through a natural microbial process called the Nitrogen Cycle, which causes it toconvert into nitrate. Nitrate is a nutrient rich food source for plants [4] and studieshave shown that plants’ uptake efficiency of nitrates ranges from 86 % to 98 %.Whatever concentration of nitrate is left in the water is not harmful to the fish, sothe water can be recirculated for fish use [6].
3 Development of the Model
The overarching goal of the model is to capture the symbiotic relationship betweenthe fish and the macrophytes. However, as evidenced by background research onaquaponic agriculture, the aerobic microbial process which converts ammonia tonitrate needs to be considered. The assumpions made are as follows: i. The aquaponic ecosystem is a closed environment. ii. The fish population increases at some natural survival rate . Deaths Births /, hindered by a carrying capacity due to the limited tank space. iii. There is additional fish decay due to increased ammonia presence in the water until it reaches a critical ammonia level where no fish survive. This can be reasonably modelled using a linear constant of . Toxic Ammonia Ammonia Level /. iv. Ammonia is present in the system exclusively due to fish waste and hence grows at a rate proportional to the fish population. It decays due to its conversion to nitrate. Aquaponic Ecosystems Using ODEs 191
v. Nitrate grows at a rate proporational to the level of ammonia, and decays due to plant uptake. vi. Plants grow at a constant rate hindered by a carrying capacity indicative of the limited surface area of the system. vii. Modelling the concentrations of ammonia and nitrate in the system will capture any other relationships between other variables in the nitrogen cycle.viii. The system is well mixed so the nitrogen cycle occurs naturally and plants have even access to nitrate. The proposed model is below, with variables F, A, N, and P representing thepopulation of fish, ammonia (in mg), nitrate (in mg) and population of plantsrespectively: PF D a1 1 F FP A F (1) KF KA AP D a2 F a3 A (2) NP D a4 A a5 NP (3) P PP D a6 1 PN (4) KP
where ai 0 8i are growth and decay rates, and KF ; KA ; KP > 0 are the carryingcapacities of fish, ammonia, and plants respectively. Equation (1) models the evolution of the fish population, using assumptions ii andiii. Equation (2) models the evolution of the ammonia concentration in the systemusing assumption iv. Equation (3) captures the growth rate of nitrate concentrationas it relates to the conversion from ammonia and the decay rate due to plant uptakeusing assumption v. The final equation (4) captures the growth rate of the plantsusing assumption vi.
4 Analyzing the Equilibria
Our aquaponic environment model (1), (2), (3) and (4) consists of four equationswith nine unknown parameters. The Jacobian matrix of this model is as follows: 2 a1 FP 3 KF Ca1 1 KF P KA KF 0 a1 1 KF F 6 7 F A A F a3 0 0 Df .F; A; N; P/ D 4 a2 0 a 5 (5) a4 5P a5 N a6 NP 0 0 a6 P 1 KP a6 N 1 KP K P P P 192 C. Bobak and H. Kunze
The system (1) (2), (3) and (4) has three equilibria:
fF D 0; A D 0; N D 0; P D Pg (6) fF D 0; A D 0; N D N; P D 0g (7) a1 a3 KF KA KP a1 a2 KF KA KP FD ;AD ; a1 a3 KP KA C a2 KF a1 a3 KA KP C a2 KF a1 a2 a4 KF KA ND ; P D KP (8) a5 .a1 a3 KA KP C a2 KF /
of which the third, in equation (8), since it represents all variables surviving in theenvironment. We focus on it in the next section.
4.1 Equilibrium (8): Coexistence
We analyzed the stability of the nontrivial equilibrium in equation (8) using theresearch literature to provide estimates for the nine parameters: a1 D 0:0124, a2 D0:1, a3 D 0:94, a4 D 3:6, a5 D 0:92, a6 D 0:056, KF D 250, KP D 300 andKA D 20 [1–4, 6]. The values selected for our carrying capacities were based on anarbitrary initial tank size of 10L, however, this can easily be scaled up or down toaccomodate systems of various sizes. Other initial values were fF.0/ D 10; A.0/ D0; N.0/ D 0; P.0/ D 0:5g, where P.0/ D 0:5 was used to represent plants whichhad not yet reached maturity in the system. Substituting the estimated parametervalues in the Jacobian gives: 2 3 2:77 9:31 0 0:59 6 0:1 0:98 0 0 7 A D 6 4 0 7 (9) 3:6 276 0:2285 0 0 0 0:014
The eigenvalues of A in this particular simulation are:
31 D 1:84 C 0:332i (10) 32 D 0:014 (11) 33 D 276 (12) 34 D 1:84 0:332i (13)
Notably, the real parts of (10) and (13) are negative, and values (11) and (12)are negative, thus an asymptotically stable equilibrium is achieved in this case. Aquaponic Ecosystems Using ODEs 193
Because (10) and (13) are complex, some spiralling behaviour is present in thesystem. Note that 32 is very close to 0, so the stability status of this equilibriumpoint may be sensitive to the parameter values chosen. Some experimentation wasdone within ranges of realistic values for the estimated parameters. In every case,all eigenvalues have negative real parts suggesting with some generality that theequilibrium case where all four variables are present in the system is asymptoticallystable. We discuss sensitivity in general in the next section. Substituting these values into the non-trivial equilibrium and subsequent solutiongiven in the previous section provides the following equilibrium points:
fF D 186:18; A D 19; N D 0:25; P D 300g (14)
The real-world context of these equilibrium points is exciting; it suggests thatfish grow to a point well bounded by their carrying capacity, (KF D 250). Ammoniatends towards an amount just below the threshold to which all the fish would die(KA D 20). This suggests that the system is stabilizing in a way which approachesthe maximum level of food production for the plants. Nitrate levels appear to bequite low, which is likely due to the superior nutritional uptake of the plants whichhas been noted in the literature. The plants themselves reach equilibrium at theircarrying capacity (KP D 300). This is exactly as anticipated based on the modelstructure. Figure 1 shows two plots of the modelled variables over time given the parameterassumptions. As evidenced here, all the variables are approaching their aboveequilibrium values.The fish and plant follow a logistic trend tending to their carryingcapacities. Ammonia also seems to be following this trend, which is due to its
Fig. 1 The population and concentration dynamics of the model over time (a) The solid linerepresents the fish population and the dotted line represents the plant population (b) The solidline represents the nitrate population and the dotted line represents the ammonia population 194 C. Bobak and H. Kunze
growth dependence on fish. Nitrate also shows an interesting trend; as the systemis being established, nitrate experiences a brief spike. However, once the plantgrowth reaches a level which requires significant nitrate sustenance, this spikereverses and the nitrate concentration levels off. These curves appear to looselyrepresent expected trends based on the aquaponics literature, suggesting the modelis reasonably capturing aquaponic behaviour.
5 Sensitivity Analysis
A sensitivity analysis for the parameters of this model was performed with the goalsof identifying any unexpected behaviour, guiding any data collection efforts, andmost importantly, to give an indication of the importance of accurately estimatingthe parameter values. A software toolbox in Matlab was used to perform sensitivity analysis ofbiological models. SensSB is freely available for academic purposes, and combinesa variety of local and global sensitivity methods, both using relative and absolutemeasures to achieve many of these goals [8, 9]. While locally analysing the sensitivity of parameters is a useful exercise, it hasan inherent reliance on the initial numerical estimation of the parameter. Globalmethods, which test the effect of a parameter while other parameters are variedsimultaneously, help avoid this stipulation [9]. Since the initial estimation of thevariables from the literature are considered weak points of the model, globalsensitivity measures were analysed. SensSB analyses global sensitivity through three main methods, all of whichare discussed in the software documentation (see [9]). In this study, DerivativeBased Global Sensitivity Measures (DBGSM), which were introduced in 2009 [5],were selected to optimize accuracy with minimal loss of computational efficiency[9]. DBGSM uses Monte Carlo sampling methods to average local derivatives to ameasure M N ij which averages sensitivity measures Sij over the parameter space.
% change in Parameters Sij D (15) % change in Variables Z N ij D M Sij dp (16) Hnp
As seen in Fig. 2, parameter 3 .a3 /, or the conversion rate of ammonia to nitrate,has high global absolute sensitivity. In fact, according to the output from SensSB,it accounts for 73.26 % of the total sensitivity in the model. Notably, the relativesensitivity by variable, which shows how sensitive each variable is to the parametersin the model, shows high relative sensitivity for parameters a2 , a3 , a6 , and KF . Inthe model, these parameters represent the growth rate of ammonia from fish waste,the conversion rate of ammonia to nitrate, the growth rate of plants and the carrying Aquaponic Ecosystems Using ODEs 195
Fig. 2 (a) shows the global absolute sensitivity of each of the estimated parameters while (b)shows the global relative sensitivity by variable. Note parameters 1–6 are a1 through a6 , parameter7 is the carrying capacity KF , parameter 8 is the carrying capacity KA , and parameter 9 is thecarrying capacity KP
capacity of fish. This suggests that some more care should be taken in estimatingvalues for these parameters, particularly a2 and a3 , which do not have reliable valuesin the literature. Using values from the previous simulation, both a2 and a3 were varied (individ-ually and consecutively) between values of 0 and 100 without loss of stability in thecoexistence equilibrium. However, exploring data from an established environmentmay help clear potential biases associated with this parameter estimation.
6 Future Research Direction
Future models of aquaponic systems should incorporate additional complexity.In particular, removing the assumption that the environment is closed to allowadditional harvesting cycle terms for fish and plants would add realism and requireadditional analysis. Here, the term harvesting refers to the biological process ofremoving some fish and/or plants, creating a discontinuity in the solution. As well, there is some indication in the literature that the assumption that thefish population decays linearly with increases in ammonia concentraion is invalid.Instead, it is expected that the fish population would not be affected by the presenceof ammonia in the system until the ammonia begins to approach a critical level [1].Instead, a non-linear polynomial relationship could be used, and may be a betterapproximation of the real world relationship. Building from the sensitivity analysis, real data sets could be used to solve theparameter estimation inverse problem to see how well this model, and more complexvariations, can accommodate real-world data as an approximation solution. 196 C. Bobak and H. Kunze
7 Conclusion
The developed four equation model appears to reasonably simulate biologicalbehaviour. Simulated solutions lead to a few conclusions which may or may nothold in the general case. The most important of these is that the model admitsan asymptotically stable equilibrium where fish, ammonia, nitrate, and plantsinterdependently coexist in the environment. An exploration of parameter spacesuggests that the stability of such an equilibrium point is robust to changes in theparameters. Aquaponics shows considerable promise as an ecological agriculture solution.It is the hope that this research can be used to further test the ability of aquaponicsolutions as viable practices in both developed and undeveloped countries to aid inconcerns over the sustainability of current food practices.
References
1. Wheaton, F.: Recirculating aquaculture systems: an overview of waste management. In: Proceedings of the 4th international conference on recirculating aquaculture, Roanoke (2002)2. Bhujel, R.C., Little, D.C., Hossain, M.A.: Reproductive performance and growth of stunted and normal Nile Tilapia (Oreochromis niloticus) broodfish at varying feeding rates. IEEE Aquac. 273, 71–79 (2007)3. Blidariu, F., Grozea, A.: Increasing the economical efficiency and sustainability of indoor fish farming by means of aquaponics – review. Anim. Sci. Biotechnol. 44(2), 1–8 (2011)4. Endut, A., Jusoh, A., Ali, N.: Nitrogen budget and effluent nitrogen components in aquaponics circulation systems. Desalinations Water Treat. 52, 744–752 (2014)5. Kucherenko, S., Rodriguez-Fernadez, M., Pantelides, C., Shah, N.: Monte carlo evaluation of derivative based global sensitivity measures. Reliab. Eng. Syst. Saf. 94, 1135–1148 (2009)6. Liang, J.L., Chien, Y.H.: Effects of feeding frequency and photoperiod on water quality and crop production in a tilapia-water spinache raft aquaponics system. Int. Biodeterior. Biodegrad. 83, 693–700 (2013)7. Pattillo, A.: Aquaponic system design and management. Aquaculture Extension, Iowa State University (2014)8. Rodriguez-Fernandez, M., Banga, J.R.: SensSB: a software toolbox for the development and sensitivity of systems biology models. Bioinformatics 26(13), 1, 2 (2009)9. Rodriguez-Fernandez, M., Banga, J.R.: SensSB – a software toolbox for sensitivity analysis in systems biology models. In: International Conference on Systems Biology (ICSB 2009) Aug 30–Sept 4, Stanford University, Palo Alto (2010) A New Measure of Robust Stablity for LinearOrdinary Impulsive Differential Equations
Kevin E.M. Church
Abstract A new measure of robust stability for linear ordinary impulsive dif-ferential equations with periodic structure is introduced, based on the impulseextension concept. This new stability measure reflects the sensitivity of the model touncertainty in what we see as the fundamental hypothesis of impulsive models: thatthe impulse effect occurs quicky enough that its duration can be entirely neglected.The measure, that we call the time-scale tolerance, E t , has the property that, if thevector of durations of impulse effect, a, satisfies jjajj < E t , then both the impulsivemodel and a family of continuous impulse extension equations (a specific functionaldifferential equation) to which it is related, will all be asymptotically stable.We review linear impulse extension equations, state theorems that describe theconvergence of their solutions to the associated impulsive solutions, and introduceall the machinery necessary in the development of the time-scale tolerance, statingtheoretical results on its existence and how it can be computed in practice. Weconclude with two illustrative examples and a discussion of the limitations of thetechniques presented, as well as elaborate on the ways they can be improved.
1 Introuction
Impulsive differential equations are frequently employed as models of biological,chemical, and physical systems, among others. The property of these equationsthat makes them attractive in applications is the impulse effect, which allows forthe inclusion of fast dynamics that might otherwise complicate the analysis of themodel, were they to be included in a continuous, as opposed to discrete, manner.Most monographs on impulsive differential equations—for example, [1, 5, 6]—state that these equations are reasonable approximations of continuous models withperturbations, if the perturbations themselves occur quickly, relative to the overalldynamics. In practice, this is often taken as an assumption of the model in question.Methods to determine how quickly these perturbations must occur for the model
K.E.M. Church ()Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa,ON K1N6N5, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 197J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_19 198 K.E.M. Church
to be a good “fit” to the associated continuous models have yet to be seen in theliterature. In these proceedings, we introduce a quantity that we call the time-scaletolerance, denote E t , for a linear, periodic impulsive differential equation that isasymptotically stable. This quantity has the property, that, if the vector of durationsof impulse effect, a, satisfies jjajj < E t , then both the impulsive model and afamily of continuous impulse extension equations (a specific functional differentialequation) to which it is related, will all be asymptotically stable. We review linear impulse extension equations, which were first introduced in [2–4], state theorems that describe the convergence of their solutions to the associatedimpulsive solutions, and introduce all the machinery necessary in the developmentof the time-scale tolerance, stating theoretical results on its existence and how itcan be computed in practice. We conclude with two illustrative examples and adiscussion of the limitations of the present techniques, and how they can be extendedto accomodate a larger class of problems.
2 Linear Impulse Extension Equations
To begin, we introduce some notation that will be present throughout this chapter.If x D fxk W k 2 Zg is a real-valued sequence, we denote xk D xkC1 xk . Thekth element of a real-valued sequence x will always be denoted xk , and we mayabuse notation and identify the sequence x with the symbol xk . Indexed families ofsequences, such as, fx j W j 2 Ug, will always have their index appear in the exponent. jIn this context, xk denotes the kth element of sequence j from the family U. Finally,our sequences will usually be bi-infinite; that is, indexed by the integers. The symboljj jj will denote a (fixed) Euclidean norm, whenever there is no ambiguity, and if Ais a set, its closure will be denoted A. Consider a linear, impulsive differential equation with impulses at fixed times
dx D A.t/x C g.t/; t ¤ k dt (1) x D Bk x C hk ; t D k :
with t 2 R, phase space ˝ Rn , A W R ! Rnn , Bk 2 Rnn , hk 2 Rn , and sequenceof impulses k for k 2 Z. We assume that A and g are locally integrable, and that thesequence of impulse times, k , is monotone increasing and unbounded. An impulse extension equation for (1), as described in Church and Smith? [3, 4]and Church [2], is defined as follows.Definition 1 Consider a linear impulsive differential equation (1).• A step sequence over k is sequence of positive real numbers a D fak W k 2 Zg such that ak 2 .0; k / for all k 2 Z. We denote Sj D Sj .a/ Œ j ; j C aj / and A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential. . . 199
S S D S.a/ j2Z Sj . The set of all step sequences will be denoted S , and is defined by
S WD fa W Z ! R ; ak 2 .0; k /g:
• The pair .'kB ; 'kh /, with sequences of functions 'kB W R RC ! Rnn and 'kh W R RC ! Rn , is a family of impulse extension for (1) if for all a 2 S and all k 2 Z, the functions 'kB .; ak / and 'kh .; ak / are integrable on Sk .a/ and satisfy the equalities Z Z 'kB .t; ak /dt D Bk ; 'kh .t; ak /dt D hk : (2) Sk .a/ Sk .a/
• Given a step sequence a 2 S and a family of impulse extensions ' D .'kB ; 'kh / for (1), the impulse extension equation associated to (1) and induced by .'; a/ is the (functional) differential equation dx A.t/x C g.t/; t … S.a/; D (3) dt A.t/x C g.t/ C 'kB .t; ak /x. k / C 'kh .t; ak /; t 2 Sk .a/:
Definition 2 Let a family of impulse extensions, ' D .'kB ; 'kh /, and a step sequencea 2 S be given. A function y W I ! Rn defined on an interval I R is a classicalsolution of the impulse extension equation (3) induced by .'; a/, if y is continuous,the sets I \ Sk .a/ are either empty or contain k , and y satisfies the differentialequation (3) almost everywhere on I. Given an initial condition
x.t0 / D x0 ; (4)
with .t0 ; x0 / 2 RRn , the function y.t/ is a solution of the initial-value problem (3)–(4) if, in addition, y.t0 / D x0 .Remark 1 Definitions 1 and 2 can be readily modified to accomodate nonlinearordinary impulsive differential equations; see Church and Smith? [4] and Church[2].Definition 3 The predictable set of the impulse extension equation (3) induced by.'; a/ is ( Z ! ) t P D R n t 2 S.a/ W det I C X 1 .s; k /'kB .s; ak /ds D 0 ; max k f k tg
where X.t; s/ is the Cauchy matrix of the hom*ogeneous system z0 D A.t/z. The following theorem states the mode in which solutions of the impulseextension equation, (3), converge to those of the impulsive differential equation, (1). 200 K.E.M. Church
Theorem 1 Suppose det.I C Bk / ¤ 0 for all k 2 Z, and let ' D .'kB ; 'kh / bea given family of impulse extensions for (1). There exists a positive sequence ofreal numbers k , depending only on A.t/ and sequence of impulse times k , with the following property. Suppose, for 2 fB; hg and each k 2 Z, there exists wk WŒ k ; kC1 Œ0; k / ! R that is continuous and vanishing at . k ; 0/, for which 1 1 'k .t; s/ k D O wk .t; s/ s (5) s e k 1
for t 2 Œ k ; k C s/ as s ! 0. The following are true:• For all t0 2 R, there exists ı > 0, such that, for a 2 S with jjajj1 < ı and all x0 2 Rn , the impulse extension equation (3) induced by .'; a/ possesses a unique classical solution, x.tI a/, satisfying the initial condition x.t0 / D x0 .• The function x.tI a/ converges pointwise to x.tI 0/, the solution the initial value problem x.t0 / D x0 for the impulsive differential equation, (1), as a ! 0.• If N R is bounded and no strictly decreasing sequence in N has an impulse time k as its limit, the above convergence is uniform for t 2 N.Proof (Outline) The existence of the sequence k follows from the Generalized k .t k /Gronwall’sR kC1inequality: we have jjX.tI k /jj e for t 2 Œ k ; kC1 , wherek D k jjA.s/jjds, and X.tI s/ is the Cauchy matrix of x0 D A.t/x. Whenjjajj1 < t0 maxf k W k < t0 g, the solution of (3)–(4) induced by .'; a/ satisfyingx.t0 I a/ D x0 , can be written as
x.tI a/ D U.tI a/x0 C xp .tI a/;
for a matrix function t 7! U.tI a/ satisfying U.t0 I a/ D I, and xp .t0 I a/ D 0 (thisfollows by Proposition 4.2 of [4]). If U.tI 0/ denotes the fundamental matrix solutionof the hom*ogeneous equation associated to (1), one can show that the inequality ˇˇ ! ˇˇ Y0 ˇˇ jjU.t; a/ U.t; 0/jj jjX.tI k /jj ˇˇ La .tI k / X. rC1 I r /La . r C ar I r / ˇˇ rDk1 !ˇˇ Y0 ˇˇ ˇˇ .I C Bk / X. rC1 I r /.I C Br / ˇˇ ˇˇ rDk1
holds, where we have assumed t0 D 0 for ease of presentation (other cases followby similar reasoning, by results from [4]), and Z minft; k Cak g La .tI k / D I C X 1 .sI k /'kB .s; ak /ds: k
It can be shown that the right-hand side of the upper bound converges to zeroas a ! 0 pointwise, as jjajj1 ! 0 (see the proof of Theorem 3.5.5. from A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential. . . 201
[2] for the main idea; condition (5) is needed). A similar inequality holds forjjxp .tI a/ xp .tI 0/jj, where xp .tI 0/ is the solution of (1) satisfying xp .t0 I 0/ D 0,and the convergence result holds for that piece of the solution as well. For uniformconvergence, it suffices to consider N to be a finite union of closed intervals withxn # x 2 N ) x … f k g. The hypotheses of Theorem 1 are simplified if the equations (1) and (3) areperiodic.Definition 4 The linear impulsive differential equation (1) is T-periodic with cimpulses per period if A.t C T/ D A.t/ and g.t C T/ D g.t/ for all t 2 R, and kCc D k C T, BkCc D Bk and hkCc D hk for all k 2 Z. The step sequence a 2 S isc-periodic, and we write a 2 Sc , if akCc D ak for all k 2 Z. The family of impulse extensions ' D .'kB ; 'kh / is .T; c/-periodic if 'kCc .t C T; s/ D 'k .t; s/ for all t 2 R,all k 2 Z, all s 2 .0; k / and 2 fB; hg.Corollary 1 Suppose the impulsive differential equation (1) is T-periodic with cimpulses per period. Let ' D .'kB ; 'kh / be a .T; c/-periodic family of impulseextensions for (1). Suppose det.I C Bk / ¤ 0 for k D 0; : : : ; c 1: Let theimpulsive differential equation (1) have a fundamental matrix X.t/ with Floquetdecomposition X.t/ D P.t/et satisfying X. 0 / D I. The conclusions of Theorem 1hold for step sequences a 2 Sc , with k jjjj. The proof of the above corollary is omitted, since it is simple to prove usingTheorem 1. For periodic impulse extension equations, we have an asymptoticFloquet theorem. A proof is available in [2], where it is listed as Theorem 3.6.16.Theorem 2 Suppose the impulsive differential equation (1) is T-periodic with cimpulses per period. Let ' D .'kB ; 'kh / be a .T; c/-periodic family of impulseextensions for (1). Let det.I C Bk / ¤ 0 for k D 0; : : : ; c 1. Then, under thehypotheses of Corollary 1 on the asymptotic criterion (5), there exists ı > 0 suchthat, if a 2 Sc satisfies jjajj < ı, any solution x.t/ of the hom*ogeneous impulseextension equation induced by .'; a/, dx A.t/x; t … S.a/ D (6) dt A.t/x C 'kB .t; ak /x. k /; t 2 Sk .a/:
can be written as a product,
x.t/ D Ua .t/x0 D Pa .t/ea t x0 ; (7)
for some x0 2 Rn , T-periodic matrix Pa .t/, and nonsingular a . Ua can benormalized so that Ua . 0 / D I, and in this case, we have a ! 0 as a ! 0, where0 is the matrix appearing in the Floquet decomposition, X.t/ D U0 .t/ D P0 .t/e0 t ,of the hom*ogeneous equation associated to the periodic impulsive differentialequation, (1), with U0 . 0 / D I. 202 K.E.M. Church
The stability of the periodic linear impulse extension equation induced by some.'; a/ is determined by the spectrum of a , just as with ordinary and impulsivedifferential equations. The main difference is that stability (and uniform stability)only holds for initial conditions in particular subsets of the predictable set, P, andsuch restrictions are in fact, optimal. For details, see [4].
3 The Time-Scale Tolerance for Linear, Periodic Impulsive Differential Equations
Stability of (3) is completely determined by the associated hom*ogeneous equa-tion (6); see [4]. As such, our investigation will now shift to hom*ogeneous,.T; c/-periodic impulsive differential equations,
dx D A.t/x; t ¤ k dt (8) x D Bk x; t D k :
and impulse extension equations for (8), induced by .'; a/ D .' B ; a/, with a 2 Sc , dx A.t/x; t … S.a/ D (9) dt A.t/x C 'kB .t; ak /x. k /; t 2 Sk .a/:
From here onward, M0 will denote the monodromy matrix for (8) satisfyingM0 D X. 0 CT; 0 /, where X.t; s/ is the Cauchy matrix for (8). We assume M0 < 1from here onward. We will comment in Sect. 3.3 on what can be done if M0 1.Definition 5 Consider a periodic hom*ogeneous impulsive differential equation, (8).Let D fk g be a c-element sequence of positive real numbers and w D fwk g bea c-element sequence of functions wk W Œ k ; kC1 Sc ! RC that are continuousand vanishing at . k ; 0/ and such that wk .; a/ is integrable on Sk .a/. A family ofperiodic impulse extensions, ' D f'k g, is uniformly exponentially .; w/-regulatedin the mean or simply .; w/-regulated if the inequality ˇˇ ˇˇ ˇˇ ˇˇ ˇˇ'k .s; a/ 1 Bk ˇˇ wk .s; a/ (10) ˇˇ ak ˇˇ ek ak 1
is satisfied, for all s 2 Sk .a/ and k D 0; : : : ; c 1. A pair .; w/ that satisfiesthe above criteria will be referred to as a uniform exponential regulator, or simplyexponential regulator. If ' is uniformly .; w/-regulated, we will write ' 2 .; w/. A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential. . . 203
Definition 6 If R D .; w/ is an exponential regulator, the .R; a/-pseudospectralradius of (8), denoted .R; a/, is defined by
.R; a/ D sup M.'; a/; (11) '2R
and M.'; a/ denotes the monodromy matrix of the impulse extension equationfor (8) induced by .'; a/.Definition 7 Suppose (8) is asymptotically stable. Let R be an exponential regula-tor. The R-stable set, denoted E s .R/, is defined as follows. ˚ E s .R/ D a 2 Sc W 8' 2 .; w/, M.'; a/ < 1 (12)
The R-time-scale tolerance is the number
E t .R/ D supf W 9a 2 E s .R/; jjajj D ; B .0/ \ Sc E s .R/g: (13)
The time-scale tolerance is defined precisely so that we have the followingelementary property, whose proof we omit.Proposition 1 Given an exponential regulator R D .; w/, the time-scale tolerancebehaves as a robust stability threshold for the impulsive system (1); if jjajj < E t .R/,then .R; a/ < 1. In other words, systems (8) and the impulse extension equation (6)induced by .'; a/ are both stable, for all ' 2 R.Theorem 3 Suppose D jjjj, as in Corollary 1. If R D .; w/ is an exponentialregulator and (8) is asymptotically stable, then E t .R / is nonzero and the map a 7!.R ; a/ satisfies
lim .R ; a/ D M0 ; a!0
where the limit is for a 2 Sc .Proof (Outline) The monodromy matrix for (6) induced by .'; a/ for ' 2 R canbe written as 0 Y Z M.'; a/ D X. kC1 I k / X 1 .sI k /k .s; a/ds kDc1 Sk .a/ Z 1 C I C X 1 .sI k /Bk ds ; ak Sk .a/
with k .s; a/ D 'k .s; a/ a1k Bk . Taking norms, each of the k terms can be boundedby inequality (10), and the upper bound is independent on the explicit choice of ',depending only on the regulator R . With the choice of given in the theorem, each 204 K.E.M. Church
intergral involving K converges to zero, while the other clearly converges to I C Bk .Therefore, M.'; a/ ! M0 uniformly for ' 2 R . The result follows. In practice, computing the time-scale tolerance is difficult. We can, thankfully,resort to conservative estimates.Theorem 4 Let R be an exponential regulator for (8). Suppose there exists acontinuous function n W Sc ! RC such that n.0/ D 0 and
jjM.'; a/ M0 jj n.a/
for all a 2 Sc and ' 2 R.1. Let M denote the -pseudospectral radius of the matrix M. The following inclusion is valid:
Ebs .R/ WD fa 2 Sc W n.a/ M0 < 1g E s .R/:
2. Let h > 0 denote the unique solution of the equation h M0 D 1. The inequality
Ebt .R/ WD supfjjajj W a 2 Bjjajj .0/ \ Ebs .R/ ¤ ;g E t .R/
is valid, and if n is monotone increasing, we have Ebt .R/ D minfjjajj W n.a/ D h; a 2 Sc g:Proof (Outline) By definition of the pseudospectral radius, we have
n.a/ M0 D supf.Z/ W Z 2 Rnn ; jjZ M0 jj n.a/g supfM.'; a/ W ' 2 R, jjM.'; a/ M0 jj n.a/g D supfM.'; a/ W ' 2 Rg D .R; a/;
which demonstrates the set inclusion. As for the inequality, that the supremum termis bounded by E t .R/ is obvious from the set inclusion. That Ebt .R/ is achieved atsome a for which n.a/ D h can be seen by noticing that, as n is continuous andincreasing, the set Ebs .R/ is star convex with basepoint 0. Consequently, maximizingthe radius of a ball in the positive orthant within this set is equivalent to minimizingthe distance to the boundary, and the latter is is precisely the level set n.a/ D h.Corollary 2 Denote X.t/ D X.tI 0 /. If c D 1, the following inequality holds forall ' 2 R D .; w/. Z w0 .s; a/ jjM.'; a/ M0 jj jjX. 1 /jj jjX 1 .s/jj ds S0 .a/ e a0 1 ˇˇ Z ˇˇ ˇˇ 1 ˇˇ C ˇˇˇˇ .X .s/ I/dsB0 ˇˇˇˇ : 1 (14) a 0 S0 .a/ A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential. . . 205
Proof (Outline) Z 1 1 M.'; a/ M0 D X. 1 / I C X 1 .s/ '.s; a/ B0 C B0 ds S0 .a/ a0 a0 X. 1 /ŒI C B0 :
Re-arranging the above, taking norms and using inequality (10) provides the result. If c ¤ 1, a similar estimate to the above holds. However, it is rather cumbersome,and the associated proof is a notationally difficult inductive argument. It is omittedfor brevity.
3.1 Example: An Exact Computation for a Scalar Equation
Consider the following scalar impulsive differential equation
x0 D x; t ¤ kT (15) x D bx; t D kT;
with parameters > 0, b > 0 and T > 0. Assume M0 D .1 b/e T < 1; so that the 1trivial solution is asymptotically stable. We choose w.t; a/ D c Ta p for parametersc and p > 0. The bound on the right-hand side of (14), denote nQ .a/, itself has anupper bound: e T a 1p T 1 e a nQ .a/ n.a; p/ WD c Ce b 1 ; T a
and n is strictly increasing in both a and the parameter p. Therefore, c b n.a;p/ n.a;1/ D lim n.a;p/ M0 D e T 1 C .1 e a / (16) p!1 a
for each finite p > 0. Solving the equation n.a ;1/ M0 D 1 for a and applyingTheorem 4, the following theorem is proven. Theorem 5 Consider the impulsive system (15). Define u WD 1b e T 1 c . IfM0 WD .1 b/e T < 1 and c < .1 M0 /e T , then, for all a > 0 satisfying theinequality 1 1 1 1 a< W e u WD a ; u u 206 K.E.M. Church
1we have M.'; a/ < 1, for all ' 2 .; w/, with w.t; a/ D c Ta p , for any p > 0,where W is the principal branch of the Lambert W function, or product logarithmfunction (i.e. the inverse of the map x 7! xex ).
3.2 Example: Control of a Pest with Age Structure
Consider the following system of impulsive differential equations. 0 10=21 5=7 X D X; t ¤ k (17) 1=4 1=7 0:7 0 X D X; t D 2k (18) 0 0:4 0 0 X D X; t D 2kC1 ; (19) 0 0:7
with sequence of impulses k D 7bk=2c C .k mod 2/ and t in units of days. Thecontinuous dynamics, (17), could describe, for example, the population of somepest organism, X D .X1 ; X2 / 0, with juvenile (X1 ) and adult (X2 ) life stages. Withboth impulsive controls included, (17), (18) and (19) is asymptotically stable, withdominant Floquet multiplier equal to 0:4200. If one or both controls are neglected,however, the trivial solution is unstable, so both controls are necessary to control thepopulation. Figure 1 provides visualizations of the subsets Ebs E s described in Theorem 4,of the R-stable sets for two uniform exponential regulators for the system (17), (18)and (19). The first regulator, which generates the smaller of the two stable sets (red pin the figure), is R D .; ak /. The second regulator is R D .; ak /. For bothregulators, D jjjj, as in Corollary 1.
3.3 Limitations
There are two main limitations of the techniques described in these proceedings.First and foremost, only linear systems are treated. The time-scale tolerance canindeed be defined for nonlinear systems of impulsive differential equations in moreabstract settings, although the definitions must all be localized around periodic orbitsor other stable objects. Some of our current research concerns these problems. A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential. . . 207
Fig. 1 Conservative approximations, Ebs , of the R-stable sets for two uniform exponential regula-tors. Arrows indicate the associated lower bounds for time-scale tolerances
Second, we treated only impulsive systems that are asymptotically stable. Thesetechniques generally fail in the presence of a center subspace; see Example 3.5.6of [2]. However, a similar approach does work if there is an unstable subspace. Forexample, if M0 > 1, one might want to know conditions on a 2 Sc under whichM.'; a/ > 1; for all ' 2 R, with R some suitable set of impulse extensions. If, forsome continuous function n satisfying n.0/ D 0, we have jjM.'; a/ M0 jj n.a/for all ' 2 R, one can verify the string of inequalities ˚ inf M.'; a/ inf M W jjM M0 jj sup jjM.'; a/ M0 jj '2R '2R
D inffM W jjM M0 jj inffx W jjM.'; a/ M0 jj x; 8' 2 Rgg inffM W jjM M0 jj n.a/g WD n.a/ M0
holds. Therefore, an appropriate lower estimate for the analoguous time-scaletolerance can be found by solving the optimization problem
Ebt .R/ WD supfjjajj W a 2 Bjjajj .0/ fa 2 Sc W n.a/ M0 > 1g ¤ ;g: 208 K.E.M. Church
References
1. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Appli- cations. Longman Scientific & Technical, Burnt Mill (1993)2. Church, K.: Applications of impulsive differential equations to the control of malaria outbreaks and introduction to impulse extension equations: a general framework to study the validity of ordinary differential equation models with discontinuities in state. M.Sc Thesis, University of Ottawa (2014)3. Church, K.E.M., Smith?, R.J.: Analysis of piecewise-continuous extensions of periodic linear impulsive differential equations with fixed, strictly inhom*ogeneous impulses. Dyn. Contin. Discret. Impuls. Syst. Ser. B: Appl. Algorithms 21, 101–119 (2014)4. Church, K.E.M., Smith?, R.J.: Existence and uniqueness of solutions of general impulse extension equations with specification to linear equations. Dyn. Contin. Discret. Impuls. Syst. Ser. B: Appl. Algorithms 22, 163–197 (2015)5. Lashmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific Publishing, Singapore (1989)6. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific Publish- ing, Singapore (1995) Coupled Lattice Boltzmann Modelingof Bidomain Type Models in CardiacElectrophysiology
S. Corre and A. Belmiloudi
Abstract In this work, a modified coupling Lattice Boltzmann Model (LBM) insimulation of cardiac electrophysiology is developed in order to capture the detailedactivities of macro- to micro-scale transport processes. The propagation of electricalactivity in the human heart is mathematically modelled by bidomain type systems.As transmembrane potential evolves, we take into account domain anisotropicalproperties using intracellular and extracellular conductivity, such as in a pacemakeror an electrocardiogram, in both parallel and perpendicular directions to thefibers. The bidomain system represents multi-scale, stiff and strongly nonlinearcoupled reaction-diffusion models that consists of a set of ordinary differentialequations coupled with a set of partial differential equations. Due to dynamicand geometry complexity, numerical simulation and implementation of bidomaintype systems are extremely challenging conceptual and computational problemsbut are very important in many real-life and biomedical applications. This papersuggests a modified LBM scheme, reliable, efficient, stable and easy to implementin the context of such bidomain systems. The numerical results demonstrate theeffectiveness and accuracy of our approach using general methods for bidomain typesystems and show good agreement with analytical solutions and numerical resultsreported in the literature.
1 Introduction
Computational cardiac electrophysiological modeling is now an important field inapplied mathematics. Indeed, nowadays, heart and cardiovascular diseases are stillthe leading cause of death and disability all over the world. That is why we needto improve our knowledge about heart behavior, and more particularly about itselectrical behavior. Consequently we want strong methods to compute electricalfluctuations in the myocardium to prevent cardiac disorders (as arrhythmias), orto study interactions between brain and heart. We modelize electrical behavior
S. Corre • A. Belmiloudi ()UEB-IRMAR, Rennes, Francee-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 209J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_20 210 S. Corre and A. Belmiloudi
of the heart in the myocardium with the bidomain system, derived from Ohm’slaw. This biophysical model of electrical cardiac activity links electrophysiologicalcell models, at small scales, and myocardial tissue mechanics, metabolism andblood flow at large scales. In mathematical viewpoint, bidomain system leads usto compute intracellular and extracellular electrical potentials i and e with takinginto account the cellular membrane dynamics U. This is a system of non-linearpartial differential equations (PDEs) coupled with ordinary differential equations(ODEs). The PDEs describe the propagation of the electrical potentials and ODEsdescribe the electrochemical processes. During last years a lot of studies aboutbidomain models have led to results about well-posedness, existence and uniquenessof solutions (see e.g., [2, 3] and the references therein), and several numericalmethods based on methods as finite difference method or finite element methodare used to solve these models (see e.g., [8] and the references therein). In thispaper, we propose a modified Lattice Boltzmann Method (LBM) which is simpleto implement, effective, accurate and well suited to bidomain systems which is acoupled nonlinear parabolic/elliptic PDEs. LBM is based on microscopic modelsand mesoscopic kinetic equations. Indeed, traditional numerical methods as finitedifference method or finite element method directly solve governing equationsfor deriving macroscopic variable, whereas LBM is based on the particle (thediscrete) distribution function and numerical solving the continuous Boltzmanntransport equation. Then the macroscopic variables of the bidomain system can berecovered from the discrete equations through the multi-scaling Chapman-Enskogexpansion procedure. LBM was originated from Boltzmann’s kinetic theory ofgases (1970s), and attracts more and more attentions for simulating complex fluidflows since 1990s. More recently, LBM has been extended successfully to simulatedifferent types of parabolic reaction-diffusion equation as Keller-Segel chemotaxismodel [10] and monodomain model in cardiac electrophysiology [4], or Poissonequation [5]. This paper is organized as follows: in Sect. 2 we recall briefly the derivationof the bidomain model. In Sect. 3 we present and describe the modified LBMmethod. In Sect. 4, the validity of this method is demonstrated by comparingthe numerical solution to the exact solution of bidomain model with a classicalFitzHugh-Nagumo model (FHN), and convergence of solution is established. Someinteresting numerical simulations to analyze the influence of some parameters onelectrical wave propagation, including the bidomain model with a modified FHNmodel, are also carried out in this section. This paper is ended by a conclusion andsome further works.
2 The Bidomain Model
The bidomain model of cardiac tissue is expressed mathematically by the followingsteady-state of coupled partial differential equations governing the electrical poten-tials (in the physical region ˝ occupied by the excitable cardiac tissue, which is an LBM for Bidomain Type Models 211
open, bounded, and connected subset of Rd , d 3 and during a time interval .0; T/) div.K i ri / D Im fis ; div.K e re / D Im fes ; (1)
where i and e are the intracellular and extracellular potentials, respectively; K i .x/and K e .x/ are the conductivity tensors describing the anisotropic intracellular andextracellular conductive media; fis .x; t/ and fes .x; t/ are the respective externallyapplied current sources. The transmembrane current density is described by Im andis given by the following expression: @ Im D .cm C I ion /; (2) @twhere is the transmembrane potential, which is defined as D i e , is theratio of the membrane surface area to the volume occupied by the tissue, cm term isthe transmembrane capacitance time unit area. The tissue is assumed to be passive,so the capacitance cm can be assumed to be not a function of the state variables.The nonlinear operator I ion .x; tI ; U/ describes the sum of transmembrane ioniccurrents across the cell membrane with U the electrophysiological ionic statevariables (which describe e.g., the dynamics of ion-channel and ion concentrationsin different cellular compartments). These variables satisfy the following ODE (withH a nonlinear operator) @U D H.x; tI ; U/: (3) @tFrom (1), (2) and (3), the bidomain model can be formulated in terms of the statevariables , e and U as follows (in Q D ˝ .0; T/) @ .cm C I ion .:I ; U// div.K i r/ D div.K i re / C fis ; @t div..K e C K i /re / D div.K i r/ C . fes C fis /; (4) @U D H.:I ; U/: @tThe operators I ion and H which describe electrophysiological behavior of thesystem have usually the following form (affine functions with respect to U) I ion .:I ; U/ D I1 .:I / C I2 .:I /U; H.:I ; U/ D H0 .:I / C .:/U: (5)
To close the system we impose the following boundary conditions
.K i r. C e //:n D i ; .K e re /:n D e on ˙ D @˝ .0; T/; (6)
where n being the outward normal to D @˝ and i and e are the intra- and extra-cellular currents per unit area applied across the boundary, and the following initialconditions (in ˝)
.t D 0/ D 0 ; U.t D 0/ D U0 : (7)
Such problems have compatibility conditions determining whether there are anysolutions to the PDEs. This is easily found by integrating the second equation of (4) 212 S. Corre and A. Belmiloudi
over the domain and using the divergence theorem with the boundary conditions (6)(a.e. in (0, T)). Then (for compatibility reasons), we require the following condition Z Z .i C e /d C . fes C fis /dx D 0: (8) ˝
Moreover, the function e is defined within a class of equivalence, regardless ofa time-dependent function. This function can be fixed, for example by setting theGauge condition (a.e. in (0, T)) Z e dx D 0: (9) ˝
Under some hypotheses for the data and parameters of the system and some regular-ity of operators I ion and H, system (4) with (6)–(7) and under the conditions (8)–(9)is a well-posed problem (for more details see [2]).
3 Numerical Method and Algorithm
In this section, a numerical method is presented for the bidomain system (4) in twospace dimensions. For this, we introduce a coupled modified LBM for solving thecoupled system of nonlinear parabolic and elliptic equations (i.e. the first and thesecond equations of (4)). Then we treat the ODE satisfied by ionic state by applyingGronwall Lemma to obtain an integral formulation, and by using a quadrature ruleto approximate the obtained integral. In the sequel, without loss of generality, weassume cm D 1 and D 1. Moreover we assume K i D Ki Id , K e D Ke Id , with Ididentity matrix and Ki , Ke constants.Remark 1 The developed LBM method has been constructed to take into accountthe case in which K i D K i .; e / and K e D K e .; e /. In order to simplifythe presentation, we have assumed in this paper that these operators are constantmatrices.
3.1 LBM for Coupled Parabolic and Elliptic Equations
In this first part, we develop and describe the modified LBM to solve the followingsystem (which corresponds to two first parts of (4))
@ div.Ki r. C e // D F.:I ; e /; @t (10) div.Ki r C .Ki C Ke /re / D G.:I ; e /;
where F and G are non linear operators. LBM for Bidomain Type Models 213
3.1.1 LBE for General Reaction-Diffusion Equations
To begin, we introduce the LBM to solve the following reaction-diffusion equationwith the macroscopic variable ˚ @ ˚.x; t/ div.Kr˚.x; t// D H.x; tI ˚/: (11) @tThe evolution equation of the LBM for (11) is given by @ h.x; tI e/ C e rh.x; tI e/ D Q.h.x; tI e// C P.x; tI e/; (12) @twhere h.x; tI e/ is the distribution function of particle moving with velocity e atposition x and time t, P is the distribution type function of particle of macroscopicexternal force H moving with velocity e and Q is the Bhatnagar-Gross-Krook (BGK)collision operator defined by Q.h/ D 1 .h.x; tI e/ heq .x; tI e// ; where heq isthe equilibrium distribution function and is the dimensionless relaxation time.LBM leads us to approximate (12) to recover reaction-diffusion equation (11) withChapman-Enskog expansion. For that, we discretize Q in time and space. Thenwe introduce a lattice size cell x and a lattice time step size t, and we definestreaming lattice speed c D x= t. The D2Q9 lattice, which involves 9 velocity vectors, is considered for appliedlattice scheme, which is the most used scheme for two-dimensional model. Figure 1shows a typical lattice node of D2Q9 model with velocities ei for various directionsdefined by 0 cos .i 1/ 2 p cos .i 92 / 2 e0 D ; eiD1;2;3I4 D c ; eiD5;6;7;8 D 2c : 0 sin .i 1/ 2 sin .i 92 / 2
Fig. 1 Particle velocities forD2Q9 LBM 214 S. Corre and A. Belmiloudi
Fig. 2 streaming process of a lattice node
For each particle on the lattice, we associate the discrete distribution functions hi eqand hi (in the mesoscopic level), and the discrete operator Hi of H for i D 0; : : : ; 8.Then, the form of the Lattice Boltzmann Equation (LBE) with an external force byintroducing BGK approximations can be written as follows 1 eq hi .x C ei t; t C t/ D hi .x; t/ hi .x; t/ hi .x; t/ t2 @ C tHi .x; t/ C H .x; t/: 2 @t i (13)
The key steps in LBM, which is directly derived from LBE (13), are the collisionand streaming processes (shown on Fig. 2) which are given by
1 eq t2 @ i .x; t/ D hi .x; t/ hi .x; t/hi .x; t/ C tHi .x; t/C 2 @t Hi .x; t/; hcol (14) hi .x C ei t; t C t/ D hcol i .x; t/: (15)
From Chapman-Enskog expansion analysis, the above LBM can recover to the eqreaction-diffusion equation (11) if we take hi D wi ˚, Hi D wi H and the initial eqdistribution at t D 0: hi .x; 0/ D hi .x; 0/. This analysis is based on the followingproperties:
8 X 8 X eq hi .x; t/ D hi .x; t/ D ˚.x; t/.macroscopic variable/; iD0 iD0 X8 8 X 8 X eq Hi .x; t/ D H.x; t/; ei hi .x; t/ D 0; ei Hi .x; t/ D 0; (16) iD0 iD0 iD0 X8 X8 eq c2 c2 ei ei hi .x; t/ D ˚.x; t/Id ; ei ei Hi .x; t/ D H.x; t/Id : iD0 3 iD0 3 LBM for Bidomain Type Models 215
Remark 2 During streaming and collision processes, in order to satisfy boundaryconditions, the boundary nodes need special treatments on distribution functions,which are essential to stability and accuracy of the method.
3.1.2 Coupled Modified LBM
To introduce our modified LBM, we have to take into account the coupled termswhich link reaction-diffusion equation and elliptic equation in the system (10). Asin [10], in order to take into account this coupling, we introduce two correction terms Si D wi .1 .Ki r/ C 2 .Ki re // and Sie D wi 1e .Ki r/ C 2e ..Ki C Ke /re / ,where the functions 1 ; 2 ; 1e and 2e are determined by Chapman-Enskog expan-sions. Then, we can solve the reaction-diffusion equation with a first LBE where thedistribution function f leads to recover . We construct exactly the same LBM thandeveloped in Sect. 3.1.1. So we choose fi D wi , Fi D wi F and D c3K 1 eq 2 t C 2 to i
satisfy previous properties (16). Finally, we add the corrector term Si as follows
1 eq fi .x C ei t; t C t/ D fi .x; t/ fi .x; t/ fi .x; t/ C tFi .x; t/ t2 @ C Fi .x; t/ C tSi .x; t/: (17) 2 @t
P 8Hence, the macroscopic variable , defined as: fi .x; t/ D .x; t/: iD0 For the elliptic equation, the LBM developed is based on the LBM employed in[5]. The first step is to introduce a new time variable r as lim Qe .x; rI t/ D e .x; t/. r!1Then, the equilibrium distribution function is defined as eq wi Qe .x; rI t/ for i ¤ 0; gi .x; r/ D .w0 1/Qe .x; rI t/ for i D 0
P 8 eq 1 P 8 eqand we can deduce that gi .x; r/ D 0; 1w0 gi .x; r/ D Qe .x; rI t/: Finally, as for iD0 iD1previous LBE (17), we add the corrector term Sie and we obtain the following LBE(to recover e )
1 eq gi .x C ei r; r C rI t/ D gi .x; rI t/ gi .x; rI t/ gi .x; rI t/ e C tGi .x; rI t/ C tSie .x; rI t/: (18)
8 1 XHence, the macroscopic variable Qe , defined as: gi .x; rI t/ D Qe .x; rI t/: 1 w0 iD0 216 S. Corre and A. Belmiloudi
3.2 Treatment of ODE
Now, we present briefly the method to solve ODE satisfy by ionic state U, withinitial condition U.x; 0/ D U0 .x/. According to (4) and the form of H given in (5)by H.x; tI ; U/ D H0 .x; tI / C U.x; t/, with assumed to be a constant, and byusing Gronwall Lemma we can deduce:
Z tC t t t U.x; t C t/ D U.x; t/e Ce H0 .x; sI .s//es ds: t
Then, according to approximation of derived integral by trapezoidal methodbetween t and t C t we obtain the following approximation of U denoted alsoby U
t U.:; t C t/ D U.:; t/e t C H0 .:; t C tI .t C t//e t CH0 .:; tI .t// : 2 (19) Finally, after non-dimentionalization, mesh definition and initialization of initialconditions, parameters and data, the proposed algorithm to solve the bidomainsystem can be summarized as follows1. Initialization: tD0.2. LBE according to time t by using (17) to compute .x; t C t/.3. Trapezoidal method by using (19) to compute U.x; t C t/.4. Loop on new time variable r: a. LBE according to time r by using (18) to compute Qe .x; r C rI t C t/ an approximation of e .x; t C t/ . b. If convergence criteria is not reached, set r D r C r and go back to 4a.5. Set e .x; t C t/ WD Qe .x; rI t C t/.6. If t ¤ T, set t WD t C t and go back to 2.
4 Numerical Simulation and Applications
To validate the capacity of our modified coupled LBM to deal with 2D bidomainsystems, several situations are numerically simulated. In the first study, we considerthe bidomain system with hom*ogeneous Neumann boundary conditions and with aclassical FitzHugh-Nagumo model (FHN), in which non linear operators are definedas
I D 1=˛1 . 3 =3 U/; H D ˛2 . ˇ1 U C ˇ2 /; (20) LBM for Bidomain Type Models 217
where ˛1 ; ˛2 ; ˇ1 ; ˇ2 are positive constant parameters respectively called excitationrate constant, recovery rate constant, recovery decay constant and excitation decayconstant. First, we consider the system with known analytical solution in orderto validate and verify the accuracy and stability of the method. And second, weconsider the system as in [8] and we estimate the influence of both constants ˛1 and˛2 on the behaviour of the system. In the last application we solve and analyze thesystem with the following modified FHN model (see e.g., [8])
I D 0:0004. C 85/ .U C . C 70/. 40// ; H D 0:63. C 85/ 0:013U: (21)Nota Bene: If the exact solution sol is known, we can measure the efficiency of ksol kL2 .˝/method with the following L2 relative error: Err D ksol kL2 .˝/ : For simplicity, we assume that the domain is a square region ˝ D Œ0I 1 Œ0I 1 ,the final time is fixed T D 1, and the cell surface to volume ratio D 1.
4.1 Benchmark Problem and Validation
In this first analysis, we investigate the accuracy and spatial convergence rateof the proposed modified LBM for which we postulate that the error estimatesof the method is of order 2 in space and of order 1 in time (for sufficientlyregular solution). We perform a convergence study on cartesian grid, by taking˛1 D 1, ˛2 D 1, ˇ1 D 1, Ki D 1 and Ke D 1, and by setting ˇ2 .x; y; t/ Dsol .x; y; t/ C 2et cos..x C y//: The initial conditions are .x; 0/ D 0; e .x; 0/ D0; U.x; 0/ D cos..x C y// and we choose I , Ii and Ie to close the problemaccording to solution and compatibility conditions. The exact solution is given by:sol .x; y; t/ D tx2 .x 1/2 y2 .y 1/2 ; esol .x; y; t/ D t.cos.x/ C cos.y//; andU sol .x; y; t/ D et cos..x C y//. To study the convergence, we have constructeda sequence of meshes with decreasing spatial step x between 1=30 and 1=200 and t D x2 . We just care about relative error on and U at t D 0:5 and t D 1(see Table 1). Indeed, the chosen convergence criteria (for the iterative method to
Table 1 Relation between relative error and lattice spacing for and Ut = 0.5 x Err ErrU t=1 x Err ErrU 1=30 0.1914 0.0005 1=30 1.7618 0.0009 1=50 0.0389 0.0002 1=50 0.4559 0.0003 1=70 0.0116 7.97e5 1=70 0.1654 0.0001 1=100 0.0050 4.48e5 1=100 0.0790 7.92e5 1=150 0.0016 1.99e5 1=150 0.0284 3.52e5 1=200 0.0007 1.12e5 1=200 0.0138 1.99e5 218 S. Corre and A. Belmiloudi
Fig. 3 Error curves for and U
approach e ) involves a constant error because this criteria is not defined in functionof lattice size cell. We present on Fig. 3 (at t D 0:5 and t D 1) the convergencecurves, log.Error/ versus x, for and U. We observe that the slope of errorcurves for passes approximately from 3 to 2:5 and the slope of error curves forU is approximately equal to 2. This shows that our numerical error estimates areagree with the postulated error estimates, and indicate the good performance of ourproposed method.
4.2 Influence of Some Parameters on Electrical Wave Propagation
This second computations have been made to test performance of our methodby analyzing the influence of some parameters on electrical wave propagationaccording to FHN models: a classical model (20) and a modified model (21). Here,we take x D 1=50 and t D x2 . LBM for Bidomain Type Models 219
4.2.1 First Data Setting
In this first application, we consider the classical FHN model (20) with fixed ˇ1 D0:5 and ˇ2 D 1 and we study the effect of parameters ˛1 and ˛2 . We assume Ki DKe D 1 and we consider the following initial conditions .x; y; 0/ D 1:28791 Csin.x/; e .x; y; 0/ D 0; U.x; y; 0/ D 0:5758: The parameters ˛1 and ˛2 control the dynamic between transmembrane potential and the ionic state U. In order to study the propagation of an electrical wavethrough the cardiac tissue, we analyze the relaxation time of . In Table 2, weobserve that ˛2 has negligible impact on this relaxation time. Conversely, Fig. 4 andTable 2 show that tiny ˛1 value leads to near-infinite slope values when potential peaks and relaxes.
Table 2 Relaxation time of according to ˛1 and ˛2 variations˛1 ˛2 Relaxation time ˛2 ˛1 Relaxation time 0.01 0.4700 0.01 0.03960.2 0.05 0.4704 0.2 0.05 0.1568 0.1 0.4716 0.1 0.2764 0.5 0.4988 0.5 0.9168
Fig. 4 .0:5; 0:5; t/ with ˛1 D 0:01 to 0:5, ˛2 D 0:2 220 S. Corre and A. Belmiloudi
60 4 −10 40 −20 3 20 −30 2 0
ρ(x, y, 1)ρe (x, y, 1)
U (x, y, 1) −40 −20 −50 1 −40 −60 0 −60 −70
−1 −80 −80 1 0 1 1 1 0.5 0 0.5 0.5 0.5 0.5 0.5 y 0 0 x 1 1 y 0 0 x y x
Fig. 5 Evolution of e , and U at t D 1 with x D 1=50
4.2.2 Second Data Setting
In this final application, we consider the modified FHN model (21) which is knownto be more adapted than the classical model (20) because of a product between and U. We assume Ki D 1:75 and Ke D 7 and we consider the following initialconditions: .x; y; 0/ D 85 C 100.1 sin.xy//; e .x; y; 0/ D 0; U.x; y; 0/ D 0:Figure 5 shows a color visualization of the simulation for , e and U at final timet D T. The analyze we have done before is still valid.Remark 3 It is known that the action potential duration (APD) and conductionvelocity (CV) could be significantly affected by variation in the values of FHNparameters. So, it will be interesting to perform comparison of the APD and CV forthe various FHM ionic-type models (and others) coupling the bidomain model, butalso to understand how sensitive APD and CV are to variability in these parameters.
5 Conclusion and Commentary
An efficient and stable coupled LBM to solve a two-dimensional bidomain modelof cardiac tissue is developed. From the Chapman-Enskog expansion analysis,the bidomain system which is a coupled of reaction-diffusion, elliptic and EDOequations, can be correctly recovered by our modified LBM. This method is easy toimplement and easy to parallelize. It is clear that, due to the multi-scale nature of thesystem, the cartesian grid used in our preliminary simulations is not very sufficientto compute in a computationally efficient manner real life clinical situation withcomplex geometry which is in general computationally expensive. Therefore, it isexpected to solve the Lattice Boltzmann system on adapted cartesian or triangularunstructured grids as e.g., in [6, 9] and the references therein. Moreover, in orderto overcome the limitations of the constraint CFL stability condition, we extendthe method to implicit or semi-implicit time schemes, e.g., by using the -method(with 2 Œ0; 1 ) or Runge-kutta methods, coupled with adaptive time steppingstrategies, as e.g. in [7] and the references therein. This coupled LBM method will LBM for Bidomain Type Models 221
be shown in a forthcoming paper for more general coupled models with realisticcomplex geometries. It would be interesting to use this developed method withobservations coming from experimental data and a more complete description ofthe biophysical model of electrical cardiac activity. In order to get even closer to amore realistic calculation, it is necessary to study in the future this method coupledwith optimization technique and robust control problems by using the approachdeveloped in [1].
Acknowledgements The authors are grateful to the referee for many constructive comments andsuggestions which have improved the presentation of this manuscript.
References
1. Belmiloudi, A.: Stabilization, Optimal and Robust Control. Theory and Applications in Biological and Physical Sciences. Springer, London (2008) 2. Belmiloudi, A.: Robust control problem of uncertain bidomain models in cardiac electrophisi- ology. J. Coupled Syst. Multiscale Dyn. 19, 332–350 (2013) 3. Bourgault, Y., et al.: Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology. Nonlinear Anal-Real 10, 458–482 (2009) 4. Campos, J.O., et al.: Lattice Boltzmann method for parallel simulations of cardiac electrophys- iology using GPUs. J. Comput. Appl. Math. 295, 70–82 (2016) 5. Chai, Z., Shi, B.C.: Novel Boltzmann model for the Poisson equation. Appl. Math. Model. 32, 2050–2058 (2008) 6. Fan, Z., et al.: Adapted unstructured LBM for flow simulation on curved surfaces. In: ACM SIGGRAPH, Los Angeles, pp. 245–254 (2005) 7. Huang, J., et al.: A fully implicit method for lattice Boltzmann equations. SIAM J. Sci. Comput. 37(5), Special Section, S291–S313 (2015) 8. Sharomi, O., Spiteri, R.: Convergence order vs. parallelism in the numerical simulation of the bidomain equations. J. Phys.: Conf. Ser. 385, 1–6 (2012) 9. Valero-Lara, P., Jansson, J.: A non-uniform Staggered Cartesian grid approach for Lattice- Boltzmann method. Procedia Comput. Sci. 51, 296–305 (2015)10. Yang, X., et al.: Coupled lattice Boltzmann method for generalized Keller-Segel chemotaxis model. Comput. Math. Appl. 12, 1653–1670 (2014) Dynamics and Bifurcations in Low-DimensionalModels of Intracranial Pressure
D. Evans, C. Drapaca, and J.P. Cusumano
Abstract Intracranial Pressure (ICP) is a physiological parameter of the brainwhich plays an important role in the diagnosis and treatment of pathologies suchas hydrocephalus and traumatic brain injury. Currently, all reliable methods for ICPmonitoring involve drilling through the skull to place a pressure probe inside thebrain. As a result, ICP is only measured in the most critical cases which requireneurosurgical intervention. Mathematical models that relate ICP to physiologicalparameters whose measurements are minimally invasive could contribute to betterdiagnostic and treatment protocols for brain disorders. Ideally, such mathematicalmodels should have the capability to predict ICP in real time from non-invasivemeasurements of other clinically relevant parameters without the need for high-risk procedures. In this paper, we examine in detail the dynamics and stability of amathematical model proposed by Ursino and Lodi in (J Appl Physiol 82(4):1256–1269, 1997) which predicts ICP from measurements of arterial blood pressure. Westudy how the equilibria vary with the model parameters and, aided by numericalsimulations, we obtain bifurcation diagrams for the system of non-linear ordinarydifferential equations. Expanding upon the work of Ursino and Lodi, we show thatthe model exhibits not only one Hopf bifurcation but also a reverse Hopf bifurcationin certain parameter regimes. In addition, we present global phase portraits of thesystem in interesting parameter configurations.
1 Introduction
Intracranial pressure (ICP) monitoring provides critical physiological informationon brain damage in patients suffering from neurological disorders such as hydro-cephalus and traumatic brain injury (TBI). The only reliable method of measuringICP involves drilling a hole directly through the skull, and inserting a catheter witha pressure probe into the brain [5]. Given the invasiveness of this approach, ICPmonitoring is only performed on patients in very critical conditions during life-saving neurosurgical interventions. Mathematical models that couple ICP dynamics
D. Evans () • C. Drapaca • J.P. CusumanoThe Pennsylvania State University, University Park, PA, USAe-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 223J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_21 224 D. Evans et al.
and other physiological parameters whose monitoring is less invasive provide apowerful alternative to the direct measurement of ICP. In addition, mathematicalmodels that also incorporate brain’s regulatory mechanisms of ICP could proveessential in designing better diagnostic and therapeutic procedures for a wide rangeof neurological diseases. Although the potential exists to create a model that couldbe used to estimate ICP in real time, the current state of the art is still far fromthis goal. A comprehensive review of mathematical models of ICP dynamics can befound by Wakeland and Goldstein [8]. One particular model proposed by Ursino and Lodi [6], predicts ICP dynamicsfrom non-invasive measurements of arterial blood pressure, which appears to bein agreement with the ICP measured in patients with TBI. The model, which wewill refer to as the Ursino-Lodi model, is a generalization of the one-compartmentmathematical model of ICP dynamics pioneered by Marmarou [4] that includesseveral blood compartments, each with variable conductances and compliances, andICP auto-regulation due to blood vessel wall tension, and viscous forces. Since thefocus of the Ursino-Lodi model was its clinical applicability, a full mathematicalanalysis of the model is still missing. Therefore, the aim of this paper is to examinein detail the dynamics and stability of the Ursino-Lodi model. We use numericalsimulations to study how the equilibria vary with the model parameters and obtainbifurcation diagrams for the system of non-linear ordinary differential equations.Our in-depth analysis shows that the model exhibits not only one Hopf bifurcation,as reported in [6], but also a reverse Hopf bifurcation in certain parameter regimes.We also present global phase portraits of the system in interesting parameterconfigurations.
2 Physiological Considerations
The human brain is a multi-phase mixture of many different materials, includingcerebrospinal fluid (CSF), brain cells, and cerebral vasculature. The majority ofwhat is considered brain tissue consists of white matter and grey matter, which areboth soft and mildy compressible materials [3]. CSF is a colorless liquid made of99 % water that fills the brain’s ventricles, the subarachnoid space (space betweenthe two deeper meninges that envelop the brain tissue: pia mater and the arachnoidmater), and intracellular space. The presence of CSF introduces damping to thebrain motion, which cushions and protects it from injury [2]. CSF is constantlyformed by the choroid plexuses of the brain’s ventricles from the cerebral blood,circulates through the brain tissue and spinal cord, and is constantly reabsorbed intothe cerebral veins located in dura mater (above the arachnoid mater) [5]. There existsan approximately constant pressure gradient from the inside of the brain (ventriclescontaining CSF) to the outside (dura matter). CSF is then forced over this pressuregradient over time, allowing it to be reabsorbed by the venous system. The rate ofabsorption is linked to the intracranial pressure as follows. Above a certain value ofICP, the relationship is nearly linear while at very low values of ICP, the absorption Dynamics and Bifurcations in Low-Dimensional Models of Intracranial Pressure 225
rate is negligible [1]. The absorption of CSF back into the bloodstream, and hence,its removal from the intracranial compartment, is the primary natural mechanism forlowering the intracranial pressure. As a result, abnormally high intracranial pressureis sometimes caused by high resistances to resorption. In addition to the brain naturally regulating pressure via the volume of CSF,the brain also contains autoregulatory mechanisms to control the flow of bloodin the cerebral arteries. In order to ensure a constant artery-to-vein difference inoxygen concentration, the brain automatically regulates the cerebral blood flow(CBF) in a process known as cerebral autoregulation. As oxygen consumption,blood pressure, and blood viscosity change, the cerebral arteries dilate or constrictto regulate the flow of blood to the brain. The volume of the cerebral arteries, thecompression of the brain tissue, and the volume of CSF within the skull are allcontrolled by ICP. The linked mechanisms of feedback and control through all ofthese processes have the potential to create instability in the steady-state behaviorof the ICP, most notably in the case of Lundberg A waves. This pathologicalphenomenon is characterized by long, sustained increases in the ICP. The ICP, whichis normally approximately constant, increases dramatically in oscillations withperiods of approximately 5–20 min, and amplitudes of about 50–100 mmHg [5].
3 Ursino-Lodi Model
The Ursino-Lodi mathematical model [6] includes both the ventricular CSF and thecerebral vasculature. The model is made of five compartments: one compartmentfor the ventricular CSF and four compartments for the blood in the cerebral arteries,capillaries, veins, and venous sinus, respectively. In addition, the model providesa relationship between pressures and volumes of cerebral blood and the pressureand volume of the ventricular CSF. Lastly, the model includes equations whichdynamically control the flow of cerebral blood, which account for the effects ofcerebral autoregulation. By modeling the effect of cerebral autoregulation, theUrsino-Lodi model has the capability to predict the oscillatory Lundberg A Waves. Applying the mass conservation principle to the ventricular component yields dPic dVa Pc Pic Pic Pvs Cic D C C Ii ; (1) dt dt Rf Rowhere Cic is the intracranial “compliance”, Va is the blood volume in the cerebralarteries and arterioles, Pc ; Pic ; and Pvs are the capillary, intracranial, and venoussinus pressures, respectively, Rf and Ro are the resistances to CSF formation and,respectively, CSF outflow, and Ii is the rate of externally injected of extracted CSFvolume. As in [4], the nonlinear compliance term Cic is assumed to be related to Picby the following formula:
1 Cic D ; (2) kE Pic 226 D. Evans et al.
where kE is brain’s elastance coefficient (a stiffness-like modulus introduced by [4]).The mass preservation principle for the compartment of cerebral capillaries yieldsthe following equation:
Pa Pc Pc Pic Pc Pic Pc Pic D C ; (3) Ra Rf Rpv Rpv
where Ra and Rpv denote the resistances of the arterial-arteriolar and respectivelyvenular vessels. The approximation made in Eq. (3) is based on the assumption thatthe CSF production rate is much less than the cerebral blood flow rate. By assuming a linear blood volume-pressure relationship Va D Ca .Pa Pic /,where Ca denotes the arterial-arteriolar compliance, the following equation can beobtained through differentiation: dVa dPa dPic dCa D Ca C .Pa Pic /: (4) dt dt dt dt
By replacing Eqs. (2) and (4) into Eq. (1), we obtain: dPic kE Pic dPa dCa Pc Pic Pic Pvs D Ca C .Pa Pic / C C Ii : dt 1 C Ca kE Pic dt dt Rf Ro (5)
Lastly, the model is completed by assuming that the cerebral autoregulation modifiesthe compliance Ca as follows:
dCa 1 D .Ca C .Gx//; (6) dt where is a time constant, G is the maximum autoregulation gain, and
.Can C Ca =2/ C .Can Ca =2/ exp.Gx=k / .Gx/ D ; (7) 1 C exp.Gx=k /
with k D Ca =4. Pa Pc By definition, the cerebral blood flow q is q D and thus the normalized Radeviation of q from its normal value qn is: q qn xD : (8) qn
The parameter Ca represents the maximum allowed change in the arterial com-pliance Ca from its basal value Can . Furthermore, it depends on whether thearterial-arteriolar vessels are contracting (x > 0) or dilating (x < 0) as follows: Ca1 ; if x < 0 Ca D (9) Ca2 ; if x > 0 : Dynamics and Bifurcations in Low-Dimensional Models of Intracranial Pressure 227
By assuming that the arteries are circular cylinders of radius r and the blood isa viscous Newtonian fluid, the Hagen-Poiseuille equation which states that thearterial resistance is inversely proportional to r4 can be used to obtain the followingexpression for Ra : 2 kR Can Ra D 2 : (10) Va
The system of non-linear ordinary differential equations (5) and (6) is the Ursino-Lodi model written in terms of the state variables Pic and Ca .
4 ICP Dynamics Predicted by Ursino-Lodi Model
The clinically measured ICP is a pulsatile periodic waveform which has a periodequal to one heartbeat. However, the Lundberg A waves happen on a time scalethat is much longer than one heartbeat, so in order to observe them we replacethe heartbeat with an averaged pressure over one period which greatly simplifiesthe analysis without loss of applicability. Notably, it ensures that the system ofdifferential equations we work with remains autonomous. The parameters of theUrsino-Lodi model are the variables that are treated as constants with respect totime. A point in the parameter space representing basal parameters accompanies themodel, which is presented as Table 1. In our numerical simulations, all parametersare set to these values unless otherwise indicated. Variations from these parameters
Table 1 Ursino-Lodi Model Parameter ValueBasal Parameters [6] Ro 526:3 mmHg s ml1 Rpv 1:24 mmHg s ml1 Rf 2:38 103 mmHg s ml1 Ca1 0:75 ml=mmHg Ca2 0:075 ml=mmHg Can 0:15 ml=mmHg kE 0:11 ml1 kR 4:91 104 mmHg3 20 s qn 12:5 ml=s G 1:5 ml mmHg1 100 % CBF change1 Pa 100 mmHg Pic 9:5 mmHg Pc 25 mmHg Pvs 6:0 mmHg Ca 0:15 ml=mmHg 228 D. Evans et al.
will be represented as unitless normalized values with respect to the values given inTable 1. The equilibrium points in the model are found at locations where PP ic D CP a D0, where the dot operator indicates the time derivative. Given positive real inputparameters, this system has consistently two equilibrium points in the domain ofPic 0, Ca 0. These two equilibria are marked with stars in Fig. 2b. What wewill refer to as the “primary” equilibrium point in this work is marked “SN” here, forstable node, and what we refer to as the “secondary” equilibrium point is marked“SP” here, for saddle point, which we will discuss shortly. Although the primaryequilibrium point is a stable node for the values of Table 1, we will soon discusshow deviations from these can change its stability. Nevertheless, we treat both equilibria equally when employing the followingtechniques. By slowly varying the initial state of the system by one model parameter(here, either Ro , kE , or G), we obtain a locus of points that correspond to the steadystate value of the system. Using the basal case as an initial guess, we use a root-solver (the MATLAB built-in function fzero) to track the equilibrium values ofPic and Ca as a function of three of the model parameters: Ro , kE , and G, forboth equilibrium points. Because there is more than one root for the equilibriumstate, we kept each solution branch separate by, upon increase of the bifurcationparameter, using the previous solution as the subsequent initial guess for the root-solver. Figures 1 and 2a show how the locations of both equilibria vary as a functionof both Ro and G. We found that varying the intracranial elastance, kE , does notaffect the equilibrium values predicted by the model for either equilibrium point.In addition, the location of the secondary equilibrium point does not change as afunction of Ro . We would like to describe how the stability of these equilibria change withrespect to the model parameters. We therefore linearize the system of equations (5)and (6) about the equilibria to obtain the linear system xP D Jx. Here, x DŒ.Pic Pic /; .Ca Ca / T is the perturbation from the equilibrium value, .Pic ; Ca /,and J is the Jacobian matrix of the two-dimensional vector field defined by the righthand sides of Eqs. (5) and (6). Then, the eigenstructure of J determines the stabilityof the equilibrium point [7]. Confirming what was previously inferred from the phase portrait, the primaryequilibrium point is a stable node, having two negative real eigenvalues in the basalcondition. Since there are no other stable equilibria in the domain, the basin ofattraction is the entire domain Pic > 0, Ca > 0. In addition, we find that thesecondary equilibrium point always occurs at a value of Pic D 0 regardless of otherinput parameters. The eigenvalues of the linearized system near this equilibrium arereal with opposite signs. Therefore, this equilibrium can be classified as a saddlepoint. It is clinically valid that the primary equilibrium is a stable node in the basalcondition. That is, given an initial state (Pic , Ca ), the system will autonomouslyregulate itself to one steady-state value. However, as previously alluded to, therewill be instances where the system no longer converges to a constant steady-statevalue, but instead converges to a stable limit cycle in the phase space, representing Dynamics and Bifurcations in Low-Dimensional Models of Intracranial Pressure 229
(a) 70 (b) 0.5 60 0.45
Ca Root value (ml/mmHg) 50 0.4P Root (mmHg)
0.35 40 0.3 30 0.25 ic
20 0.2 10 0.15 0 0.1 0 5 10 15 20 25 0 5 10 15 20 25 Ro (normalized) R (normalized) o
(c) 34 (d) 0.25 33 Ca Root (ml/mmHg) 0.24 32 0.23 P Root (mmHg)
31 0.22
30 0.21 ic
29 0.2
28 0.19
27 0.18 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 G (normalized) G (normalized)
Fig. 1 Location of the primary equilibrium point. (a) P ic versus Ro , (b) Ca versus Ro , (c) Picversus G, (d) Ca versus G
(a) 0.137 (b) 90 Equilibria Trajectories 80 0.136 70 ic
0.135 Intracranial Pressure: P
0.134 60
0.133 50
0.132 40
0.131 30
0.13 20
0.129 10
0.128 0 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 G (normalized) Arterial Compliance: Ca
Fig. 2 (a) Location of Ca on the secondary equilibrium point as a function of G (P ic is unaffected).(b) Phase portrait of the system in the basal condition as per Table 1. Here, the primary equilibriumpoint is a stable node (SN), while the secondary equilibrium point is a saddle point (SP) 230 D. Evans et al.
the clinically observed case of Lundberg A Waves. Mathematically, the transitionbetween the two cases implies the occurrence of a Hopf bifurcation, where a stablelimit cycle develops around the equilibrium point, which has then lost its stability.The point at which this transition occurs is known as the Hopf bifurcation point,which occurs where the pair of eigenvalues are pure imaginary with zero real part.Therefore, in order to find where this bifurcation takes place, we numerically searchfor cases where the eigenvalues are pure imaginary. We use a fourth-order finitedifference approximation of the Jacobian evaluated at the equilibrium point, andevaluate the eigenvalues of the resulting matrix. These relationships are shown inFigs. 3a–c. If we follow the eigenstructure of the primary equilibrium point as Ro increases(as shown in Fig. 3a), we notice that the equilibrium passes through several stability
(a) (b) 0.15 0.04 0.1 0.02 0.05 0 0 −0.02 λλ
−0.04 −0.05
−0.06 Real λ1 −0.1 Real λ1 Imag λ1 Imag λ1 −0.08 Real λ2 −0.15 Real λ2 Imag λ2 Imag λ2
−0.1 −0.2 0 5 10 15 20 25 1 2 3 4 5 6 Ro (normalized) kE
(c) (d) 1 0.25 Real λ1 0.2 Imag λ1 0 Real λ2 Imag λ2 0.15 −1 Real λ1 0.1 Imag λ1 −2 λ
Real λ2λ
0.05 Imag λ2
−3 0 −4 −0.05
−0.1 −5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 G (normalized) G (normalized)
Fig. 3 Eigenvalue plots of equilibrium solution branches, showing changes in stability withparameter variations. The Hopf bifurcation points are marked with stars. (a) Primary equilibrium: versus Ro . Hopf bifurcation occurs at Ro 9:793, and reverse Hopf bifurcation occurs atRo 17:430. (b) Primary equilibrium: versus kE (with increased Ro D 8) Hopf bifurcationoccurs at kE 2:615. (c) Primary equilibrium: versus G (with increased Ro D 8 and kE D 2:1).Hopf bifurcation occurs at G 2:072. (d) Secondary equilibrium: versus G. The imaginaryparts are not visible in the plot because they are nearly coincident with the real part of 1 , whichtakes very small positive values Dynamics and Bifurcations in Low-Dimensional Models of Intracranial Pressure 231
classifications. The equilibrium begins as a stable node when Ro is unchanged.As Ro increases, the eigenvalues transition from negative and real, to a complexconjugate pair with negative real part, changing the equilibrium to a stable focus.Increase Ro further, and we reach the Hopf bifurcation point, where at this instant,the eigenvalues are a pure imaginary complex conjugate pair. The Hopf bifurcationalso marks the creation of a limit cycle around the equilibrium point. After thebifurcation point, the eigenvalues are still a complex conjugate pair, but this timethey have a positive real part, which means the equilibrium is an unstable nodein this region. A further increase in Ro causes the reverse to take place, takingthe equilibrium back through a reverse Hopf bifurcation, extinguishing the limitcycle, and returning it to a stable focus, and eventually a stable node. It is in theregion between the forward and reverse Hopf bifurcation points that we observe alimit cycle in the system. A slightly similar situation occurs when the bifurcationparameters are kE and G (Fig. 3b, c). While increasing these bifurcation parametersproduces the forward Hopf bifurcation in the same way, further increases do notcause a reverse Hopf bifurcation. Instead, the eigenvalues of the unstable focus meetand become a pair of positive real eigenvalues, indicating the equilibrium’s changeto an unstable node (Fig. 4). For the sake of completeness, we have also analyzed the stability of the saddlepoint equilibrium. However, we have not observed any changes in the stability ofthis equilibrium point. Increasing Ro and kE do not change the eigenvalues at all,and while increasing G changes the values of the eigenvalues, it preserves the overallstructure of two real eigenvalues with differing signs (see Fig. 3d).
(a) 90 Equilibria (b) 110 Equilibria Trajectories Trajectories 80 100 90 ic
ic
70Intracranial Pressure: P
Intracranial Pressure: P
80 60 70 50 60 40 50 40 30 30 UN 20 20 10 10 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Arterial Compliance: Ca Arterial Compliance: C a
Fig. 4 Phase portraits exhibiting limit cycles, where the primary equilibrium is an unstable node(UN), and the secondary equilibrium is a saddle point (SP). (a) Normalized kE D 2:1, Ro D 10,and G D 4. (b) Normalized kE D 8 and Ro D 8 232 D. Evans et al.
5 Conclusions and Further Work
Through the use of numerical simulations, we have shown how varying theparameters G, kE , and Ro in the Ursino-Lodi model changes the value of theequilibrium intracranial pressure and arterial compliance. We have also shown howvarying these parameters can force the primary equilibrium to undergo a Hopfbifurcation, creating a stable limit cycle around the now unstable equilibrium point.In addition to confirming the claims made by Ursino and Lodi, we have also shownthe existence of a reverse Hopf bifurcation at very high levels of Ro To the authors’knowledge, no interesting bifurcation phenomena is witnessed by varying the modelparameters not mentioned here, hence their exclusion from the work. We intend touse the work presented here when developing new models of intracranial pressuredynamics, to offer additional insight on the stability structure of the steady-statesolutions.
References
1. Cutler, R.W.P., Page, L., Galicich, J., Watters, G.V.: Formation and absorption of cerebrospinal fluid in man. Brain 91(4), 707–720 (1968)2. Goldsmith, W.: The state of head injury biomechanics: past, present, and future: part 1. Crit. Rev. Biomed. Eng. 29(5&6), 441–600 (2001)3. Kyriacou, S.K., Mohamed, A., Miller, K., Neff, S.: Brain mechanics for neurosurgery: modeling issues. Biomech. Model. Mechanobiol. 1(2), 151–164 (2002)4. Marmarou, A., Shulman, K., Rosende, R.M.: A nonlinear analysis of the cerebrospinal fluid system and intracranial pressure dynamics. J. Neurosurg. 48(3), 332–344 (1978)5. Suarez, J.I.: Critical Care Neurology and Neurosurgery. Springer, New York (2004)6. Ursino, M., Lodi, C.A.: A simple mathematical model of the interaction between intracranial pressure and cerebral hemodynamics. J. Appl. Physiol. 82(4), 1256–1269 (1997)7. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer, New York (1996)8. Wakeland, W., Goldstein, B.: A review of physiological simulation models of intracranial pressure dynamics. Comput. Biol. Med. 38(9), 1024–1041 (2008) Persistent hom*ology for AnalyzingEnvironmental Lake Monitoring Data
Benjamin A. Fraser, Mark P. Wachowiak, and Renata Wachowiak-Smolíková
Abstract Topological data analysis (TDA) is a new method for analyzing large,high-dimensional, heterogeneous, and noisy data that are characteristic of modernscientific and engineering applications. One major tool in TDA is persistent hom*ol-ogy, wherein a filtration of a simplicial complex is generated from point cloudsand subsequently analyzed for topological features. Betti numbers are computedacross varying spatial resolutions, based on a proximity parameter R, where the n-thBetti number equals the rank of the n-th hom*ology group. In this paper, persistenthom*ology is applied to lake environmental monitoring data collected from a sondesensor attached to a commercial cruise vessel, and to weather station observations.A modified form of the witness complex described by de Silva is used in an attemptto eliminate the need for persistence and thus to reduce computation time. Frompreliminary results, witness complexes are very promising in capturing the shapeof the data and for detecting patterns. It is therefore proposed that TDA, combinedwith standard statistical techniques and interactive visualizations, enable insightsinto observations collected from environmental monitoring sensors.
1 Introduction
Improvements in sensor network technology and sensor mechanisms have facilitatedacquisition of vasts amount of data. In the case of biomedical and environmentalmonitoring, these data tend to be complex, heterogeneous, noisy, unstructured, andhigh-dimensional [3]. Furthermore, data are frequently acquired as long vectors,where only a small number of coordinates are meaningful to specific questionsat hand, but it is often not known a priori which coordinates are relevant [1].Discovering meaningful structural or temporal patterns in these data – that is, toobtain knowledge about the data’s large-scale organization – is very difficult, evenfor domain experts. New, innovative computational methods and visual data miningtools are therefore needed [3]. Among recently proposed techniques, topologicaldata analysis (TDA) is very promising. TDA provides robust feature definitions,
B.A. Fraser () • M.P. Wachowiak • R. Wachowiak-SmolíkováNipissing University, North Bay, ON, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 233J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_22 234 B.A. Fraser et al.
emphasizing the global aspects of the data and shape recognition in point cloudsof high dimensions. Specifically, hom*ology provides the basis for computingsimplicial complexes (a mesh that represents a space in specific ways), in whichthe topology and geometry of the point clouds are separated [10]. Topologicalmethods for data analysis were described in the seminal paper by Carlsson [1] andsubsequently expanded. They are useful in studying connectivity information, suchas the classification of loops and high-dimensional surfaces that may implicitly existin the data. Unlike geometric methods, TDA is not sensitive to properties such ascurvature, and, because the focus is on the global structure of the data, numericalvalues of distance functions are less important. These features allow researchers tostudy complex data where there is an unclear or incomplete idea of what specificmetrics are to be extracted [1]. TDA has been applied to a variety of scientificproblems, most notably in biomedicine, including the organization of genomic datafrom many different sources in the study of breast cancer [6], and for computer-aided diagnosis of pulmonary embolism [7]. TDA is also used for business andsocial analysis [4]. Analytics and advanced computational methods are increasingly important inenvironmental data analysis [2, 5]. New institutes and research initiatives focusexclusively on this topic (e.g. the Institute for Environmental Analytics at theUniversity of Reading – www.the-iea.org). In the current paper, we adapt TDAto the analysis of environmental sensor data. Specifically, we modify the witnesscomplex filtration described by de Silva [8] to eliminate the need for persistence,and consequently, to reduce computation time. We analyze a large number of datavectors in R6 representing various properties (e.g. time, temperature, air pressure,etc.) collected from sensors attached to a commercial cruise vessel, with thegoal of obtaining insights into the complex physical phenomena characterizingdynamic lake conditions. In another example, data from weather station sensorsare analyzed. After normalization of a Euclidean distance matrix for these vectors,a mesh or triangulation is constructed which accurately represents the topologyof this data set. Although the data cannot be characterized as “big data” (verylarge and heterogeneous data sets that generally cannot be processed or analyzedusing standard computational or statistical techniques), the readings are sufficientlynumerous and high-dimensional that constructing such a mesh on the full set ofpoints would be prohibitively costly. We must therefore choose a subset that isrepresentative of the full set and retains its topology. The resulting analyses areexpected to complement visual and statistical methods to enrich insights and tofacilitate discovery of various lake phenomena. The next section describes the TDAalgorithm as well as our adaptations, followed by a preliminary investigation ofapplying TDA to environmental lake monitoring data. Persistent hom*ology for Analyzing Environmental Lake Monitoring Data 235
2 Algorithm
We construct a simplicial complex on a representative subset (landmarks) of the fulldata set using the remaining points as witnesses to decide whether edges will beadded between them or not. A different choice of filtration of the witness complexis used than that described by de Silva [8], and the two approaches are comparedfor their reliance on persistent versus simplicial hom*ology to accurately recover theBetti number profile of the data.
2.1 Simplicial Complexes
A k-simplex is a complete graph on k 1 vertices. Every clique that is a subgraphof a simplex is a face of that simplex.Definition 2.1 A simplicial complex is a collection K of simplices such that:• Every face of a simplex in K is in K• The intersection of any two simplices in K is a face of each of themIn particular, we are interested in the Vietoris-Rips complex [11], the largestsimplicial complex on a given set of vertices and edges.Definition 2.2 Given a vertex set V, the Vietoris-Rips complex VR.V; R/ containsthe p-simplex Œa0 ; : : : ; ap iff for every edge Œej ; ek , 0 j k p, jej ek j RConstructing a VR complex for given V and R then amounts to inserting edgesbetween vertices that are within R of each other, and subsequently determining allcliques in the graph, and including them as simplices.
2.2 Landmarks and Witnesses
Given a set X of N data points, we need to choose a subset of those points consistingof n landmarks that accurately represents the shape of the point cloud. Morecomplex shapes may require more landmarks for accurate representation, whereas acircle could be represented by only four points. Since the remaining N n points willact as witnesses, it is important not to select too large a proportion of the original N.Ideally, N should be quite large, so that a large number of landmarks can be selected.In practice, the ratio N=n should range from 30 to 100, depending on the size of theoriginal data. 236 B.A. Fraser et al.
Fig. 1 The red, green andblue points are weakwitnesses to the triangles oftheir respective colour, and tothe edges of the innertriangle, but there is nowitness to the inner triangleitself
The minmax [8] method of landmark selection is employed, which attempts tochoose landmarks as evenly spaced as possible throughout the data. We proceed asfollows:• Choose a random point to be first in a set of landmarks L• Add each subsequent point x 2 X to L to be such that minfdist.x; p/ 8 p 2 Lg is maximized Let the points in P D X n L be the set of potential witness points.Definition 2.3 ([8]) A point w 2 P is a weak witness to a p-simplex A D.a0 ; a1 ; : : : ap / in L if jw aj < jw bj 8 a 2 A and b 2 L n A. It is a strongwitness if jw a0 j D jw a1 j D : : : D jw ap j as well.
Example 2.1 Take three vertices of an equilateral triangle, as well as the threemidpoints of its edges, as landmarks. Consider the three witnesses which lie justoutside the central triangle formed by the midpoints (Fig. 1). These are weakwitnesses to its edges, but none of them is a weak witness to the triangle itself;they are weak witnesses to the outer triangles corresponding to their colour. We may wish to include a simplex, such as the triangle in the preceding example,in the case where all of its subsimplices are included. This motivates the followingdefinition of a witness complex that will be employed, where witnesses are usedto determine the 1-skeleton, or neighbourhood graph of edges, and then the higherdimensional simplices will be added by Vietoris-Rips expansion [11]. Persistent hom*ology for Analyzing Environmental Lake Monitoring Data 237
2.3 Witness Complexes
Two witness complex filtrations are described below. The first is from De Silva[8], the second is the proposed adaptation of this construction. A filtration of awitness complex is a nested sequence of increasing subsets of a simplicial complex.There are three inputs: a parameter v that determines how many landmarks canbe witnessed simultaneously by a single point, the distance matrices between datapoints, and a value R which ranges across an interval and creates the filtration. Take D to be the n N Euclidean distance matrix (on the n landmarks and N fulldata set). Take E to be the n n distance matrix between landmarks, and E0 to bethe n jPj distance matrix (between the n landmarks and jPj potential witnesses).Both witness complexes have the n landmarks as their 0-simplices.
For W.DI R; v/ (De Silva, [8]), R 2 Œ0; 1/, and for W .E; E0 I R; v/, R 2 Œ0; 1 : • if v D 0, then for i D 1; 2; : : : ; N define mi D 0 and for i D 1; 2; : : : ; jPj define ni D 0 • if v > 0, then for i D 1; 2; : : : ; N define mi to be the v-th smallest entry of the i-th column of D, and for i D 1; 2; : : : ; jPj define ni to be the v-th smallest entry of the i-th column of E0 – the edge D Œab belongs to W.DI R; v/ iff there exists a witness i 2 f1; 2; : : : ; Ng such that max.D.a; i/; D.b; i// R C mi – the edge D Œab belongs to W .E; E0 I R; v/ iff there exists a witness i 2 f1; 2; : : : ; jPjg such that max.E0 .a; i/; E0 .b; i// ni and E.a; b/ R • the p simplex D Œa0 a1 : : : ap belongs to the witness complex iff all of its edges also belong to the witness complex
Example 2.2 The reason for splitting the distance matrix D into E and E0 in ourconstruction is that landmarks should not act as witnesses. Consider a point cloudwhere nearly all the vectors are tightly clustered, and there are a few widely spacedoutliers (Fig. 2). All of these widely spaced outliers are likely to be chosen aslandmarks by minmax, and if they can act as witnesses to each other, they can all beconnected to each other by edges, resulting in a complex which does not accuratelyrepresent the data’s shape. But if landmarks and witnesses are mutually exclusive,then the problem of outlier selection will be mitigated (albeit, each outlier selectedas a landmark will still affect b0 ). 238 B.A. Fraser et al.
Fig. 2 An example of awitness complex W.DI 0; 2/where outliers were selectedas landmarks (red), and thenacted as witnesses to eachother. Featuresunrepresentative of the dataare found
If landmarks are not taken as witnesses, observe the following equality:W.DI 0; v/ D W .E; E0 I 1; v/. Further, W.DI R; v/ is the same as the Vietoris-Rips complex for v D 0, and W .E; E0 I R; v/ is the same for v n. Belowwe explore these relations, and why appropriately chosen values of v can makepersistent hom*ology unnecessary. At the point in the filtrations where W.DI 0; v/ D W .E; E0 I 1; v/, each witnesspoint is the center of a smallest possible closed ball which contains at least vlandmarks. Any landmarks that all fall within one such ball are joined by edges.This is the intermediate stage, say, W , between completely disconnected verticesand a fully connected graph in which we are interested.• W.DI R; v/; R 2 Œ0; 1/ describes the sequence of simplicial complexes from W up to the complete graph on n vertices. It does this by expanding the closed balls around each witness point until all of the landmarks are contained in one.• W .E; E0 I R; v/; R 2 Œ0; 1 describes the sequence of simplicial complexes from the set of disconnected 0-simplices to W . It does this by expanding closed balls centered on the landmarks as an additional condition for connection by an edge. The rationale behind our definition of W .E; E0 I R; v/ is that we find joininglandmarks to each other by considering their own distances from each other ratherthan from a witness to be a more natural progression, and that we consider W tobe a suitable stopping point, where the simplicial complex accurately represents theshape of the data. (In fact, as will be evident from the examples that follow, thereappears to be no need to compute simplicial complexes that are more connectedthan W .)Example 2.3 One reason that persistence is necessary for W.DI R; v/ is that only vvalues up to v D 2 are considered. Consider the following example: the four cornersof a square centered on the origin are chosen as landmark points, and infinitelymany witness points are taken uniformly from the interior of the square, excludingthe origin (Fig. 3). Then at R D 0, W.D; R; 2/ is nearly correct, but there exists nowitness which has opposite corners across the diagonal as its two nearest landmarks: Persistent hom*ology for Analyzing Environmental Lake Monitoring Data 239
Fig. 3 The 1-skeletonresulting from W.D; 0; 2/,with the corners of a squareas landmarks. Without awitness precisely at theorigin, a 2-dimensional holeis detected. At R D 0:1, thehole is filled
as such, they are not joined by edges, and the Betti number b1 is found to be 1 ratherthan 0, as it should be. As such, it is necessary to increment R by at least a smallamount to fill the hole. This issue may have been avoided by choosing v D 3. In fact, with data ofdimension d, choosing v D d C 1 generally suffices. Witness complexes are used toreduce the number of cells: in 3 dimensions, a clique on 10 vertices is superfluous.However, a clique on 4 vertices is not. We can recover an accurate Betti profile ofour data by increasing v slightly and still keep the number of cells relatively low.And by eliminating the need for persistence (since hom*ology need be computedonly once), more landmarks can be selected. Finally, it remains to be explained why the parameter R should be used to createa filtration at all, if persistence is not used. The answer is that outliers can still be aproblem, and W may not actually be correct.Example 2.4 Take as our data set a large number of points from 23 of a circle,and two additional points which are very near to each other from the center ofthe missing arc – these are the outliers. Minmax selects one of these outliers asa landmark, as well as some evenly distributed ones along the 23 arc. An idealsimplicial complex should only connect the landmarks along the 23 arc, but withv D 3, the other outlier which was not taken as a landmark witnesses the two endsof that arc and connects them, completing the circle (Fig. 4). Clearly, this problem could have been avoided by taking the finished complex tobe at some R < 1, as the edges in the 23 arc would have been formed by R 0:1,but the green edges needed R 0:3. In general, the more landmarks that are taken,the closer they will be to each other, and the complex should be complete at a lowervalue of R. Any edges which emerge at higher values are more likely to be artifactsof outliers being witnessed, as in the above example. Therefore, we propose thatthe filtration be calculated normally, but when the rate of edge addition (relative to 240 B.A. Fraser et al.
Fig. 4 For W .E; E0 I 1; 3/,the black point witnesses thegreen edges, which connectan outlier point and bridge achasm in the data, whichshould not have been done
the increase in R) begins to fall off dramatically, that the complex be consideredcomplete and its Betti profile found.
3 Application
The proposed methods are applied to three types of data. The first is an artificiallygenerated sphere point cloud, to mirror the experiments in [8]. The second is lakemonitoring data, and the last are vectors of various observed weather properties.
3.1 Sphere Data
The first artificial data set consists of N D 3200 points chosen uniformly fromthe surface of S2 . Using minmax selection, 64 landmarks were chosen. BothW .E; E0 I 1; 3/ and W .E; E0 I 1; 4/ (Fig. 5) recovered the correct Betti profile.b0 ; b1 ; b2 / D .1; 0; 1/ in each of 100 trials. Then 12,500 points were chosen uniformly from within S2 having four interiorspherical voids, ranging in diameter from 0.2 to 0.5. Again, using minmax, 125landmarks were chosen. W .E; E0 I 1; 3/ (Fig. 5) recovered the correct Betti profile.b0 ; b1 ; b2 / D .1; 0; 4/ in each of 10 trials. Persistent hom*ology for Analyzing Environmental Lake Monitoring Data 241
Fig. 5 Simplicial complexes constructed on points taken from S2 (left) and D2 with 4 sphericalvoids in it (right)
3.2 Lake Monitoring Data
The TDA approach developed in this paper was applied to lake monitoringdata to complement existing statistical and visualization analysis tools [9] (seevisual.nipissingu.ca/Commanda2). Data were collected from Lake Nipissing (46ı160 1200 N 79ı 470 2400 W) via a sonde sensor attached to a commercial cruisevessel (the Chief Commanda II). The vectors are in six dimensions: temperature(ı C), specific conductivity ( s/cm), dissolved oxygen concentration (mg/L), pH,chlorophyll RFU (relative fluorescence units), and total algae RFU. A 3-dimensionalwitness complex W .E; E0 I R; 4/ was constructed on a subseet of 36 landmarks of216 data vectors from Sept. 4, 2011. Betti profiles of .5; 1; 0/ or .5; 0; 1/ were found, but are considered to beunreliable due to the random nature of landmark selection. Removing variousdimensions can simplify locating the source of the feature by constructing the samecomplex on the same data set, minus one of its coordinates. For instance, eliminatingchlorophyll RFU made the detection of these features rarer, so we expect it is acontributing factor, while eliminating total algae RFU resulted in these featuresbeing detected more frequently. Consequently, the total algae property appears tobe irrelevant, as it interferes with the topological feature. In this case, reducing the coordinates to specific conductivity, pH, and chloro-phyll RFU resulted in reliably reproducible 3-dimensional holes appearing in thewitness complex W .E; E0 I 1; 4/ (Fig. 6a). Further research is needed to understandthese results, and to determine whether these features indicate some interactionbetween these properties of the lake. 242 B.A. Fraser et al.
Fig. 6 (a) Reliably detected 3-dim hole on dimension-reduced lake data; (b) b1 increasing by 2 atR D 0:76 in weather data, a result of outliers
3.3 Weather Data
TDA was applied to data collected from a Nipissing University weather sta-tion to complement existing web-based analysis and interactive visualizations(geovisage.nipissingu.ca). Six-dimensional weather data (photosynthetically activeradiation (PAR) (mol/J), rainfall (mm), relative humidity (%), soil moisture (%),soil temperature (ı C), and wind speed (m/s)) for Temiskaming Shores, Ontario(approximately 47ı 310 N 79ı 410 W) were sampled at 5-min intervals, resultingin 8640 vectors for the month of April, 2013. A 2-complex W .E; E0 I 1; 3/ wasconstructed on a subset of 86 landmarks, and during every such trial there wereedges added for large values of R, some even at R > 0:9. These edge additionsresulted in an increase of the found value of b1 . This is an instance of outliersaffecting the result (Fig. 6b), as the majority of the complex was already completeby R D 0:35, and the Betti profile found at that point is considered more accurate. Another strategy for isolating features is to build the complex on different timescales. For instance, a Betti profile of .b0 ; b1 / D .1; 5/ was most often found for themonth of April. It may be helpful to perform the same analysis on individual weeks,or days, to determine whether the appearance of certain features can be related toweather events, such as a thunderstorm, at those particular times.
4 Conclusions
This paper presents an improvement to the persistent hom*ology paradigm for TDAthat adapts and builds on the work of de Silva. By employing witness complexes,and with less dependence upon persistence, we are better able to represent the shapeof complex data, with improved computational efficiency. Based on its success withartificial data sets and preliminary results from real environmental data, we propose Persistent hom*ology for Analyzing Environmental Lake Monitoring Data 243
that simplicial hom*ology on this modified version of a witness complex is effectivein approximating the shape of complex data, and is potentially useful as an adjuncttool for identifying patterns in these data. Further investigation with methods suchas dimension reduction, and determining the parts of the data that give rise to atopological feature would aid in understanding how properties interact. Future workwill concentrate on improving algorithm efficiency, and, as described above, in moredetailed temporal analysis, and in temporal and spatial comparisons of the featuresobtained from TDA to determine how these features reflect physical phenomena.Finally, TDA and traditional computational and statistical techniques, as well asinteractive visualizations, will be integrated to gain a more complete picture ofdynamic environmental conditions.
Acknowledgements M. P. Wachowiak is supported by NSERC Grant #386586-2011.
References
1. Carlsson, G.: Topology and data. Bull. Am. Math. Soc. 46, 255–308 (2009) 2. Dutta, R., Li, C., Smith, D., Das, A., Aryal, J.: Big data architecture for environmental analytics. In: Denzer, R., Argent, R.M., Schimak, G., Hr̆ebíc̆ek, J. (eds.) Environmental Software Systems. Infrastructures, Services and Applications, vol. 448, pp. 578–588. Springer International Publishing, Cham (2015) 3. Holzinger, A.: Extravaganza tutorial on hot ideas for interactive knowledge discovery and data mining in biomedical informatics. In: Ślȩzak, D., Tan, A.-H., Peters, J.F., Schwabe, L. (eds.) Brain Informatics and Health. LNCS, vol. 8609, pp. 502–515. Springer, Heidelberg (2014) 4. Lum, P.Y., Singh, G., Lehman, A., Ishkanov, T., Vejdemo-Johansson, M., Alagappan, M., Carlsson, J., Carlsson, G.: Extracting insights from the shape of complex data using topology. Nature 3 (2013). doi:10.1038/srep01236 5. Namieśnik, J., Wardencki, W.: Monitoring and analytics of atmospheric air pollution. Pol. J. Environ. Stud. 11(3), 211–216 (2002) 6. Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. 108(17), 7265–7270 (2011) 7. Rucco, M., Falsetti, L., Herman, D., Petrossian, T., Merelli, E., Nitti, C., Salvi, A.: Using Topological Data Analysis for diagnosis pulmonary embolism. http://arxiv.org/abs/1409.5020 [physics.med-ph] 8. de Silva, V., Carlsson, G.: Topological estimation using witness complexes. In: Proceedings of the Symposium on Point-Based Graphics, pp. 157–166. Eurographics Association, Aire-la- Ville (2004) 9. Wachowiak, M.P., Wachowiak-Smolikova, R., Dobbs, B.T., Abbott, J., Walters, D.: Interactive web-based visualization for lake monitoring in community-based participatory research. Environ. Pollut. 4(2), 42–54 (2015)10. Zomorodian, A.: Computational topology. In: Algorithms and Theory of Computation Hand- book. Applied Algorithms and Data Structures Series, p. 3. Chapman & Hall/CRC, Boca Raton (2010)11. Zomorodian, A.: Fast construction of the Vietoris-Rips complex. Comput. Graph. 34, 263–271 (2010) Estimating Escherichia coli ContaminationSpread in Ground Beef Production Usinga Discrete Probability Model
Petko M. Kitanov and Allan R. Willms
Abstract Human illness due to contamination of food by pathogenic strains ofEscherichia coli is a serious public health concern and can cause significanteconomic losses in the food industry. Undercooked ground beef is the primarymeans of transmission of pathogenic E. coli to humans. In the Western world, mostground beef is produced in large facilities where many carcasses are butchered andvarious pieces of them are ground together in large batches. Assuming that thesource of contamination is a single carcass, the primary determinant of how manybatches of ground beef from a particular production cycle are affected is the mannerin which pieces of that carcass are spread about in the raw sources that contribute tothe ground beef batches. Assuming that ground beef from a particular batch has beenidentified at the consumer end as contaminated by E. coli, we model the probabilitythat previous and subsequent batches generated in the same production cycle arealso contaminated. This model may help the beef industry to identify the likelihoodof contamination in other batches and potentially save money by not needing tocook or recall unaffected batches of ground beef.
1 Introduction
Pathogenic Escherichia coli strains, particularly O157:H7, cause illness in humans.Scallan et al. [8] indicate that a significant part of all cases of acquired food-borneillness in the U.S.A. is caused by the pathogenic strains of E. coli. These strains canbe transmitted to humans through various food products including produce, dairy,and meat. The primary transmission of these strains from cattle to humans is byconsumption of beef, especially under-cooked ground beef.
P.M. KitanovUniversity of Ottawa, Ottawa, ON, K1N 6N5, Canada,e-mail: [emailprotected]. Willms ()University of Guelph, Guelph, ON, N1G 2W1, Canada,e-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 245J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_23 246 P.M. Kitanov and A.R. Willms
In the western world, since ground beef production is concentrated in largemeat processing plants, an E. coli contamination event at one of these plants canpotentially affect many people over a wide area. Food safety regulations in NorthAmerica require the removal of all production and raw sources associated withan identified contamination event. This leads to the recalling of large amounts ofbeef, much of it likely uncontaminated, and consequently large economic losses forthe beef industry along with damage to reputation. This paper presents a discreteprobability model for estimation of the likelihood that sequential batches of groundbeef produced in a large plant are contaminated with pathogenic E. coli given thatone batch is contaminated. There have been many studies regarding the transmission of pathogenic E.coli and its impact on human health. For example, risk assessment for E. coli inground beef and burgers in different countries has been quantified [4–7, 9]. A riskassessment model for E. coli O157:H7 in ground beef and beef cuts in Canada ispresented in [10]. One study, [11], used data from a large E.coli outbreak in theU.S.A. to identify possible sources of contamination. The differing conditions inproduction and meat processing plants, distribution networks, and cooking methods,etc., make risk assessment of contamination a very difficult task. The primarymethod used in the above studies is statistical analysis of empirical data. Stochasticmodels for outbreak and transmission of E. coli O157:H7 infection in cattle, arepresented in [12] and [15]. Aslam et al. [1, 2] and Bell [3] have studied the sources of contamination bypathogenic E. coli in a beef-packing plant. E. coli in meat products, originatesmainly from the hides of the incoming animals and is transferred to the trimmings,and subsequently the ground beef, during the dressing of carcasses and carcassbreaking. Various decontamination treatments, such as antimicrobial solutions andpasteurization, have been implemented at some beef processing plants. Although itwas found in [14] that these treatments significantly reduce the risk of contamina-tion, the hazard can not be completely removed. Also, there are still no effectivemethods for quickly screening large amounts of ground beef in big productionfacilities. All of these uncertainties justify using probabilistic methods for controland estimation of E. coli contamination in the production of ground beef.
2 The Model
Consider a large ground beef production facility. The ground beef is produced inbatches of several tonnes each. The input to these batches are several raw sources,typically a “lean” fresh source and a “fat” fresh source, but also often a frozen sourceand other sources such as BLBT (Boneless Lean Beef Trimmings). The batch iswell-mixed, so that if any contamination is present on any of the raw source materialthat is input into the batch, the entire batch is deemed to be contaminated. Estimating E. coli Spread in Beef Production 247
2.1 Basic Assumptions
The primary assumption is that the contamination is due to a single carcass,often associated with the fat layer on that carcass. Spread of the contaminant inthe production process is assumed to be due to division and dispersion of thecontaminated carcass portions; transfer via physical contact with other pieces andmachinery surfaces is assumed to be negligible. Further, due to the temperatureat which production occurs, it is assumed that the contaminant does not growappreciably. The manner in which a carcass is spread over some region in a raw source ishighly dependent on the production process—the way in which trim is added to theraw source bins, etc. We make a number of simplifying assumptions:• for each raw source, the number and mass of all pieces from all carcasses is the same, and the manner of spread of the pieces from each carcass throughout the raw source is the same• material in a raw source bin is not mixed, or what mixing occurs is captured by the carcass distribution function,• the material in the raw source bins can be ordered and is used as input to the ground beef production batches in that orderThis last assumption allows us to define a “mass location” in each raw source; massfrom a particular raw source used in batch number b comes from locations justprior to those for the mass from the same source used in batch number b C 1. Thislocation is a discrete variable increasing in increments of the mass of a piece, whichby the first assumption is always the same in a given raw source. Although theseassumptions are all invalid to some degree, they are useful for making a tractablemodel that we believe to have relevancy to the problem. Further, replacing themwith more realistic assumptions would require considerably more information onthe production process. Suppose there are B batches of ground beef produced in a production cycle andS raw sources used as input to these batches, not all of which need be used inany particular batch. It is assumed that a particular batch of ground beef has beenidentified as being contaminated. This batch is referred to as the “hot” batch, and itis identified as batch number h. The origin of this contamination is due to a singlehot raw source and in turn, to a single hot carcass in this raw source. This modelcomputes probabilities of contamination due to the same origin for all other batchesin the production cycle.
2.2 Carcass Spread in a Raw Source
Let ps be the number of pieces, each of mass as contributed by each carcass in rawsource s. Let Cs be the total number of carcasses in the raw source, then the total 248 P.M. Kitanov and A.R. Willms
number of pieces in the raw source is Ns D Cs ps . Since we are assuming the rawsource material is ordered, we assign piece position numbers 1 to Ns for this source.The total mass in the raw source is Ms D Cs ps as . The pieces from each carcass are distributed throughout this raw source in somemanner, the same for each carcass. Here we consider a piece-wise linear distributionfunction that is even around its centre: 8 C ˆ .jnjLi1 /Hi C.Li jnj/Hi1 ˆ < Li Li1 if Li1 < jnj < Li , 1 i K; C F.n/ D H CH (1) ˆ 2 i i if jnj D Li , 0 1 K; :̂ 0 if jnj > LK :
This function’s domain has 2K pieces, distributed symmetrically around zero,divided by the constants 0 D L0 < L1 < L2 < < LK , where each Li , 1 i K,is a positive multiple of ps , and Hi˙ 0, 0 i K, are the limiting values of F atLi from the left () and right (C), respectively. Necessarily, H0 D H0C , HKC D 0,and
X K C .Li Li1 /.Hi C Hi1 / D 1: iD1
This class of functions includes the uniform distribution, obtained with K D 1, andH0˙ D H1 D 2L1 1 . If the parameters are chosen such that Hi D HiC , 1 i K, thenthe function F is continuous. Each raw source s may have a different distributionfunction Fs , and thus a different set of parameters K, L, and H. We assume that thedistribution centres, c of each carcass are uniformly spread through the source: 1 Ns Ms c D c ps ; c 2 Z; 1 c Cs D D : (2) 2 ps ps as
Here c is the carcass number and the discrete probability density function for carcassc in this raw source is Fs .n c /. The expected fraction of a carcass c present inany set of piece positions R f1; 2; : : : ; Ns g is X Qsc .R/ D Fs .n c /: (3) n2R
Equivalently, Fs .n c / may be interpreted as the density function for a particularpiece of this carcass, and Qsc .R/ as the probability that this piece is located in theset R. The piece-wise linear nature of this density function allows it to satisfy therequirement that the fraction of all carcasses at a particular piece position must addto p1s so that the expected number of pieces at that location will be one. Estimating E. coli Spread in Beef Production 249
2.3 Batch Contamination Probability
Let msb be the mass input from source s to batch b, and let Msb be the mass inputfrom source s to batches prior to batch b (both assumed to be multiples of as ). Let nsband Nsb be the number of pieces from source s input to batch b and input to batchesprior to b, respectively, that is, msb D nsb as and Msb D Nsb as . The sequential piecepositions in raw source s that are input to batch b are then the integers in the interval
Bsb D ŒNsb C 1; Nsb C nsb :
For source s, denote the probability of carcass c being absent from the set Bsb asAsc .Bsb /. Modelling the selection of pieces as being independent of the other piecesthat have already been selected from this carcass and other carcasses, this absenceprobability is
Asc .Bsb / D .1 Qsc .Bsb //ps ; (4)
where Qsc is defined by (3). Consider two batches, h and j, which receive input from source s, and a particularcarcass with distribution centred at c . The probability that this carcass has at leastone piece that is input to batch h and at least one that is input to batch j is Prob.c from s in h & j/ D 1 Asc .Bsh / C Asc .Bsj / Asc .Bsh [ Bsj / : (5)
The last term is present because it is included in both the previous terms but shouldonly be counted once. Given that batch h is the hot batch and assuming the contamination is due toa hot carcass in raw source s, then the probability that a particular carcass c fromsource s is the hot carcass is a uniform probability depending on the total number ofcarcasses, Csh from source s input to h:
1 ps Prob.c from s is hot/ D : (6) Csh nsh
The right hand expression in (6) is an upper bound on the probability since thenumber of distinct carcasses present, Csh , will likely be more than nsh =ps , especiallyif the spread of each carcass is large. A reasonable estimate for Csh might be toinclude all carcasses c whose expected number of pieces in batch h is at least one,that is Qsc .Bsh / 1=ps. The contamination in batch h may be due to any of the raw sources that wereinput for this batch and these raw sources may have varying degrees of relativesusceptibility to being contaminated. For example, the susceptibility factor, gs , forraw source number s, might be near zero if the source is frozen material, and mightbe one if the source is fresh material from the plant. Let fs be the fraction of fat in 250 P.M. Kitanov and A.R. Willms
raw source s. The probability that raw source s is the origin of the contamination inbatch h is the weighted fraction of the input mass, the weights being the product ofthe susceptibility factors and the fat fractions:
gs fs msh Prob.s is hot/ D PS : (7) kD1 gk fk mkh
The probability that batch j is also contaminated given that batch h is contami-nated is then found by summing over all raw sources:
Prob. j hot j h/ D X S X Cs Prob.s is hot/ Prob.c from s is hot/ Prob.c from s in h & j/: (8) sD1 cD1
This expression can be evaluated using (4), (5), (6), and (7). One aspect that has not been accounted for is the possibility that one carcassmay be present in more than one raw source. This could occur, for example, in aproduction facility where one fresh lean bin is filled and removed from the trimmingline while a particular carcass has only partially been trimmed. In this case piecesfrom one carcass will occur at the “top” of one raw source bin and the “bottom”of the next, thus potentially spreading the contamination over a much wider area, ifthis happens to be the hot carcass. The model could be modified to account for thispossibility if desired, but would require information about the order and means offilling of the various raw sources.
3 Example
To illustrate the above model we present one example. These data are fictitiousbut based on typical values one might encounter in a large ground beef productionfacility. In this example, there are a total of S D 7 sources (three frozen lean, twofresh lean, and two fresh fat), and a total of B D 14 one-tonne batches of groundbeef are produced. Relevant information for the sources is provided in Table 1. Theuniform distribution for F (K D 1, H0˙ D H1 D 1=.2L1/) was used for all rawsources in this example. The mass from each source used in each batch is providedin Table 2. Using the above model, and letting each batch be the hot batch in turn,probabilities of contamination for each batch were computed. The probabilitythat a particular carcass c in source s is the hot carcass, Prob.c from s is hot/,was computed using the suggestion following (6), that is, counting the numberof carcasses in source s that are present in batch h as all those carcasses c with Estimating E. coli Spread in Beef Production 251
Table 1 Model parameters for the raw sourcesSource gs fs ps as (kg) L1s Cs Ms (kg)I (frozen lean) 0.2 0.05 25 0.6 400 134 2000II (frozen lean) 0.2 0.09 25 0.6 400 134 2000III (frozen lean) 0.2 0.07 25 0.6 400 134 2000IV (fresh lean) 0.8 0.10 20 0.7 350 179 2500V (fresh lean) 0.8 0.08 20 0.7 350 215 3000VI (fresh fat) 1.0 0.40 50 0.5 500 80 2000VII (fresh fat) 1.0 0.45 50 0.5 500 80 2000
Table 2 Source input mass, msb , (kg) and total fat percentage for each batch Source Frozen lean Fresh lean Fresh fatBatch I II III IV V VI VII Fat %1 312 136 552 252 384 52 564 253 114 404 260 222 254 262 239 231 268 255 201 205 89 293 212 256 320 180 292 100 108 157 407 105 284 204 158 390 456 154 159 300 205 325 170 1510 209 211 543 37 1011 293 132 536 39 1012 318 94 540 48 1013 479 454 67 1014 701 226 73 10
Qsc .Bsh / 1=ps. Complete results are given in Table 3 and a selection of these areplotted in Fig. 1. The likelihood of other batches being contaminated is highly dependent on thesource input. In the example, if one of the early batches is the hot batch, thenthe likelihood of contamination is only nonzero for batches near the hot batch.Conversely, if one of the last batches is the hot batch, then a larger number ofother batches have a nonzero probability of being contaminated. This is due tothe source configuration shown in Tables 1 and 2. Since the contamination is mostlikely to be carried in the fatty sources (VI and VII) followed by the lean freshsource (IV and V), the distribution of these sources across the batches is a primarycontributor to the contamination probability distribution. The other factor we foundto be very important was the values of L1s . If these spread indicators were small,then the contamination was much more confined to nearby batches to the hot batch. 252 P.M. Kitanov and A.R. Willms
Table 3 Probability of contamination for each batch in percent. Each column corresponds to adifferent hot batch Hot batchBatch 1 2 3 4 5 6 7 8 9 10 11 12 13 141 99 47 6 0 0 0 0 0 0 0 0 0 0 02 62 99 34 13 0 0 0 0 0 0 0 0 0 03 5 46 100 65 35 3 0 0 0 0 0 0 0 04 0 21 79 100 72 47 19 0 0 0 0 0 0 05 0 0 38 66 100 81 47 18 2 0 0 0 0 06 0 0 3 34 63 98 69 34 12 0 0 0 0 07 0 0 0 12 24 48 100 66 32 6 6 5 2 08 0 0 0 0 11 24 66 100 60 24 14 15 15 139 0 0 0 0 1 8 37 66 100 63 34 26 29 3010 0 0 0 0 0 0 17 39 72 100 66 30 32 3411 0 0 0 0 0 0 13 28 49 60 100 62 35 3812 0 0 0 0 0 0 8 23 39 20 50 100 64 4313 0 0 0 0 0 0 3 18 34 17 21 57 100 7414 0 0 0 0 0 0 0 12 28 14 18 27 67 100
100
80 Probability %
60
40
20
0 2 5 8 11 14 Batch
Fig. 1 Probability of contamination for each batch given that a fixed batch is contaminated (hot).The five separate curves correspond to batches 2,5,8,11, and 14 being the hot batch
The overall result then, is the obvious observation that if one wishes to restrictcontamination, then one should restrict the spread of each carcass within the rawsource, and restrict the spread of the raw source across batches. In other words, tryto make it so that one carcass is only present in one batch. Estimating E. coli Spread in Beef Production 253
4 Conclusion
We believe that the proposed model may help to reduce economic losses in the beefindustry. To our knowledge, this is a novel method for estimating of the probabilityof E. coli contamination in the production of ground beef. Due to the necessarilycautious food safety regulations, in order for this model to be adopted by regulatoryagencies, it would need to be further refined to account for plant-specific processes,and its predictions thoroughly tested. Most of the input to the model is easily obtained from production records, oreasily estimated (such as the average size of pieces). The most difficult input valuesto estimate are the spread parameters, K, L, and H ˙ , however, some knowledgeof these values could be obtained either through detailed observations of the plantprocesses or by using genetic sampling experiments in the raw sources as well asthe final ground beef product. A useful next step would be to use some type ofinnocuous surrogate for E. coli that could be applied to pieces from a particularcarcass and then identified in the final product batches. This would allow for betterdetermination of carcass spread within a raw source and would provide data fordirect validation of the model. Further development of the model, allowing for each carcass to have differentsize pieces, and allowing for carcasses to be present in more than one raw source, iscurrently being completed [13]. This extended work also incorporates genetic typingdata from a production facility to help inform the choice of carcass distributionfunction.
Acknowledgements The second author was supported by a grant from the Applied LivestockGenomics Program of Genome Alberta.
References
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6. Duffy, G., Cummins, E., Nally, P., O’Brien, S., Butler, F.: A review of quantitative microbial risk assessment in the management of Escherichia coli O157:H7 on beef. Meat Sci. 74(1), 76–88 (2006) 7. Ebel, E., Schlosser, W., Kause, J., Orloski, K., Roberts, T., Narrod, C., Malcolm, S., Coleman, M., Powell, M.: Draft risk assessment of the public health impact of Escherichia coli O157:H7 in ground beef. J. Food Prot. 67(9), 1991–1999 (2004) 8. Scallan, E., Hoekstra, R.M., Angulo, F.J., Tauxe, R.V., Widdowson, M.A., Roy, S.L., Jones, J.L., Griffin, P.M.: Foodborne illness acquired in the United States – major pathogens. Emerg. Infect. Dis. 17(1), 7–15 (2011) 9. Signorini, M., Tarabla, H.: Quantitative risk assessment for verocytotoxigenic Escherichia coli in ground beef hamburgers in Argentina. Int. J. Food Microbiol. 132(2–3), 153–161 (2009)10. Smith, B.A., Fazil, A., Lammerding, A.M.: A risk assessment model for Escherichia coli O157:H7 in ground beef and beef cuts in canada: evaluating the effects of interventions. Food Control 29, 364–381 (2013)11. Tuttle, J., Gomez, T., Doyle, M.P., Wells, J.G., Zhao, T., Tauxe, R.V., Griffin, P.M.: Lessons from a large outbreak of Escherichia coli O157:H7 infections: insights into the infectious dose and method of widespread contamination of hamburger patties. Epidemiol. Infect. 122(2), 185– 192 (1999)12. Wang, X., Gautam, R., Pinedo, P.J., Allen, L.J.S., Ivanek, R.: A stochastic model for transmission, extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices. J. Math. Biol. (2013). doi:10.1007/s00285-013- 0707-113. Willms, A.R., Kitanov, P.M.: Probability of Escherichia coli contamination spread in ground beef production (2016, submitted)14. Yang, X., Badoni, M., Youssef, M.K., Gill, C.O.: Enhanced control of microbiological contamination of product at a large beef packing plant. J. Food Prot. 75(1), 144–149 (2012)15. Zhang, X.S., Chase-Topping, M.E., McKendrick, I.J., Savill, N.J., Woolhouse, M.E.J.: Spread of Escherichia coli O157:H7 infection among Scottish cattle farms: Stochastic models and model selection. Epidemics 2, 11–20 (2010) The Impact of Movement on Disease Dynamicsin a Multi-city Compartmental Model IncludingResidency Patch
Diána Knipl
Abstract The impact of population dispersal between two cities on the spread of adisease is investigated analytically. A general SIRS model is presented that tracksthe place of residence of individuals, allowing for different movement rates of localresidents and visitors in a city. Provided the basic reproduction number is greaterthan one, we demonstrate in our model that increasing the travel volumes of someinfected groups may result in the extinction of a disease, even though the diseasecannot be eliminated in each city when the cities are isolated.
1 Introduction
The spatial spread of infectious diseases has been observed many times in history.Most recent examples include the 2002–2003 SARS epidemic in Asia and the globalspread of the 2009 pandemic influenza A(H1N1). The Middle East RespiratorySyndrome coronavirus (MERS-CoV) outbreak emerged in 2012, and West Africa iscurrently witnessing the extensive Ebola virus (EBOV) outbreak, that pose a globalthreat. There is an increasing interest in the mathematical modelling literature forthe spread of epidemics between discrete geographical locations (patches, or cities).Such metapopulation models incorporate single or multiple species occupyingmultiple spatial patches that are connected by movement dependent or independentof disease status. Such models have been discussed for an array of infectiousdiseases including measles and influenza, by Arino and coauthors [2–5], Sattenspiel
D. Knipl ()Agent-Based Modelling Laboratory, 331 Lumbers, York University, 4700 Keele St., Toronto, ON,M3J 1P3, CanadaMTA–SZTE Analysis and Stochastic Research Group, University of Szeged, Aradi vértanúktere 1, Szeged, H-6720, HungaryCurrent Institution: Department of Mathematics, University College London, Gower Street,London, WC1E 6BT, UKe-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 255J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_24 256 D. Knipl
and coauthors [14, 15], Wang and coauthors [9, 13, 19, 20]. The work of Arino [1],and Arino and van den Driessche [6] provide a thorough review of the literature. When considering intervention strategies for epidemic models, our attention isfocused on the basic reproduction number R 0 , which is the expected number ofsecondary cases generated by a typical infected host introduced into a susceptiblepopulation. This quantity serves as a threshold parameter for disease elimination; ifR 0 < 1 then the disease dies out when a small number of infected individuals isintroduced whereas if R 0 > 1 then the disease can persist in the population. Theabove mentioned works illustrate that in metapopulation models R 0 often arisesas a complicated formula of the model parameters. Such models include multipleinfected classes, and individuals’ movement makes it challenging to compute thenumber of new infections generated by an infected case, and to understand thedependence of R 0 on the movement rates. To calculate R 0 in metapopulationmodels, the next-generation method is used (see Diekmann et al. [7]). The models in [4, 5, 14, 15] include residency patch, that is, these models keeptrack of the patch of origin of an individual as well as where an individual isat a given time (either as resident, or as visitor). There are many reasons whyindividuals should be distinguished in an epidemic model by their residentialstatuses; visitors and local residents may have very different contact rates andmixing patterns, but more significantly, these groups are different in their travel ratesbecause in reality, a large part of outbound travels from a city are return trips. In theabove works, the basic reproduction number was calculated and its dependence onthe movement rates was studied numerically. Some complicated behavior of R 0in these parameters was highlighted in [1, 2, 4]: numerical simulations suggestthat when the infection is present in same patches but absent in others withoutmovement, then travel with small rates can allow for disease persistence in themetapopulation although higher travel rates can drive the disease to extinction. In this work we present a demographic SIRS epidemic metapopulation modelin two cities, and analytically investigate the impact of individuals’ movementbetween the two cities on the disease dynamics. In each city we distinguish residentsfrom visitors, and consider the general situation when individuals with differentdisease statuses and residential statuses have different movement rates. In ouranalysis we utilize the concept of the target reproduction number, developed byShuai et al. in [16, 17]. This quantity measures the effort required to eliminateinfectious diseases, when an intervention strategy is targeted at single entriesor sets of such entries of a next-generation matrix. Focusing on the control ofinfected individuals’ movement between the two cities—an intervention strategyoften applied in pandemic situations—we give conditions and describe how thetravel rate of a specific group or some of these groups should be changed to preventan outbreak. The Impact of Movement in an Epidemic Model with Residency Patch 257
2 Model Formulation
We formulate a dynamical model to describe the spread of an infectious diseaseamong two cities. We divide the entire populations of the two cities into the disjointclasses Sjm , Ijm , Rmj , j 2 f1; 2g, m 2 fr; vg, where the letters S, I, and R representthe compartments of susceptible, infected, and recovered individuals, respectively.Lower index j 2 f1; 2g specifies the current city, upper index m 2 fr; vg denotesthe residential status of the individual in the current city (resident or visitor). Anindividual who is currently in city j and has residential status v, originally belongsto city k hence we say that this individual has origin in city k (k 2 f1; 2g; k ¤ j). LetSjm .t/, Ijm .t/, Rm j .t/, j 2 f1; 2g, m 2 fr; vg be the number of individuals belongingto Sjm , Ijm , Rmj respectively, at time t. The transmission rate between a susceptibleindividual with residential status m and an infected individual with residential statusn in region j ( j 2 f1; 2g; m; n 2 fr; vg) is denoted by ˇjmn , and disease transmission ismodelled by standard incidence. Model parameter j is the recovery rate of infectedindividuals in city j, and dj is the natural mortality rate of all individuals with originin city j. Recovered individuals with residential status m in city j lose disease-induced immunity by rate jm . For the total population of residents and visitorscurrently being in city j we use the notations Njr and Njv , and let Njo denote thetotal population with origin in j. It holds that
Njr D Sjr C Ijr C Rrj ; Njv D Sjv C Ijv C Rvj ; Njo D Sjr C Ijr C Rrj C Skv C Ikv C Rvk ; k ¤ j:
For the recruitment term j into the susceptible resident population we assume thatj is a function of the populations Njr and Nkv (k ¤ j), that is, the populations withorigin in city j. We denote by mSm Im Rm kj , mkj , and mkj the travel rate of susceptible,infected, and recovered individuals, respectively, with residential status m in cityj travelling to city k. Based on the assumptions formulated above, we obtain thefollowing system of differential equations for the disease transmission in city j:
dSrj Sr I r Sr I v dt D j .Njr ; Nkv / ˇjrr N r CN j j v ˇj rv j j r r r Sr r Sv v N r CN v dj Sj C j Rj mkj Sj C mjk Sk ; j j j j dIjr Sr I r Sr I v Iv v dt D ˇjrr N r CN j j v C ˇj rv N r CN v .j C dj /Ij mkj Ij C mjk Ik ; j j r Ir r j j j j dRrj Rv v dt D j Ijr .jr C dj /Rrj mRr kj Rj C mjk Rk ; r dSvj Sv I r Sv I v dt D ˇjvr N r CN j j vv j j v ˇj N r CN v dk Sjv C jv Rvj mSv v kj Sj C mjk Sk ; Sr r j j j j dIjv Sv I r Svj Ijv dt D ˇjvr N r CN j j vv v C ˇj Njr CNjv .j C dk /Ijv mIv v kj Ij C mjk Ik ; Ir r j j dRvj dt D j Ijv .jv C dk /Rrj mRv v kj Rj C mjk Rk : Rr r
(1) 258 D. Knipl
Standard arguments from the theory of differential equations guarantee that thesystem (1) is well posed. The function forming the right hand side of the systemis Lipschitz continuous, which implies the existence of a unique solution. Thederivative of each system variable is nonnegative when the variable is zero, hencesolutions remain nonnegative for nonnegative initial data. For the dynamics of thetotal population with origin in city j, we obtain the equation
dNjo D j .Njr ; Nkv / dj .Njr C Nkv /; k ¤ j: dt
If j .Njr ; Nkv / D dj .Njr C Nkv / then the population with origin in j is constant. Forconstant recruitment term j it is easy to derive that NO oj D j =dj gives the uniqueequilibrium of Njo . With fixed N1o and N2o it is obvious from nonnegativity that thesolutions of the system (1) are bounded. The model is at an equilibrium if the timederivatives in the system (1) are zero. At a disease-free equilibrium it holds thatI1r D I1v D I2r D I2v D 0 that implies Rr1 D Rv1 D Rr2 D Rv2 D 0. Thus at a DFE S1r ,S1v , S2r , S2v satisfy
1 .N1r ; N2v / d1 S1r mSr r Sv v 21 S1 C m12 S2 D 0; d2 S1v m21 S1v C mSr Sv r 12 S2 D 0; v Sv v 2 .N2 ; N1 / d2 S2 m12 S2 C m21 S1 r r Sr r D 0; d1 S2v mSv v Sr r 12 S2 C m21 S1 D 0:
Hence if N1o and N2o are fixed then using that 1 D d1 N1o and 2 D d2 N2o , it followsthat r 1 S1 d1 C mSr21 mSv 12 d1 N1o D ; S2v mSr 21 d1 C mSv12 0 r 1 S2 d2 C mSr12 mSv 21 d2 N2o D : S1v mSr 12 d2 C mSv21 0
The following result is proved.Proposition 1 Assume that the total populations with origin in city 1 and withorigin in city 2 are constant. Then there is a unique DFE in the model (1) where
.dj CmSv jk /dj Nj o Sjr D .dj Cmkj /.dj Cmjk /mSv Sr Sv Sr ; jk mkj mSr d N o Sjv D ; j; k 2 f1; 2g; k ¤ j; jk k k .dk CmSr jk /.dk Cmkj /mkj mjk Sv Sv Sr
Ijr D Ijv D 0; Rrj D Rvj D 0; and Njr D Sjr ; Njv D Sjv :
For the stability of the DFE in the full model (1) we linearize the subsystem of (1)that consists of the equations for I1r , I1v , I2r , and I2v —the infected subsystem—about The Impact of Movement in an Epidemic Model with Residency Patch 259
the DFE, and give the Jacobian J, as
J D B G M: 2 ˇ1rr N1r ˇ1rv N1r 3 N1r CN1v N1r CN1v 0 0 2 Ir 3 6 7 m21 0 0 mIv 12 6 ˇ1vr N1v ˇ1vv N1v 7 6 0 0 7 6 0 mIv Ir 0 7 N1r CN1v N1r CN1v 6 21 m12 7; BD6 ˇ2rv N2r 7 ; M D 4 6 0 0 ˇ2rr N2r 7 0 m21 m12 Iv Ir 0 5 4 N2r CN2v N2r CN2v 5 0 0 ˇ2vr N2v ˇ2vv N2v mIr 21 0 0 mIv 12 N2r CN2v N2r CN2v
and G D diag.1 C d1 ; 1 C d2 ; 2 C d2 ; 2 C d1 / DW diag.gr1 ; gv1 ; gr2 ; gv2 /. Let s.A/denote the maximum real part of all eigenvalues of any square matrix A, and .A/denote the dominant eigenvalue of any square matrix A. We say that a square matrixA has the Z-sign pattern if all entries of A are nonpositive except possibly those inthe diagonal. If A1 0 holds then A is a non-singular M-matrix (several definitionsexist for M-matrices, see [8, Theorem 5.1]). By [18, Lemma 1] the stability ofthe DFE is determined by the eigenvalues of J; more precisely, the DFE is locallyasymptotically stable if s.J/ < 0, meaning that all eigenvalues have negative realpart, and the DFE is unstable if s.J/ > 0, when there is an eigenvalue with positivereal part. The proof of the next proposition follows by similar arguments as those inthe proof of [18, Theorem 2].Proposition 2 Consider a splitting F V of the Jacobian of the infected subsystemabout the DFE, where F is a nonnegative matrix and V is a non-singular M-matrix.Then, it holds that s.J/ < 0 if and only if .FV 1 / < 1, s.J/ D 0 if and only if.FV 1 / D 1, and s.J/ > 0 if and only if .FV 1 / > 1. The stability of the DFE is often characterized through the basic reproductionnumber R 0 , that is defined as the dominant eigenvalue of the next-generationmatrix (NGM). The concept of the NGM was initially introduced by Diekmannet al. [7]. This matrix is computed as K0 WD F0 V01 , where F0 equals B, thetransmission matrix describing new infections, and V0 is defined as G C M, thetransition matrix for the transitions between and out of infected classes. F0 andV0 satisfy the conditions of Proposition 2, hence R 0 D .K0 / D .F0 V01 / is athreshold quantity for the stability of the DFE. We obtain the following corollaryfrom Proposition 2.Corollary 1 Consider a splitting F V of the Jacobian of the infected subsystemabout the DFE, where F is a nonnegative matrix and V is a non-singular M-matrix.Then, it holds that .FV 1 / < 1 if and only if R 0 < 1, .FV 1 / D 1 if and only ifR 0 D 1, and .FV 1 / > 1 if and only if R 0 > 1. With other words, for any splitting F V of the Jacobian where F is nonnegativeand V is a non-singular M-matrix, there arises an alternative NGM by FV 1 ;moreover, .FV 1 / and R 0 agree at the threshold value for the stability of theDFE. In the next section we will investigate the impact of movement on the disease 260 D. Knipl
dynamics by constructing some alternative next-generation matrices and utilizingthe method of Shuai et al. [16] to measure the effort required to control the disease.
3 Main Results
Using the definition of G and the transmission matrix F0 , we introduce the quantities
ˇ1rr N1r Cˇ1vr N1v ˇ1rv N1r Cˇ1vv N1v R r1 D gr1 .N1r CN1v / ; R v1 D gv1 .N1r CN1v / ; ˇ2rr N2r Cˇ2vr N2v ˇ2rv N2r Cˇ2vv N2v R r2 D gr2 .N2r CN2v / ; R v2 D gv2 .N2r CN2v / ;
where R mj denotes the expected number of new cases in city j when a single infectedindividual with residential status m who doesn’t travel is introduced into city j. Consider the matrices F1 D B M C diag.mIr 21 ; m21 ; m12 ; m12 / and V1 D Iv Ir Iv
G C diag.m21 ; m21 ; m12 ; m12 /. Then J D F1 V1 gives another splitting of the Ir Iv Ir Iv
Jacobian, moreover F1 0 and V1 is a non-singular M-matrix. We obtain thefollowing theorem.Theorem 1 If R r1 > 1, R v1 > 1, R r2 > 1, and R v2 > 1 then the DFE isunstable when the cities are isolated, and movement cannot stabilize the DFE. Ifthe inequalities are reversed then the DFE is stable when the cities are isolated, andmovement cannot destabilize the DFE.Proof Consider the splitting J D F1 V1 . As V1 is a diagonal matrix, one easilycomputes the alternative NGM 2 ˇ1rr N1r ˇ1rv N1r mIv 3 .mIr v v v 0 12 v 21 Cg1 /.N1 CN1 / .mIv 21 Cg1 /.N1 CN1 / .mIv 12 Cg2 / r r r 6 7 6 ˇ1vr N1v ˇ1vv N1v mIr 7 6 v v v 12 0 7 D6 .mIr 21 Cg1 /.N1 CN1 / .mIv 21 Cg1 /.N1 CN1 / .mIr 12 Cg2 / 7: r r r rK1 D F1 V11 6 mIv ˇ2 N2 rr r ˇ2 N2 rv r 7 6 0 21 v v v v 7 4 .mIv 21 Cg1 / .m12 Cg2 /.N2 CN2 / .m12 Cg2 /.N2 CN2 / 5 Ir r r Iv r
mIr ˇ2vr N2v ˇ2vv N2v .mIr 21 0 v v v 21 Cg1 / .mIr 12 Cg2 /.N2 CN2 / .m12 Cg2 /.N2 CN2 / r r r Iv r
A standard result for nonnegative matrices (see, e.g., [12, Theorem 1.1]) says thatthe dominant eigenvalue of a nonnegative matrix is bounded below and above bythe minimum and maximum of its column sums. We look at the column sums of K1to give upper and lower bounds on the dominant eigenvalue. The column sum in the ˇ1rr N1r Cˇ1vr N1v CmIr 21first column is .mIr v , and using basic calculus we derive that 21 Cg1 /.N1 CN1 / r r
ˇ1rr N1r Cˇ1vr N1v CmIr ˇ1rr N1r Cˇ1vr N1v ˇ1rr N1r Cˇ1vr N1v 1< .mIr v 21 gr1 .N1r CN1v / if .N1r CN1v / gr1 > 0 , R r1 > 1; 21 Cg1 /.N1 CN1 / r r
ˇ1rr N1r Cˇ1vr N1v ˇ1rr N1r Cˇ1vr N1v CmIr ˇ1rr N1r Cˇ1vr N1v gr1 .N1r CN1v / .mIr v 21 <1 if .N1r CN1v / gr1 < 0 , R r1 < 1: 21 Cg1 /.N1 CN1 / r r The Impact of Movement in an Epidemic Model with Residency Patch 261
Similar results follow for the second, third, and fourth columns. Thus if R r1 > 1,R v1 > 1, R r2 > 1, and R v2 > 1 hold then all column sums are greater than 1 for anymIr Iv Ir Iv 21 , m21 , m12 , and m12 , that implies by Proposition 2 that the dominant eigenvalueof K1 is greater than 1 and the DFE is unstable. On the other hand, if the aboveinequalities are reversed then the column sums are less than 1 for any movementrates and the DFE is stable by .K1 / < 1. t u Next, we investigate some cases when changing the movement rates of somegroups can stabilize the DFE. We construct the matrix K2 WD F2 V21 , where F2 isformed as we let ŒF2 1;1 D ŒF1 1;1 gr1 and ŒF2 i;j D ŒF1 i;j if .i; j/ ¤ .1; 1/, and v vV2 D diag.mIr21 ; g1 C rrmIv ; gr2 C mIr 12 ; g2 C m12 /. V2 is a non-singular M-matrix and Iv 21 ˇ N rF2 is nonnegative if N r1CN1v > gr1 . This condition is equivalent to when the number of 1 1new infections amongst residents of city 1 is less than 1, when an infected residentwho doesn’t travel is introduced into city 1. The alternative NGM is computed as 2 3 ˇ1rr N1r gr1 ˇ1rv N1r mIv v v v 0 12 v 6 mIr 7 r Ir .N 21 1 CN 1 / m21 .mIv 21 Cg r 1 /.N1 CN1 / .mIv 12 Cg2 / 6 ˇ1vr N1v ˇ1vv N1v mIr 7 6 v v v 12 0 7K2 D F2 V2 D 6 1 7; Ir r m21 .N1 CN1 / .mIv r 21 Cg1 /.N1 CN1 / .mIr 12 Cg2 / r 6 mIv rr r ˇ2 N2 ˇ2rv N2r 7 6 0 21 v 7 4 .mIv 21 Cg1 / v .mIr r r v 12 Cg2vr/.Nv2 CN2 / .m12 Cg2 /.N2 CN2 / 5 Iv v vv v r ˇ2 N2 ˇ2 N2 1 0 .mIr r r v .mIv v r v 12 Cg2 /.N2 CN2 / 12 Cg2 /.N2 CN2 /
which is irreducible. Denote by L2 the matrix that is formed by replacing ŒK2 1;1 andŒK2 2;1 in K2 by 0. Observe that K2 converges to L2 as mIr 21 goes to infinity. We showthat the disease can be eliminated by controlling only the travel rate of the residentsof a single city. ˇ1rr N1rTheorem 2 Assume that R 0 > 1, that is, the DFE is unstable. If N1r CN1v > gr1 and ˇ N rr r.L2 / < 1 then increasing mIr 21 can stabilize the DFE. In particular, if N r1CN1v > gr1 1 1and R v1 < 1, R r2 < 1, and R v2 < 1, then increasing mIr 21 can stabilize the DFE.
Proof We utilize some terminology and results from [16, 17]. Let S Df.1; 1/; .2; 1/g, and define the 4 4 matrix K2S as ŒK2S i;j D ŒK2 i;j if .i; j/ 2 S and 0 Sotherwise. Note that S identifies the set of elements in K2 that depend on mIr 21 , and K2 Ircontains elements of K2 that are subject to change when m21 is targeted. Followingthe terminology of [16] it is thus meaningful to refer to S as the target set and to K2Sas the target matrix. Note that L2 D K2 K2S , hence the condition .L2 / < 1 impliesthat .K2 K2S / < 1, that is, the controllability condition holds and it is possible tostabilize the DFE by controlling only the elements in S [16]. We compute T S D .K2S .I K2 C K2S /1 /, the number referred to as the targetreproduction number in [16]. Here I denotes the 4 4 identity matrix. Let .mIr 21 / D c
m21 T S , where we denote by .m21 / the controlled travel rate of infected residents Ir Ir c
of city 1 travelling to city 2. It follows from Corollary 1 and [16, Theorem 2.1] byR 0 > 1 that .mIr 21 / > m21 . The matrix K2 , constructed as we replace m21 in K2 by c Ir c Ir 262 D. Knipl
.mIr 21 / , satisfies .K2 / D 1 by [16, Theorem 2.2], which means that the disease can c c
be eradicated by increasing mIr 21 . Note that the conditions R v1 < 1, R r2 < 1, and R v2 < 1 ensure that .L2 / <1. Indeed, it is easy to see that the column sums of the second, third, and fourthcolumns in L2 are less than 1 for any travel rates, and the column sum in the firstcolumn is 1. We now show that 1 is not an eigenvalue of L2 , which together with[12, Theorem 1.1] implies that the dominant eigenvalue of L2 is less than 1. Assumethat 1 is an eigenvalue of L2 , and consider a left eigenvector v D Œv1 ; v2 ; v3 ; v4 corresponding to 1. It holds that
v L2 D 1 v;
and we deduce that
v4 D v1 ; max.v1 ; v2 ; v3 / > v2 ; max.v2 ; v3 ; v4 / > v3 ; max.v1 ; v3 ; v4 / > v4 :
From the fourth inequality and v4 D v1 it follows that v3 > v4 , which together withthe third inequality implies v2 > v3 > v4 , but v1 > v2 by the second inequality, acontradiction to v1 D v4 . The proof is complete. t u To reveal the impact of visitors’ travel, a result analogous to Theorem 2 can beformulated. The proof of the following theorem follows by similar arguments tothose in Theorem 2. ˇ1vv N1vTheorem 3 Assume that R 0 > 1, that is, the DFE is unstable. If N1r CN1v > gv1 , andR r1 < 1, R r2 < 1, and R v2 < 1, then increasing mIv 21 can stabilize the DFE.
Lastly, we give conditions under which controlling outbound travel from onecity is sufficient for disease elimination. Consider two matrices F3 and V3 , definedas ŒF3 1;1 D ŒF1 1;1 gr1 , ŒF3 2;2 D ŒF1 2;2 gv1 , and ŒF3 i;j D ŒF1 i;j otherwise, and vV3 D diag.mIr 21 ; m 21 ; g2 C m12 ; g2rrCr m12 /. V3 is a non-singular M-matrix and F3 is Iv r Ir Iv vv v ˇ N ˇ Nnonnegative if N r1CN1v > gv1 and N r1CN1v > gr1 . The following theorem concerns about 1 1 1 1whether changing the movement rates of the current population of one city can leadto disease eradication. ˇ1rr N1rTheorem 4 Assume that R 0 > 1, that is, the DFE is unstable. If N1r CN1v > g1 and r
ˇ1vv N1v > gv1 but R r2 < 1 and R v2 < 1, then increasing mIrN1r CN1v 21 and mIv 21 can stabilizethe DFE.Proof The proof is similar to the proof of Theorem 2. We compute the alternativeNGM 2 3 ˇ1rr N1r gr1 ˇ1rv N1r mIv v v 0 12 v 6 21 1 vr 1 v mIr .N r CN / mIr 21 m Iv r 21 .N1 CN1 / .mIv 12 Cg2 / 7 6 ˇ1vv N1v gv1 mIr 7 6 mIr ˇ.N1 rNCN 1 12 0 7K3 D F3 V31 D6 6 21 1 v 1/ mIv r 21 .N1 CN1 / v mIv 21 .mIr 12rrCg r 2/ 7; ˇ2 N2r ˇ2rv N2r 7 6 0 1 .m12 Cg2 /.N2 CN2v / v v 7 4 Ir r r .m12 Cg2 /.N2 CN2 / 5 Iv r ˇ2vr N2v vv v ˇ2 N2 1 0 .mIr r r v .mIv v r v 12 Cg2 /.N2 CN2 / 12 Cg2 /.N2 CN2 / The Impact of Movement in an Epidemic Model with Residency Patch 263
which is irreducible, and define the target set U by identifying the entries of K3 thatdepend on mIr 21 and/or m21 . We let U D f.1; 1/; .1; 2/; .2; 1/; .2; 2/g, and define the Iv
4 4 target matrix K3 as ŒK3U i;j D ŒK3 i;j if .i; j/ 2 U and 0 otherwise. Note that U
.K3 / > 1 holds by R 0 > 1. However, the result in [12, Theorem 1.1] on the upperbound of the dominant eigenvalue implies that .K3 K3U / 1. Assume that .K3 K3U / D 1, that is, 1 is an eigenvalue of K3 K3U . Then thereis a left eigenvector v D Œv1 ; v2 ; v3 ; v4 such that
v .K3 K3U / D 1 v
holds. We derive that
v4 D v1 ; v3 D v2 ; max.v2 ; v3 ; v4 / > v3 ; max.v1 ; v3 ; v4 / > v4 ;
so v4 > v3 and v3 > v4 hold by the third and fourth inequalities, a contradiction. Weshowed that .K3 K3U / < 1, which means that there is a potential to control mIr 21 andmIv 21 in a way such that the dominant eigenvalue of the controlled matrix drops below1 (by decreasing targeted entries of K3 to values close to 0). This condition alsoallows us to compute the target reproduction number T U D .K3U .I K3 CK3U /1 /. By [16, Theorem 2.2], the controlled matrix K3c satisfies .K3c / D 1 where K3c isformed by replacing ŒK3 i;j by ŒK3 i;j =T U if .i; j/ 2 U, that is achieved by replacing 21 by .m21 / D m21 T U , and m21 by .m21 / D m21 T U . Note that T U > 1 by [16,mIr Ir c Ir Iv Iv c Ir
Theorem 2.2], which means that the disease can be eradicated by increasing mIr 21 andmIv 21 . t u In the case of transmission coefficients equal for all populations present in a city,recovery rates equal for all populations and death rates equal for all populations,R r1 and R v1 reduce to ˇ1 =. C d/, and R r2 and R v2 reduce to ˇ2 =. C d/. Notethat these quantities give the expected number of secondary infections generated bya single infected case in city 1 and city 2, respectively, in the absence of movementbetween the cities. Hence the local reproduction numbers in city 1 and city 2 can bedefined as we consider our model without dispersal:
ˇ1 ˇ2 R loc 1 D ; R loc 2 D : Cd Cd
We derive the following results from Theorems 1 and 4.Corollary 2 Suppose that ˇ1mn D ˇ1 and ˇ2mn D ˇ2 for all m; n 2 fr; vg, j D and dj D d for all j 2 f1; 2g. Then, the DFE is unstable when the cities are isolatedand R loc 1 > 1, R 2 > 1, and movement cannot stabilize the DFE. In the case loc
when R 1 < 1 and R loc loc 2 < 1, the DFE is stable when the cities are risolated, and ˇ Nmovement cannot destabilize the DFE. If the DFE is unstable and N r 1CN1 v > . C d/, 1 1 ˇ1 N1vN1r CN1v> . C d/ but R loc2 < 1, then increasing the movement rates of individualsin city 1 can stabilize the DFE. 264 D. Knipl
4 Discussion
A two-city compartmental epidemic model was considered to reveal the impact ofpopulation dispersal on disease persistence. This general SIRS model is applicablefor an array of infectious diseases, and it can also be reduced to simpler models (SIS,SIR models) by setting parameters (or their inverses) to zero. In the model setup wedistinguish local residents from temporary visitors in each city, that results in fourinfected classes in the model. We demonstrated that controlling the movement ofone or two infected groups can be sufficient for preventing a disease outbreak. Itwas discussed in [11] that the role of different inflow rates of residents and visitorsinto a city is not necessarily significant in regards of the total epidemic burden, butit is of particular importance for pandemic preparedness, when it comes to assessingthe risk for each group to import the infection to a disease-free city. Modelling the spatial spread of diseases in metapopulations remains a complextask. This paper does not concern with models that include multiple species, hencemore analysis is needed to quantify the effect of movement between patches in suchmodels, which are useful in investigating vector-borne diseases and their controlstrategies. Combining some intervention measures—like the mutual control ofdispersal rates and transmission rates—requires less effort for disease elimination,hence there is a potential to incorporate the results of this work into systematic riskassessment analyses, as described in [10].
Acknowledgements D. Knipl acknowledges the support by the Cimplex project funded bythe European Commission in the area “FET Proactive: Global Systems Science” (GSS), as aResearch and Innovation Action, under the H2020 Framework programme, Grant agreementnumber 641191.
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Alma Mašić and Hermann J. Eberl
Abstract We consider a previously introduced mathematical model of chemostatwith suspended and wall attached growth and exchange of biomass via biofilmdetachment and reattachment. In this study we investigate the role of the specificchoice of a biomass detachment criterion. We find that this choice does greatly affectoutput parameters such as biomass in the system, but it does not affect stronglyeffluent concentration and hence the prediction of reactor performance.
1 Introduction
Bacterial biofilms are layers on immersed surfaces that form wherever environ-mental conditions sustain microbial growth. They play an important role in severalenvironmental engineering applications, most notably in wastewater treatment [12],where several technologies have been designed based on biofilm properties. Animportant part of the biofilm life cycle is detachment, or dispersal, the transfer ofbiomass from the biofilm into the aqueous phase [3, 10]. Several triggers for thisphenomenon have been identified, including external factors, such as shear forces, orinternal factors, such as weakening of the EPS matrix and quorum sensing signalingmechanisms [11, 13]. Several attempts to incorporate detachment into mesoscalemathematical models of biofilms have been suggested in the literature, both in thetraditional 1D Wanner-Gujer biofilm model and its simplification, e.g. [2, 8, 11, 16],and in two- and three-dimensional biofilm models, e.g. [4–6, 18]. It is clear that the mesoscopic description of biofilm detachment strongly affectsthe mesoscopic biofilm structure predicted by these models. What is less clear isto which extent the mesoscopic description of biofilm detachment affects globalparameters that assess reactor performance. For the setting of biofilms in a porous
A. MasicEAWAG, Überlandstrasse 133, 8600 Dübendorf, Switzerlande-mail: [emailprotected]. Eberl ()University of Guelph, 50 Stone Rd E, Guelph, ON N1G2W1, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 267J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_25 268 A. Mašić and H.J. Eberl
medium, a partial answer was given in [1], where a multi-scale model was obtainedby upscaling a Wanner-Gujer type biofilm model with four different detachmentcriteria to the reactor scale; it was found that the particular choice of the mesoscopicdetachment rate does not affect the macroscopic description. Here we are interestedin the question whether for a continuous stirred tank reactor (CSTR) with wallattached and suspended growth the particular description of biofilm detachmentaffects the effluent substrate concentration predicted by the model. Models of thistype arise in the modeling of wastewater treatment processes where both biofilmsand nonsessile bacteria are present and contribute to the substrate degradationprocess. In the focus of our interest is the case where the reactor per se is designedas a biofilm reactor, in which suspended biomass occurs as a side effect through theexchange of biomass between biofilm and aqueous phase.
2 Mathematical Model
We cast a simple model for a continuous stirred tank reactor with suspended andwall attached growth in terms of the dependent variables substrate concentrationS [gm3 ], suspended biomass u [g] and biofilm thickness [m]. Following [9], itreads
uu .S/ J.S; / SP D D.Sin S/ ; (1) V V uP D u.u .S/ D ku / C Ad./ ˛u; (2) ˛u P D v.; t/ C d./: (3) A
Here, D [d1 ] is the dilution rate, Sin [gm3 ] the inflow substrate concentration, [-] the yield coefficient, ku [d1 ] the cell death rate for suspended bacteria, and ˛[d1 ] is the rate at which suspended bacteria attach to the biofilm, V [m3 ] is thereactor volume and A [m2 ] the colonizable surface area. The biomass density in thebiofilm is [gm3 ]. In (3), the function v D v.z; t/ [md1 ] denotes the growth induced velocity of thebiomass at a location z in the biofilm. Due to the incompressibility assumption thatthe biomass density is constant across the biofilm, biofilm expansion is essentiallyequivalent to biomass growth. Velocity v is obtained as the integral of the biomassproduction rate Z z v.z; t/ D . .C.// k /d; (4) 0
where k [d1 ] is the cell death rate for biofilm bacteria. Biofilm Detachment in a Chemostat with Wall Attachment 269
The substrate dependent bacterial growth rates are defined via Monod kinet-ics, i.e.
max u S max C.z/ u .S/ D ; .C.z// D ; (5) Ku C S K C C.z/
where max u ; max [d1 ] are the maximum specific growth rates, Ku ; K [gm3 ] thehalf-saturation concentrations and C.z/ [gm3 ] denotes the substrate concentrationin the biofilm at thickness z [m] from the substratum. It is obtained as the solutionof the two-point boundary value problem Dc C00 .z/ D .C.z//; C0 .0/ D 0; C./ D S: (6)
Here Dc [m2 d1 ] is the diffusion coefficient. The boundary condition at thesubstratum, z D 0, describes that substrate does not leave the reactor through thewalls, while the boundary condition at z D implies that external mass transferresistance at the biofilm/water interface is neglected. In (6) we used that substratediffusion is a much faster process than biofilm growth, i.e. that (6) can be consideredin a quasi-steady state. In (1), the sink J [gd1 ] denotes the substrate flux from the aqueous phase intothe biofilm, i.e.
dC J.S; / D ADc ./: (7) dz
Detachment of biomass from the biofilm is described by the volumetric detach-ment rate d./ [d1 ], which we assume to be differentiable. A frequently used detachment rate expression in biofilm modeling is to assumethat d is proportional to ,
d./ D E1 ; (8)
leading to a quadratic sink term in (3); E1 [d1 m1 ] is the erosion or detachmentparameter. This traditional detachment model was the only one used in [9]. Anotherdetachment rate function that is found in the literature is to assume a constant rate, i.e
d./ D E2 ; (9)
which leads to a first order sink term with erosion parameter E2 [d1 ]. It is often assumed that the detachment rate depends also on the hydraulicconditions in the reactor, which determine shear forces acting on the biofilm.Therefore, one can correlate E1;2 with D, see below in Sect. 3. We assume all model parameters to be positive. 270 A. Mašić and H.J. Eberl
First we formally re-write our model as an ordinary initial value problem. Notethat integrating (6) once and using the boundary conditions gives Z dC ./ D .C.z//dz: (10) dz Dc 0
We define ( R Dc 0 .C.z//dz; >0 j.; S/ WD (11) 0 D 0:
Note that C.z/ is indirectly a function of S due to the boundary condition in (6),therefore also j is a function of S. Then (1), (2), (3), (4), (5), (6), and (7) becomes
1 uu .S/ SP D D.Sin S/ C ADc j.; S/ (12) V uP D u.u .S/ D ku / C Ad./ ˛u (13) Dc ˛u P D j.; S/ k C d./: (14) A
While the function j.; S/ is not known explicitly in terms of elementaryfunctions, important qualitative properties are known, which we repeat here from[9]:Lemma 1 The function j.; S/ for 0, S 0 is well-defined, non-negative anddifferentiable. It has the following properties:(a) j.; 0/ D 0, j.0; / D 0, @j(b) @S .0; S/ D 0,(c)
S @j S .0; S/ ; (15) K C S @ K max where D Dc and K is the half-saturation coefficient from (5).Proposition 1 The initial value problem of (12), (13) and (14) with non-negativeinitial data possesses a unique non-negative solution for all t > 0. There is no timeinterval .t1 ; t2 /, 0 < t1 < t2 , over which a non-trivial solution exists with eitheru 0 or 0.Proof (sketch) The function j.; S/, and thus the system (12), (13) and (14) is well-defined. The tangent criterion of [15] in the usual way can be used to confirm thepositive invariance of the non-negative cone. In the non-negative cone the systemsatisfies a Lipschitz condition, which implies existence and uniqueness. An upperestimate for the solutions can be constructed by the differential inequality techniques Biofilm Detachment in a Chemostat with Wall Attachment 271
from a linear combination of the model equations, more specifically, an upper boundcan be derived for VS.t/ C u.t/ C A.t/ which then, using non-negativity, impliesupper estimates on each of S.t/; u.t/; .t/. The last statement in the assertion followsdirectly from (13) and (14). t uProposition 2 The washout equilibrium .Sin ; 0; 0/ always exists. It is unstable ifat least one of u .Sin / D ku ˛ and Dc @ @j .0; Sin / d.0/ k is positive;if both expressions are negative then the washoutequilibrium is stable if WD @j u .Sin / D ku ˛ Dc @ .0; Sin / d.0/ k ˛d.0/ > 0 and unstable ifthe inequality is reversed.Proof It is easily verified that the trivial equilibrium E0 D .Sin ; 0; 0/ always exists.To determine the stability of the equilibrium we calculate the Jacobian J.Sin ; 0; 0/of the right hand side of (12), (13) and (14). Using Lemma 1(a),(b) we find 0 1 uV .S / c @j in V @ .0; S / AD in D B C J.Sin ; 0; 0/ D @ 0 u .Sin / D ku ˛ Ad.0/ A (16) ˛ Dc @j 0 A @ .0; S in / d.0/ k
The eigenvalues are D < 0 as well as the eigenvalues of the 2 2 sub-matrix ! j22 j23 u .Sin / D ku ˛ Ad.0/ WD ˛ Dc @j (17) j32 j33 A @ .0; Sin / d.0/ k
We distinguish now between three cases: (i) if j22 > 0; j33 > 0 then the trace of thissub-matrix is positive, hence at least one eigenvalue has positive real part and theequilibrium is unstable. (ii) if j22 < 0; j33 > 0 or vice versa, then the sub-matrix has anegative determinant and hence one positive and one negative eigenvalue, implyinginstability. (iii) if j22 < 0; j33 < 0 then the trace of the sub-matrix is negative and itsdeterminant is obtained as Dc @j D u .S / D ku ˛ in .0; S / d.0/ k ˛d.0/; in (18) @
i.e. depends on parameters. Positive implies stability, negative implies insta-bility. While for d.0/ D 0, as in detachment rate function (8), stability is automatic,this is not necessarily true if d.0/ > 0, as in detachment rate function (9). t uRemark 1 Weaker, but easier to evaluate and to apply stability criteria can be @jobtained by replacing @ .0; S/ in Proposition 2 with the estimates in Lemma 1(c): [i] max S inThe inequality K CSin d.0/ k > 0 is sufficient (but not necessary) for instability max S inof the washout equilibrium. [ii] If u .Sin /Dku ˛ < 0 and K d.0/k < 0 max Sinthen the inequality .u .S in / D ku ˛/. KCSin d.0/ k / > ˛d.0/ is sufficient 272 A. Mašić and H.J. Eberl
(but not necessary) for stability of the washout equilibrium, whereas the inequality max S in.u .Sin / D ku ˛/. K d.0/ k / < ˛d.0/ is sufficient for its instability.Remark 2 For detachment rate functions with d.0/ D 0, such as (8), the aboveanalysis simplifies. The eigenvalues of J.Sin ; 0; 0/ are then its diagonal elements andcase (iii) above will always be stable. The stability results for cases (i), (ii) are thesame but follow immediately from the sign of the diagonal entries. This implies thatthe answer to the question whether a biofilm can be established in the CSTR withwall attached and suspended growth can depend on the detachment criterion usedin the modeling study. Whereas in models with d.0/ D 0 the specific detachmentrate coefficient does not affect the outcome, it does so in detachment models withd.0/ > 0. This is also consistent with the results of the upscaling study of [1] forporous medium systems. Non-trivial steady states, in which both wall attached and suspended biomass arepresent are much more difficult to analyse, even in the algebraically much simplerFreter model, in which no substrate gradients in the wall depositions are accountedfor [7, 14]. Therefore, we do not expect any insightful results in pursuing this lineof investigation and turn to a numerical study instead.
3 Simulation Results
Typical simulation results are shown in Fig. 1, using the parameters in Table 1.In one case the bulk substrate concentration Sin is chosen low enough so that thebacteria cannot be sustained; in the other case it is high enough such that both abiofilm and a suspended population attain a positive equilibrium value. The vastmajority of biomass in the system is sessile. These simulations were conducted withthe detachment rate function (8). In Fig. 2 we show for both detachment rate function (8) and (9) the steady statevalues for ; u; S for different detachment parameters E1 and E2 , which have beencorrelated with the dilution rate. Motivated by [11] we chose the relationship as 0:58 D EQ i D Ei ; (19) D0
where by D0 we denote a reference dilution rate such that for D D D0 this flow ratedependent criterion is equivalent to a detachment with a given rate constant Ei andfor D > D0 we have EQ i > Ei , while for D < D0 we have EQ i < Ei . Note that anincreased dilution rate implies a more plentiful substrate supply. The main observation is that the choice of detachment rate function and erosionconstant does not have an effect on the substrate concentration in outflow, i.e. thesubstrate removal performance of the reactor: For smaller dilution rates D < 50, theeffluent substrate concentration increases almost linearly with D, for larger D > 50, Biofilm Detachment in a Chemostat with Wall Attachment 273
−6 x 10 1 Aρλ 0.1 u
0.8
suspended and biofilm biomass (g) substrate concentration (g/m )3
0.6
0.0999 0.4
0.2
0.0998 0 0 5 10 15 20 25 0 5 10 15 20 25 t (days) t (days)
10 Aρλ 0.08 u substrate concentration in reactor (g/m ) 3
8 suspended and biofilm biomass (g)
0.06 6
0.04 4
0.02 2
0 0 0 5 10 15 0 5 10 15 t (days) t (days)
Fig. 1 A typical simulation of model (12), (13) and (14). Case 1 (top): The washout equilibriumis stable, case 2 (bottom): the washout equilibrium is unstable. In both simulations the detachmentcriterion (8) was used 274 A. Mašić and H.J. Eberl
Table 1 Model parameters used in the simulationsSymbol Parameter Value Reference˛ Attachment rate 1/day AssumedDc Diffusion coefficient 104 m2 /day [17]E1 Erosion parameter 1000=.mday) Assumed Yield of biomass from substrate 0:63 - [17]K , Ku Half-saturation coefficients 4 g/m3 [17]k , ku Death rates 0:4/day [17]max , u max Maximum specific growth rates 6/day [17] Biofilm biomass density 10;000 g/m3 [17]
the increase becomes sublinear. The results obtained for different base dilutionrates D0 , hence different detachment rates Ei , appear indistinguishable for eachdetachment criterion (8) and (9). Also comparing the results between the simulationexperiments for (8) and (9), respectively, shows that for both detachment criteria thesame values of S are found. The biofilm thickness, however, depends strongly on the detachment rate func-tion, and so does the suspended biomass density and by extension the amount ofbiomass in the effluent of the reactor. An explanation for this phenomenon is that in a thick enough biofilm, as is typicalfor many biofilm based wastewater treatment applications, only a small part of thebiofilm is active whereas a major part does not contribute to reactor performancedue to substrate limitations in the inner layers of the film. An increased detachmentforce decreases the biofilm thickness but as long as it remains above the thicknessof the active layer this does not have an effect on performance. If the detachment isstrong enough such that the resulting biofilm becomes too thin, this statement doesnot hold anymore.
4 Conclusion
We have investigated the question whether the choice of a mesoscale biofilmdetachment model affects the prediction of reactor performance in a macroscalebiofilm reactor model. We have found that for the question whether a biofilm can be established orwhether biomass is washed out the choice of detachment rate can matter: Thedetachment coefficient does not affect stability of the washout equilibrium in case ofa detachment rate function that vanishes in the absence of biofilm, i.e. if d.0/ D 0.However, if detachment rate functions with d.0/ > 0 are used, then the coefficientsof the detachment model can affect stability of the washout equilibrium. In biofilm based reactor technology, the washout equilibrium is of minorimportance, because reactors are designed such that sufficiently thick biofilms are Biofilm Detachment in a Chemostat with Wall Attachment 275
−3 x 10 25 0.015 1 D =1 0 D =10 0 D0=40 D0=100 20 E=1000
0.01substrate in reactor (g/m3)
suspended biomass (g)
biofilm thickness (m) 15
0.5
10
0.005
D =1 D0=1 5 0 D0=10 D =10 0 D0=40 D0=40 D0=100 D =100 0 E=1000 E=1000 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 D (1/day) D (1/day) D (1/day)
× 10 -3 25 0.008 D 0=1 D 0=10 D 0=40
20 D 0=100 2 E=0.05 substrate in reactor (g/m3)
suspended biomass (g)
biofilm thickness (m)
15
0.004
10 1
D 0=1 D 0=1 5 D 0=10 D 0=10 D 0=40 D 0=40 D 0=100 D 0=100 E=0.05 E=0.05 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 D (1/day) D (1/day) D (1/day)
Fig. 2 Steady states attained in dependence of dilution rate D. Top: detachment criterion (8),reprinted from [9] with permission; bottom: detachment criterion (9)
obtained. In our simulations we show that in such situations with well developed,sufficiently thick biofilms the particular choice of a mesoscopic detachment ratefunction does not quantitatively affect macroscopic reactor performance as mea-sured in terms of effluent substrate concentration, but it does affect the predictedbiofilm thickness. 276 A. Mašić and H.J. Eberl
Acknowledgements HJE was supported by NSERC Canada through the Discovery Grant Pro-gram and through the Canada Research Chairs Program.
References
1. Abbas, F., Eberl, H.J.: Investigation of the role of mesoscale detachment rate expressions in a macroscale model of a porous medium biofilm Reactor, Int. J. Biomath. Biostat. 2, 123–143 (2013) 2. Abbas, F., Sudarsan, R., Eberl, H.J.: Longtime behaviour of one-dimensional biofilm models with shear dependent detachment rates. Math. Biosci. Eng. 9, 215–239 (2012) 3. Bester, E., Edwards, E.A., Wolfaardt, G.M.: Planktonic cell yield is linked to biofilm development. Can. J. Microbiol. 55, 1195–1206 (2009) 4. Duddu, R., Chopp, D.L., Moran, B: A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioeng. 103, 92–104 (2008) 5. Emerenini, B.O., Hense, B.A., Kuttler, C., Eberl, H.J.: A mathematical model of quorum sensing induced biofilm detachment. PLoS ONE 10(7), e0132385 (2015) 6. Hunt, S.M., Hamilton, M.A., Sears, J.T., Harkin, G., Reno, J: A computer investigation of chemically mediated detachment in bacterial biofilms. J. Microbiol. 149, 1155–1163 (2003) 7. Jones, D., Kojouharov, H.V., Le, D., Smith, H.L.: The Freter model: a simple model of biofilm formation. Math. Biol. 47, 137–152 (2003) 8. Kommendal, R., Bakke, R.: Modeling Pseudomonas aeruginosa biofilm detachment. HIT Working Paper no 3/2003 (2003) 9. Masic, A., Eberl, H.J.: Persistence in a single species CSTR model with suspended flocs and wall attached biofilms. Bull. Math. Biol. 74, 1001–1024 (2012)10. Morgenroth, E.: Detachment: an often-overlooked phenomenon in biofilm research and modeling. In: Wuertz, S., et al. (eds.) Biofilms in Wastewater Treatment, pp 246–290. IWA Publishing, London (2003)11. Rittmann, B.E.: The effect of shear stress on biofilm loss rate. Biotechnol. Bioeng. 24, 501–506 (1982)12. Rittmann, B.E, McCarty, P.L.: Environmental Biotechnology. McGraw-Hill, Boston (2001)13. Solano, C., Echeverz, M., Lasa, I.: Biofilm dispersion and quorum sensing. Curr. Opin. Microbiol. 18, 96–104 (2014)14. Stemmons, E.D., Smith, H.L.: Competition in a chemostat with wall attachment. SIAM J. Appl. Math. 61, 567–595 (2000)15. Walter, W.: Gewöhnliche Differentialgleichungen, 7th edn. Springer-Verlag, Berlin (2000)16. Wanner, O., Gujer, W.: A multispecies biofilm model. Biotechnol. Bioeng. 28, 314–328 (1986)17. Wanner, O., Eberl, H., Morgenroth, E., Noguera, D.R., Picioreanu, C., Rittmann, B., van Loosdrecht, M.: Mathematical modeling of biofilms. Scientific and Technical Report No.18. IWA Publishing, London (2006)18. Xavier, J.B., Picioreanu, C., van Loosdrecht, M.C.M: A general description of detachment for multidimensional modeling of biofilms. Biotechnol. Bioeng. 91, 651–669 (2005) Application of CFD Modelling to the Restorationof Eutrophic Lakes
A. Najafi-Nejad-Nasser, S.S. Li, and C.N. Mulligan
Abstract Eutrophication has been a worldwide lake pollution problem, caused bythe presence of excessive nutrients in lakes. The nutrients can come from an externalor internal source. The release of phosphorous (P) from resuspended sedimentsfrom the lake bottom is a significant internal source. This paper discusses howto effectively control such a release. We considered using artificial circulationtechnique, and carried out CFD modelling of circulation triggered by air-bubbleinjection into the lake water. The simulations are based on the RANS equations. Wepredicted distributed water and air-bubble velocities, as well as air volume fraction.The predictions compare well with experimental data. Turbulent eddy motions causeoxygenated surface water to flow downward and effectively mix with bottom water.The air bubbles directly enhance the dissolved oxygen level. Both mechanismswould inhibit the release of P from bottom sediments. Using proper methods forinterphasal forces and turbulence closure is the key to success.
1 Introduction
Lake eutrophication is a nutrient-enrichment scenario where an excess amount ofnutrients enters a lake and causes a drastic growth of algae. The subsequent deathof algae typically forms a thin greenish layer on the lake surface. Eutrophicationreduces light penetration into the lower water column and re-oxygenation of waterthrough air circulation [2]. Dead algae would become food for bacteria. The foodconsumption process uses oxygen, and thus causes the dissolved oxygen level of thelake water to drop, leading to fish kills [2]. Also, eutrophication causes undesirableodour and taste [6, 9]. Lake eutrophication occurred worldwide. It has been a major environmentalissue [10]. Thus, it is worth a while to develop strategies for the effective control oftrophic state related to nutrient loading. The aim of this paper is: (1) to characterisewater circulation, turbulent mixing, and oxygen concentration, induced by airbubble injection; (2) to determine the accuracy of interphase-force and turbulence
A. Najafi-Nejad-Nasser () • S.S. Li • C.N. MulliganConcordia University, 1455 de Maisonneuve Blvd. W., Montreal, QC, H3G 1M8, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 277J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_26 278 A. Najafi-Nejad-Nasser et al.
closure methods through a comparison between computational results and availablelaboratory data. In phytoplankton growth and dynamics, P is a key element. It is the limitingnutrient in fresh water systems. Correspondently, existing strategies for eutrophi-cation control and/or remediation have mostly focused on reducing P load [5, 17].Various efforts were made to reduce external sources of nutrients loading [12, 21]. However, in the 1970s, studies [13, 20] revealed that P released from lake bottomsediments could continue as internal loading. Internal loading significantly delayedthe recovery of eutrophic lakes even after the control of external loading [14]. Underspecific conditions, internal loading contributed up to 80 % of total P in lakes [11].For example, in Lake Ockeechobe in Florida, internal loading was in the same orderof magnitude as external sources [19]. Thus, it is of equal importance to be able toeffectively control the release of P from bottom sediments to the lake water. One plausible technique to control internal loading is to introduce bubbleplumes to the lake water. Previous studies have identified some relevant parametersassociated with this technique [8, 9, 15, 16, 24]. Kim et al. [9] investigatedexperimentally the effects of bubble size and diffusing area (geometric parameters)on lake destratification. Destratification tends to promote air circulation and improvethe dissolved oxygen level. Kim et al. [9] suggested that destratification efficiencyis proportional to the bubble diffusing area and inversely proportional to the bubblediameter and overall tank area. Rensen and Roig [15] studied the non-stationarybehaviour of flow by conducting experiments of 2D bubble plumes in a confinedtank. According to Imteaz and Aseada [8], the number of ports, air flow rate, andbubbler starting time were important parameters for optimal bubbling operation. Using the modelling approach, Sahoo and Luktenia [16] studied 1-D bubbleplumes. They reported that bubbles of close to 1 mm in radius gave a higher oxygentransfer rate and mechanical efficiency than larger bubbles. Yum et al. [24] simulatedtwo-phase bubble plumes, using experimental data for model calibration/verifi-cation. Their work led to the derivation of relationships between stratificationefficiency, plume spacing, and destratification number. Since the calibration dataare from a small tank under controlled environment, uncertainties about therelationships’ suitability for application to field conditions exist. More advancedmodelling studies are needed to quantify the beneficial effects of bubble plumes onwater quality. This paper tackles the problem by numerically solving the RANS equations onunstructured finite volume mesh. The solution methods entail the use of suitablemodels for turbulence closure. Previously, a number of investigators have assessedthe suitability of turbulence closure models for different applications [7, 18, 22]. Application of CFD Modelling to the Restoration of Eutrophic Lakes 279
2 Computational Model
2.1 Model Domain
The model domain is a cylinder (Fig. 1a) with a height h D 40 cm, and a diameterFG D 50 cm. The top of the cylinder is open to air. The bottom has a circular holeat the middle. This hole has a diameter D D 6 cm. Through this hole (inlet), airbubbles of a given diameter are injected into the otherwise stagnant water containedin the cylinder. This model domain is used because experimental results are availableto allow a direct comparison. For efficient computation of water and air bubblemotions, we consider three geometric configurations of computational domain: (1)Computational domain A: a simple 2D plane cutting through the centreline of thecylinder (Fig. 1c); (2) Computational domain B: an axis-symmetrical domain cuttingthrough the centreline (Fig. 1b); (3) Computational domain C: a 10ı wedge of thecylinder (Fig. 1d). Domains A and B are covered by quadrilateral mesh generatedusing a cell size of 1 mm. The mesh contains 401,802 and 199,562 computing nodes,respectively. Domain C is covered by quadrilateral mesh created using a cell size of5 mm. The total computing nodes are 33,580.
Fig. 1 A diagram of the model cylinder containing water. Water and air-bubble motions areinduced by the injection of air bubbles 280 A. Najafi-Nejad-Nasser et al.
2.2 Solution Method-Eulerian Approach
The turbulent motions of the liquid phase (water) and gas phase (air bubble) weresimulated with the Euler-Euler method using ANSYS Fluent. The motions aredescribed using separate momentum equations. The two phases are related througha momentum exchange term. The interphase momentum transfer is due to interfacialforces acting and interactions between water and air bubbles (such as lift force anddrag force) [3, 4, 23]. Hence, a proper solution for the bubble columns dependents onthe correct modelling of interphase forces and turbulence models. In this paper, wecompare the performance of different interphase force models and two turbulenceclosure models (the k- model and the SST k-! model). All the simulations useunsteady formulation. Phase Coupled SIMPLE (PC-SIMPLE) algorithm is used forpressure velocity coupling. This algorithm is an extension of the SIMPLE algorithmto multiphase problems. The velocity solutions are obtained in a segregated fashion,and coupled by the liquid and gas phases [4].
2.3 Initial and Boundary Conditions
At time t D 0, imposed initial conditions are as follows: The free water surface islocated at the equilibrium position. The volume fraction of water ˛w is equal to onebelow the free surface (for x2 D h). Water is stagnant or the velocity componentsu1 and u2 are zero in the entire computational domain. Kinematic and dynamicconditions are imposed at the boundaries of the computational domains (Fig. 1),including(1) Inlet (at the bottom of the cylinder)(2) Outlet (on the top of the cylinder)(3) Solid side walls of the cylinder(4) Axis , OM in Fig. 1b (Runs 2–10, listed in Table 2)(5) Symmetry (for 3D simulation, Run 11) At the inlet (D=2 < x1 < D=2 and x2 = 0, Fig. 1), air bubbles of a givendiameter d (d < D) enter the domain continuously during the simulation time periodT. The direction of the entering velocity is upward, and the magnitude is uo (or u2 Duo ; u1 D 0). At the inlet, the volume fraction of water ˛w is taken as zero. At the outlet (at x2 D h, Fig. 1), fluids are exposed to the atmosphere.Accordingly, the pressure equals to the atmospheric pressure. The volume fractionof water ˛w is set to zero. At the solid walls, no-slip condition is applied for computational domains A,B, and C. For the three cases, the wall distance of the first cell from the walls isyC D 0:33, yC D 0:45, and yC D 0:81, respectively. Thus, no-slip condition isvalid. This condition means that both the tangential and normal components of thefluid velocity are set to zero. Application of CFD Modelling to the Restoration of Eutrophic Lakes 281
2.4 Simulations
Model parameters and their values used in simulations are summarised in Table 1. Atotal of 11 runs (Runs 1–11, Table 2) were carried out using the Eulerian approach.Runs 1–6 and Runs 10–11 differ from each other in the choice of turbulencedispersion force and lift force; these runs use the k- model for turbulence closure.Runs 7–9 use the SSTk-! model. The time period of all the simulations was t = 10.7 s. This is long enough since itis more than two times of the advection time scale. All the 11 runs produced finite volume solutions to the RANS equations. Theresults are presented and discussed in the next section, along with comparisons withavailable experimental data.
3 Results and Discussion
Under given conditions (Tables 1 and 2), Runs 1–11 produce water velocities.As an example, velocity vectors and corresponding flow streamlines at a state ofequilibrium for Run 3 are plotted in Figs. 2 and 3, respectively. A strong jet isseen to occur in the central region (Fig. 2), as a direct response to bubble injection.Water motions are visible in the entire domain. The jet flow entrains water from both
Table 1 Control parameters and their values Parameter Value Parameter Value Time step (s) 0.001 Initial time (s) 0 Cell size for domains A and B (mm) 1 Simulation time period (s) 10.7 Air velocity at the inlet (m/s) 0.085 Number of time step 11,000 Water velocity at the inlet (m/s) 0 Pressure at outlet (Pa) 0 Flow rate at the inlet (m3 /s) 2:4 104 Surface diameter (m) 0.5 Bubble size at the inlet (mm) 3 Depth (m) 0.4 Air volume fraction at the inlet 1 Depth of water (m) 0.4 Convergence criteria 106 Inlet pipe diameter (m) 0.06
Table 2 Solution domain, water-bubble interaction method, and turbulence closure model Run 1 2 3 4 5 6 7 8 9 10 11 Domain A B B B B B B B B B C Drag SNa SN SN SN SN SN SN SN SN SN SN Lift (–) (–) La L Ta T (–) T T T (–) Dispersion (–) (–) LDBa Sa Ba LDB (–) LDB S S (–) Closure k k k k k k SST SST SST k ka SST SSTk !, SN Schiller-Nauman, L Legendre, LDB Lopez-de-Bertodano, S Simonin, TTomiyama, B Burns et al. [4] 282 A. Najafi-Nejad-Nasser et al.
Fig. 2 Water velocity vectors for Run 3 (Table 2)
Fig. 3 Water flow streamlines for Run 3
sides in the lower water column and creates eddies (Fig. 3). These eddies producediverging flows from the centre in the upper water column. Water flow converges tocompensate the upward motion at the centre. These flow features are realistic. Also,there are upward and downward motions on both the left and right sides of the waterbody. These flow patterns would enhance renewal of bottom water with oxygenatedsurface water. For Run 3 (Fig. 2), the maximum velocity has a magnitude of nearly 5.6 times ofthe initial velocity, uo , of bubbles entering the water column. The maximum velocityhas a magnitude of 5.4uo for Run 5, being the lowest among Runs 1–11. For theother runs, the maximum velocity has a magnitude ranging from 5.5uo (for Run 4)to 8.5uo (for Run 7). Application of CFD Modelling to the Restoration of Eutrophic Lakes 283
The flow streamlines (Fig. 3) show clockwise and counterclockwise eddymotions on the right and left sides of the water body. These eddy motions penetratethe entire water depth, meaning that aeration is effective in producing exchange ofwater masses. For Runs 1–11, the radius of significant influence is larger than eighttimes the inlet radius D. Water circulation occurs over virtually the whole width ofthe water body. In other words, artificial circulation can effectively be created byinjecting air bubbles. The vertical component of water velocities at three selected heights (0.05, 0.1 and0.2 m) above the bottom varies with radial distance, r, from the centre (Fig. 4a–c).The velocity component has peak values at the centre (or r D 0), decreasing rapidlywith r. It drops to zero at r D 0:1 m for Runs 1, 2, 7 (not shown) and 11 (Fig. 4a–c). The velocity component drops to zero at r D 0:12 m, and the flow directionchanges from upward to downward for Runs 3–6 and 8–10 (Fig. 4a–c, not shownfor Runs 1–7, 9, 10). In summary, Runs 3–6 and 8–10 predict upward flow within0 < r < 0:12 m; Runs 1–2, 7 and 11 predict upward flow within 0 < r < 0:1; this istrue at different heights above the bottom. These predictions indicate that the resultsare sensitive to model setup (Table 2). Experimental data [1] show similar features;water velocity intensifies with increasing height above the inlet but weakens withincreasing radial distance. Bubble rising velocity, ua , varies with r (Fig. 5a–c). The values are extracted fromthe model results at model time of t D 10:7 s for the same heights as in Fig. 4a–c. uadecreases from peak values at the centre with increasing r; this feature was observedfrom experiments [1]. In Table 3, bubble rising velocities at different radial distances are compared withexperimental data [1]. Runs 11 produces the smallest relative error. The results of air and water velocities for Run 3 are better than those for Run2, especially at larger r distances (not shown). The reason is that the former runconsiders the effect of interfacial forces. On average (at five different heights abovethe bottom and different radial distances from the centreline), this consideration hasimproved the predictions of ua by 7.3 %, uw by 6.8 %, and VFa by 62 %. At thespecific height z = 0.3 m, the consideration has reduced the average relative error inthe predictions of ua from 18.8 % to 3.2 %.
a b cuw (m/s)
0.4 0.4 0.4
0 0 0
0 0.1 0.2 r (m) 0 0.1 0.2 r (m) 0 0.1 0.2 r (m) Run8 EXP Run11
Fig. 4 Water velocity .uw / distribution with radial distance, r. (a) z D 0.05 (m). (b) z D 0.1 (m).(c) z D 0.2 (m) 284 A. Najafi-Nejad-Nasser et al.
a b c 1 1 1 ua (m/s)
0.5 0.5 0.5
0 0 0 0 0.1 0.2 r (m) 0 0.1 0.2 r (m) 0 0.1 0.2 r (m) Run8 EXP Run11
Fig. 5 Distribution of air velocity .ua / with radial distance, r. (a) z D 0.05 (m). (b) z D 0.1 (m).(c) z D 0.2 (m)
Table 3 Relative errors of computed air velocity (ua ), and average of water turbulent kineticenergy (K) Run 1 2 3 4 5 6 7 8 9 10 11 ua (%) 35:3 23:6 19:6 20:1 19:8 17:8 25:9 14:4 20:1 16:2 12:0 K(J/kg) 0:019 0:007 0:006 0:006 0:006 0:006 0:005 0:004 0:004 0:006 0:002
a b c 0.04K (J/kg)
0.01 0.02 0.02 0.005 0.01
0 0 0 0 0.1 0.2 r (m) 0 0.1 0.2 r (m) 0 0.1 0.2 r (m) Run8 Run11
Fig. 6 Distribution of water turbulent kinetic energy .K/ with r. (a) z D 0.05 (m). (b) z D 0.1 (m).(c) z D 0.2 (m)
Variations in turbulent kinetic energy, K, of water flow with radial distance r atthree different heights from the bottom are illustrated in Fig. 6a–c. As bubbles rise,part of their kinetic energy is converted into turbulent kinetic energy. The maximumK values occur slightly off the centreline. K increases with increasing height. Ata given height, it decreases with increasing r. Overall, air bubbles entering intothe water body cause water motions and turbulence. Water and bubble motionshelp prevent sedentary condition, which is known to contribute to water-qualitydegradation. The averages (over the whole water body) of K values are listed inTable 3. Distributions of predicted air volume fraction, VFa , with r are shown in Fig. 7a–c. Values of VFa are extracted from the model results for Runs 1 to 11 at model timeof t D 10:7s for the same heights as in Fig. 4a–c. The snapshots show that at a givenheight, VFa decreases from peak values with increasing r. The peak values occurslightly off the centre for Runs 1–11 (Fig. 7a–c, not shown for Runs 1–7, 9, 10). Application of CFD Modelling to the Restoration of Eutrophic Lakes 285
a b c 0.3VF3 (m/s)
0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 0.05 r (m) 0 0.05 r (m) 0 0.05 r (m) Run8 EXP Run11
Fig. 7 Distribution of air volume fraction, VFa , with radial distance r. (a) z D 0.05 (m). (b) z D 0.1(m). (c) z D 0.2 (m)
A strong lake stratifications known to prohibit fluid motions and mixing in thevertical. However, the consideration of density stratification effects is beyond thescope of this paper.
4 Conclusions
This paper discusses artificial circulation in lakes, induced by injecting air bubbles,for the control of eutrophication. We have simulated artificial circulation using CFDmodelling techniques, and reached the following conclusions: (1) The injectiontriggers turbulent motions of water and bubbles, which feature a strong upwardflow above the injection location and energetic turbulent eddies on both sides ofthe upward flow. (2) These large scale eddies enhance renewal of bottom water withoxygenated surface water, which helps improve the dissolved oxygen level in thelower water column. (3) Air bubbles entering the water column produce turbulentkinetic energy; this source of energy will maintain small scale eddy motions in thelake water, with beneficial mixing effects. (4) The dissolved oxygen level in the lakewater is improved as a direct response to air bubbles entering the water column. (5)From the computational perspective, a 2D axisymmetric computational domain isrecommended for two reasons: (a) it offers high computational efficiency, relative toa 3D domain; and (b) it appears to be sufficient to capture measured characteristicsof air and water velocities. (6) Model predictions of water velocity, air velocity, airvolume fraction agree well with experimental data. The best agreement is obtainedwith the use of the Schiller-Nauman model for drag, the Tomiyama model for lift,the Lopez-de-Bertodano model for turbulent dispersion, and the k model forturbulence closure. 286 A. Najafi-Nejad-Nasser et al.
Acknowledgements This study received financial support through Discovery Grants held by S.S.Li and C.N. Mulligan
References
1. Anagbo, P.E., Brimacombe, J.K.: Plume characteristics and liquid circulation in gas injection through a porous plug. Metall. Trans. B 21B, 637–648 (1990) 2. Ansari, A.A., Gill, S.S., Lanza, G.R., Rast, W.: Eutrophication: Causes, Consequences and Control. Springer, Dordrecht/Heidelberg/London/New York (2011) 3. Azzopardi, B., Zhao, D., Yan, Y., Morvan, H., Mudde, R.F., Lo, S.: Hydrodynamics of Gas- liquid Reactors: Normal Operation and Upset Conditions. Wiley. Chichester, United Kingdom, (2011) 4. ANSYS: Fluent 6.3 User’s Guide (2006) 5. Dai, L., Pan, G.: The effects of red soil in removing P from water column and reducing P release from sediment in Lake Taihu. Water Sci. Technol. 65(5), 1052–1058 (2014) 6. Environment Canada: Canadian guidance framework for the management of P in freshwater systems. Ecosyst. Health: Sci.-Based Solut. 1–8, 1–114 (2004) 7. Hjarne, J., Chernoray, V., Larsson, J., Lofdahl, L.: Numerical validations of secondary flows and loss development downstream of a highly loaded low pressure turbine outlet guide Vane cascade. ASME Turbo Expo 2007, 723–733 (2007) 8. Imteaz, M.A., Asaeda, T.: Artificial mixing of lake water by bubble plume and effects of bubbling operations on algal bloom. Water Res. 34(6),1919–1929 (2000) 9. Kim, S.H., Kim, J.Y., Park, H., Park, N.S.: Effects of bubble size and diffusing area on destrafication efficiency in bubble plumes of two-layer stratification. ASCE J. Hydraul. Eng. 136, 106–115 (2010)10. Matsui, S., Ide, S., Ando, M.: Lake reservoirs: reflecting waters of sustainable use. Water Sci. Technol. 32(7),221–224 (1955)11. Penn, M.R., Auer, M.T., Doerr, S.M., Driscoll, C.T., Brooks, C.M., Effler, S.W.: Seasonality in P release rates from the sediments of a hypereutrophic lake under a matrix Of Ph and redox conditions. Can. J. Fish. Aquat. Sci. 57, 1033–104 (2000)12. Phillips, G., Bramwell, A., Pitt, J., Stansfield, J., Perrow, M.: Practical application of 25 Years’ research into the management of shallow lakes. Hydrobiologia 395/396 61–76 (1999)13. Qunhe, W., Renduo, Z., Shan, H., Hengjun, Z.: Effects of bacteria on nitrogen and P release from river sediment. Environ. Sci. 20, 404–412 (2008)14. Reddy, K.R., Q’connor, G.A., Schelske, C.L.: P Biogeochemistry of Subtropical Ecosystems, 1st edn., p. 101. CRC Press, Boca Raton (1999)15. Rensen, J., Roig, V.: Experimental study of the unsteady structure of a confined bubble plume. Int. J. Multiph. Flow 27, 1431–1449 (2001)16. Sahoo, G.B., Luketina, D.: Modeling of bubble plume design and oxygen transfer for reservoir restoration. Water Res. 37, 393–401 (2003)17. Schauser, I., Chorus, I.: Assessment of internal and external lake restoration measures for two Berlin lakes. Lake Reserv. Manag. 23, 366–376 (2007)18. Schuler, M., Zehnder, F., Weigand, B., von Wolfersdorf, J., Neumann, S.O.: The effect of turning vanes on pressure loss and heat transfer of a Ribbed rectangular two-pass internal cooling channel. ASME J. Turbomach. 133(2), 0211017–(1–10) (2011)19. Sharpley, A.N., Chapra, S.C., Wedepohl, R., Sims, J.T., Daniel, T.C., Reddy, K.R.: Managing agricultural P for protecting of surface waters: issues and options. J. Environ. Qual. 23(3), 437–451 (1994)20. Sondergaard, M., Jensen, J.P., Jappesen, E.: Role of sediment and internal loading of P in shallow lakes. Hydrobiologia 506–509, 135–145 (2003) Application of CFD Modelling to the Restoration of Eutrophic Lakes 287
21. Van Der Molen, D.T., Boers, P.C.M.: Eutrophication Control in the Netherlands. Hydrobiologia 136, 403–409 (1999)22. Wang, X., Naji, H., Mezrhab, A.: Computational investigation of different models when predicting airflow in an enclosure. ASME Press. Vessels Piping Div. Conf. 4, 179–188 (2008)23. Yeoh, G.H., Tu, J.: Computational Techniques for Multiphase Flows: basics and applications. Elsevier, Butterworth-Heinmann, United Kingdom, (2009)24. Yum, K., Kim, S.H., Park, H.: Effects of plume spacing and flowrate on destrafication efficiency of air diffusers. Water Res. 42, 3249–3262 (2008) On the Co-infection of Malariaand Schistosomiasis
Kazeem O. Okosun and Robert Smith?
Abstract Mathematical models for co-infection of diseases (that is, the simulta-neous infection of an individual by multiple diseases) are sorely lacking in theliterature. Here we present a mathematical model for the co-infection of malaria andschistosomiasis. We derive reproduction numbers for malaria and schistosomiasisindependently, then combine these to determine the effects of disease interactions.Sensitivity indices show that malaria infection may be associated with an increasedrate of schistosomiasis infection. However, schistosomiasis infection is not asso-ciated with an increased rate of malaria infection. Therefore, whenever there isco-infection of malaria and schistosomiasis in the community, our model suggeststhat control measures for each disease should be administered concurrently foreffective control.
1 Introduction
Malaria and schistosomiasis often overlap in tropical and subtropical countries,imposing tremendous disease burdens [4, 8, 14]. The substantial epidemiologicaloverlap of these two parasitic infections invariably results in frequent co-infections[7, 18]. The challenges facing the development of a highly effective malaria vaccinehave generated interest in understanding the interactions between malaria and co-endemic helminth infections, such as those caused by Schistosoma, that couldimpair vaccine efficacy by modulating host-immune responses to Plasmodiuminfection and treatment [13, 14]. Both malaria and schistosomiasis are endemic tomost African nations. However, the extent to which schistosomiasis modifies therate of febrile malaria remains unclear.
K.O. OkosunDepartment of Mathematics, Vaal University of Technology, Vanderbijlpark, South Africae-mail: [emailprotected]. Smith? ()University of Ottawa, Ottawa, ON, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 289J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_27 290 K.O. Okosun and R. Smith?
Mathematical modelling has been an important tool in understanding the dynam-ics of disease transmission and also in the decision-making processes regardingintervention mechanisms for disease control. For example, Ross [12] developedthe first mathematical models of malaria transmission. His focus was on mosquitocontrol, and he showed that, for the disease to be eliminated, the mosquitopopulation should be brought below a certain threshold. Another classical resultis due to Anderson and May [1], who derived a malaria model with the assumptionthat acquired immunity in malaria is independent of exposure duration. There is an urgent need for co-infection models for infectious diseases, particu-larly those that mix neglected tropical diseases with “the big three” (HIV, TB andmalaria) [8]. Recently, the authors in [10] proposed a model for schistosomiasis andHIV/AIDS co-dynamics, while the co-infection dynamics of malaria and cholerawere studied in [11]. However, few studies have been carried out on the co-infection of schistosomiasis. To the best of our knowledge, no work has been doneto investigate the malaria–schistosomiasis co-infection dynamics. In this paper, we formulate and analyse an SIR (susceptible, infected andrecovered) model for malaria–schistosomiasis co-infection, in order to understandthe effect that controlling for one disease may have on the other.
2 Model Formulation
Our model subdivides the total human population, denoted by Nh , into subpop-ulations of susceptible humans Sh , individuals infected only with malaria Im ,individuals infected with only schistosomiasis Isc , individuals infected with bothmalaria and schistosomiasis Cms , individuals who have recovered from malaria Rmand individuals who have recovered from schistosomiasis Rs . The total mosquitovector population, denoted by Nv , is subdivided into susceptible mosquitoes Sv andmosquitoes infected with malaria Iv . Similarly, the total snail vector population,denoted by Nsv , is subdivided into susceptible snails Ssv and snails infected withschistosomiasis Isv . Thus Nh D Sh C Im C Is C Cms C Rs C Rm , Nv D Sv C Iv andNsv D Ssv C Isv : The model is given by the following system of ordinary differential equations.
Sh0 D h C "Rs C ˛Rm ˇ1 Sh 1 Sh h Sh Im0 D ˇ1 Sh 1 Im . C h C /Im 0 Isc D 1 Sh ˇ1 Isc .! C h C /Isc 0 Cms D ˇ1 Isc C 1 Im .ı C h C C /Cms R0m D Im .˛ C h /Rm C ıCms R0s D !Isc ." C h /Rs C .1 /ıCms On the Co-infection of Malaria and Schistosomiasis 291
Sv0 D v ˇ2 Sv v Sv Iv0 D ˇ2 Sv v Iv 0 (1) Ssv D s 2 Ssv sv Ssv 0 Isv D 2 Ssv sv Isv ;
with the transmission rates given by
ˇh Iv Isv ˇv .Im C Cms / s .Isc C Cms / ˇ1 D ; 1 D ; ˇ2 D ; 2 D : Nh Nh Nh Nh
Birth rates for humans, mosquitoes and snails are, respectively, h , v andsv , while the corresponding mortality rates are h , v and sv . Here is theschistosomiasis-related death rate and is the malaria-related death rate. Theimmunity-waning rates for malaria and schistosomiasis are ˛ and " respectively,while the recovery rates from malaria, schistosomiasis and co-infection are , !and ı respectively. The term ı accounts for the portion of co-infected individualswho recover from malaria, while .1 /ı accounts for co-infected individualswho recover from schistosomiasis; thus (satisfying 0 1) represents thelikelihood of individuals to recover from malaria first. Note that all parametersmight in practice vary with time; however, we shall take variations in our criticalparameters into account with a sensitivity analysis.
3 Analysis of Malaria–Schistosomiasis Co-infection Model
The malaria–schistosomiasis model (1) has a disease-free equilibrium, given by ! h v sE 0 D .Sh ; Im ; Isc ; Cms ; Rm ; Rs ; Sv ; Iv ; Ssv ; Isv /D ; 0; 0; 0; 0; 0; ; 0; ;0 : h v sv
The linear stability of E 0 can be established using the next-generation method[17] on the system (1). It follows that the reproduction number of the malaria–schistosomiasis model (1), denoted by R msc , is given by
R msc D maxfR sc ; R 0m g ;
where s v ˇh ˇv h R 0m D h 2v . C C h / s s s h R sc D : h .m C ! C h /2sv 292 K.O. Okosun and R. Smith?
Note that the reproduction number produced by the next-generation method pro-duces a threshold quantity and not necessarily the average number of secondaryinfections [9]. We thus have the following theorem.Theorem 1 The disease-free equilibrium E 0 is locally asymptotically stable when-ever R msc < 1 and unstable otherwise.
3.1 Impact of Disease Interactions
To analyse the effects of schistosomiasis on malaria and vice versa, we begin byexpressing R sc in terms of R 0m . We solve for h to get
D1 R 20m h D ; D2 D3 R 20m
where
D1 D h 2v . C / ; D2 D v ˇh ˇv ; D3 D h 2v :
Substituting into the expression for R sc , we obtain s s s D1 R 20m R sc D : (2) Œ. C !/D2 C .D1 . C !/D3 /R 20m h 2sv
Differentiating R sc with respect to R 0m leads to r s 1 0m D R2 . C !/D2 Œ.C!/D C.D s.C!/D @R sc 2 1 2 2 3 /R0m h sv D : (3) @R 0m Œ. C !/D2 R 0m C .D1 . C !/D3 /R 30m
Similarly, expressing h in terms of R sc , we get
D4 R 2sc h D ; (4) D5 D6 R 2sc
where
D4 D h 2sv . C !/ ; D5 D s s ; D6 D h 2sv : On the Co-infection of Malaria and Schistosomiasis 293
Substituting into the expression for R 0m , we obtain s D4 ˇh ˇv v R 2sc R 0m D : (5) Œ. C /D5 C .D4 . C /D6 /R 2sc h 2sv
3.2 Sensitivity Indices of Rsc when Expressed in Terms of R0m
We next derive the sensitivity of Rsc in (2) (i.e., when expressed in terms of R0m )to each of the 13 different parameters. However, the expression for the sensitivityindices for some of the parameters are complex, so we evaluate the sensitivityindices of these parameters at the baseline parameter values as given in Table 1.Since the effect of immunity in the control of re-infection is not entirely known[6], we have assumed the schistosomiasis immunity waning rate. Due to a lack ofdata in the literature, assumptions were made for the recovery rate of co-infectedindividuals, ı, recovery rate of schistosomiasis-infected individuals, !, and the rateof recovery from malaria for co-infected individuals, .
Table 1 Parameters in the co-infection modelParameter Description value Ref Malaria-induced death 0.05–0.1 day1 [16]ˇh Malaria transmissibility to humans 0.034 day1 [2]ˇv Malaria transmissibility to mosquitoes 0.09 day1 [2] Schistosomiasis transmissibility to 0.406 day1 [15] humanss Schistosomiasis transmissibility to 0.615 day1 [3] snailsh Natural death rate in humans 0.00004 day1 [2]v Natural death rate in mosquitoes 1/15–0.143 day1 [2]sv Natural death rate in snails 0.000569 day1 [3, 15]˛ Malaria immunity waning rate 1/(60*365) day1 [2]" Schistosomiasis immunity waning rate 0.013 day1 Assumedh Human birth rate 800 people/day [3]v Mosquitoes birth rate 1000 mosquitoes/day [2]s Snail birth rate 100 snails/day [5]ı Recovery rate of co-infected individual 0.35 day1 Assumed! Recovery rate of 0.0181 day1 Assumed schistosomiasis-infected individual Recovery rate of malaria-infected 1/(2*365) day1 [2] individual Co-infected proportion who recover 0.1 Assumed from malaria only Schistosomiasis-induced death 0.0039 day1 [3] 294 K.O. Okosun and R. Smith?
Table 2 Sensitivity indices of Rsc expressed in terms of R0m Sensitivity index Sensitivity index Parameter Description if R0m < 1 if R0m > 11 sv Snail mortality 1 12 v Mosquito mortality 0:56 0:073 s Schistosomiasis transmissibility to snails 0:5 0:54 s Snail birth rate 0:5 0:55 ˇh Malaria transmissibility to humans 0:28 0:036 ˇv Malaria transmissibility to mosquitoes 0:28 0:037 v Mosquito birth rate 0:28 0:038 h Human birth rate 0:22 0:479 Malaria-induced death 0:12 0:3110 ! Recovery from schistosomiasis 0:10 0:2611 m Schistosomiasis-induced death 0:02 0:0512 Recovery from malaria rate 0:003 0:0084
The sensitivity index of Rsc with respect to , for example, is
@Rsc Rsc D 0:5 : (6) @ Rsc
The detailed sensitivity indices of Rsc resulting from the evaluation of the otherparameters of the model are shown in Table 2. Table 2 shows the parameters, arranged from the most sensitive to the least.For R0m < 1, the most sensitive parameters are the snail mortality rate, themosquito mortality rate, the transmissibility of schistosomiasis to snails and thesnail birth rate (sv , v , s and s , respectively). Since Rsvsc D 1, increasing(or decreasing) the snail mortality rate sv by 10 % decreases (or increases) Rsc by10 %; similarly, increasing (or decreasing) the mosquito mortality rate, v , by 10 %increases (or decreases) Rsc by 5:6 %. In the same way, increasing (or decreasing)the transmissibility of schistosomiasis to snails, s , increases (or decreases) Rscby 5 %. As the malaria parameters ˇh , ˇv and v increase/decrease by 10 %, thereproduction number of schistosomiasis, Rsc , decreases by 2:8 % in all three cases. For R0m > 1, the most sensitive parameters are the snail mortality rate, the rateof a snail getting infected with schistosomiasis, the snail birth rate, the human birthrate, malaria-induced death and recovery from schistosomiasis (sv , s , s , h , ,!, respectively). Since Rssc D 0:5, increasing (or decreasing) by 10 % increases (ordecreases) Rsc by 5 %; similarly, increasing (or decreasing) the recovery rate, !, by10 % increases (or decreases) Rsc by 2:6 %. Also, as the malaria parameters ˇh , ˇvand v increase/decrease by 10 %, the reproduction number of schistosomiasis, Rsc ,decreases by only 0:3 % in all three cases. On the Co-infection of Malaria and Schistosomiasis 295
Table 3 Sensitivity indices of R0m expressed in terms of Rsc Sensitivity index Sensitivity index Parameter Description if Rsc < 1 if Rsc > 11 ˇv Malaria transmissibility to mosquitoes 0:5 0:52 v Mosquito birth rate 0:5 0:53 Schistosomiasis transmissibility to humans 0:5 0:54 s Schistosomiasis transmissibility to snails 0:5 0:55 s Snail birth rate 0:5 0:56 Malaria-induced death 0:49 0:497 ! Recovery from schistosomiasis 0:41 0:418 m Schistosomiasis-induced death 0:09 0:099 Recovery from malaria 0:01 0:0110 sv Snail mortality 0:0000002 0:00000711 h Human birth rate 0:0000001 0:000004
It is clear that Rsc is sensitive to changes in R0m . That is, the sensitivity of Rsc toparameter variations depends on R0m ; whenever, R0m < 1, Rsc is less sensitive to themalaria parameters.
3.3 Sensitivity Indices of R0m when Expressed in Terms of Rsc
Similar to the previous subsection, we derive the sensitivity of R0m in (5) (i.e. whenexpressed in terms of Rsc ) to each of the different parameters. The sensitivity indexof R0m with respect to ˇh , for example, is
@R0m ˇh ˇRh0m D 0:5 : (7) @ˇh R0m
The detailed sensitivity indices of R0m resulting from the evaluation to the otherparameters of the model are shown in Table 3. It is clearly seen from Table 3that the malaria reproduction number, R0m , is not sensitive to any variation in theschistosomiasis reproduction number Rsc .
4 Numerical Simulations
Table 1 lists the parameter descriptions and values used in the numerical simulationof the co-infection model. 296 K.O. Okosun and R. Smith?
(a) (b) = 0.4
3 = 0.4 70 = 0.2
= 0.1 Malaria Infected Individuals
2.5 60
Co Infected Individuals 2 = 0.3 50
1.5 40 = 0.2 1
30 0.5
20 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (days) Time (days)
4 x 10(c) 2 (d) 600
h = 0.234 1.8 h = 0.234 500 Schistosomiasis Infected Individuals
1.6
1.4 = 0.134 Co Infected Individuals h = 0.134 400 h 1.2
1 300 = 0.034 h 0.8
0.6 200
0.4 = 0.034 h 100 0.2
0 20 40 60 80 100 0 20 40 60 80 100 Time (days) Time (days)
Fig. 1 Simulations of the malaria–schistosomiasis model showing the effect of varying transmis-sion rates
Figure 1a,b shows the effect of varying the schistosomiasis transmission param-eter on the number of individuals infected with malaria, Im , and the number ofco-infected individuals, Cms . This illustrates that effective control of schistosomiasiswould enhance the control of malaria. Conversely, Fig. 1c,d shows the effect ofvarying the malaria transmission parameter ˇh on the number of individuals infectedwith schistosomiasis, Isc , and the number of co-infected individuals. This illustratesthat effective control of malaria would enhance control of co-infection but have onlyminimal effect on schistosomiasis prevalence. Figure 2 shows the effect of varying the death rate of mosquitoes v (for exam-ple, through spraying) on the number of individuals infected with schistosomiasisand the number of co-infected individuals. As the mosquitoes are controlled, thenumber of individuals infected with malaria falls dramatically, as does the numberof co-infected individuals, while the number of schistosomiasis-infected individualsonly decreases slightly. On the Co-infection of Malaria and Schistosomiasis 297
(a) 4.5 (b) 100
4 90 μv = 0.07 80 Malaria Infected Individuals
3.5
Co Infected Individuals 70 3 60 2.5 μv = 0.09 μv =0.07 50 2 40 1.5 μ =0.09 v 30 μ = 0.143 1 v μ =0.143 v 20 0.5 10 0 0 20 40 60 80 100 0 20 40 60 80 100 Time (days) Time (days)
(c) 11000 μ =0.09 v
10000 μ =0.07 v Schistosomiasis Infected Individuals
9000
8000
7000 μ =0.143 v 6000
5000
4000
3000
2000
1000
0 20 40 60 80 100 Time (days)
Fig. 2 Simulations of the malaria–schistosomiasis model showing the effect of varying themosquito death rate
5 Concluding Remarks
In this paper, we formulated and analysed a deterministic model for the transmissionof malaria–schistosomiasis co-infection. We derived basic reproduction numbers foreach infection and determined the sensitivity of each reproduction number to allparameters. Our analysis shows that malaria infection may be associated with anincreased rate of schistosomiasis infection. However, in our model, schistosomiasisinfection is not associated with an increased rate of malaria infection. Therefore,whenever there is co-infection of malaria and schistosomiasis in the community,our model suggests that control measures for both diseases should be administeredconcurrently for effective control.
Acknowledgements The authors are grateful to an anonymous reviewer whose commentsenhanced the manuscript. OKO acknowledges the Vaal University of Technology Research Officeand the National Research Foundation (NRF), South Africa, through the KIC Grant ID 97192 forthe financial support to attend and present this paper at the AMMCS-CAIM 2015 meeting. RS?is supported by an NSERC Discovery Grant. For citation purposes, please note that the questionmark in “Smith?” is part of the author’s name. 298 K.O. Okosun and R. Smith?
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Vardayani Ratti, Peter G. Kevan, and Hermann J. Eberl
Abstract In this paper, we present a general discrete-continuous modeling frame-work to study the effect of swarming on the dynamics of a honeybee colonyinfested with varroa mite and Acute Bee Paralysis Virus (ABPV) . Two scenarios arestudied under which swarming takes place i.e., swarming due to overcrowding andswarming at fixed time intervals. For this purpose, we use an existing mathematicalmodel in the literature. The dependent variables in the model are uninfected bees,infected bees, virus carrying mites and total mites that infest the colony. The modelis studied in variable coefficients, in particular, step functions with each season asa constant in time. It is observed that the percentage of healthy bees leaving withthe swarm has a great impact on the strength and survival of the parent colony. Acolony, that otherwise dies off due to virus, survives as a properly working colonyif the percentage of the mites leaving the parent colony is above a critical value.
1 Introduction
A honey bee colony consists of a single reproductive queen, 20,000–60,000 adultworkers, 10,000–30,000 individuals at brood stage (eggs, larvae and pupae), andseveral hundred drones [13]. A large population of workers is needed to carry out thetasks of the bee colony, including cleaning, brood rearing, guarding the hive etc. Inorder to maintain a high reproduction level in the colony, the bees start preparing forswarming. Swarming is the natural method of reproduction of honeybee colonies.In the process of swarming, the original single colony reproduces to two or morecolonies. During swarming, almost 50–70 % of the worker bees leave the parentcolony with the queen (old mated queen in case of the first swarm) to a new site[3, 14]. The colony starts preparing for the swarm 1 month before the swarm isissued. Swarming normally takes place in mid-spring [14]. The timing of swarmingvaries with seasons and location of the hives. Since swarming affects pollinationand honey production, it has an impact on agriculture and economy as well. The
V. Ratti () • P.G. Kevan • H.J. EberlUniversity of Guelph, 50 Stone Road E, Guelph, ON, N1G 2W1, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 299J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_28 300 V. Ratti et al.
main symptom of swarming is the preparation of brood cups and queen rearing.The causes of swarming could be (i) large colony size, (ii) high proportion ofyoung worker bees, (iii) reduced queen pheromone due to overcrowding, and (iv)abundance of pollen and nectar leading to overcrowding. There is a common causebehind all the symptoms i.e., overcrowding in the colony. When a swarm issues, the parasites present in the parent colony are dividedamong the parent colony and the new colony formed after swarming [3]. Thismay reduce the disease infestation in the parent colony. We investigate the effectof swarming on the colony infested with a parasite Varroa destructor and AcuteBee Paralysis Virus. Varroa destructor is an ectoparasitic mite that not only feedson the bees but also carries and transmits fatal viruses in the colony. The mite feedson bees’ haemolymph by piercing their inter-segmental membrane and transmits thevirus while feeding on them. When a virus carrying mite feeds on an infected bee,it releases the virus into the bee’s haemolymph. Thus, the uninfected bee becomesinfected and mite becomes virus free. When a virus free mite feeds on an infectedbee, it begins to carry virus. There are 20 bee viruses known so far, out of whichat least 14 viruses are reported to be associated with mites [5, 9]. These viruseshave different routes of transmission and different levels of virulence. One of themost common virus is the Acute Bee Paralysis Virus. It has also been implicatedin colony losses [4, 5, 15]. The bees infected by this virus are unable to fly, loosetheir body hair and tremble uncontrollably. Unlike the Deformed Wing Virus, wherebrood infested with the virus develop into an adult sick bee, the brood infected withABPV does not survive to the adult stage and dies immediately. There have been several SIR-type mathematical models developed for honeybee-varroa mite-virus systems [2, 11–13]. There have also been models on someaspects of swarming such as the use of bee dance, design of nest selection anddecision-making processes [1, 8, 10]. However, none of these models studied thecombined effect of swarming and the varroa-virus infestation in a honeybee colony.In this paper, we provide a general framework of difference-differential equationsto investigate how swarming affects the fate of the honeybee colony infested withvarroa mites and virus. We also use numerical simulations to study (i) swarmingdue to overcrowding (ii) swarming after a fixed time interval mimicking the naturaldeath cycle of a queen bee. We vary the proportion of healthy bees leaving theparent colony and observe its effect on the dynamics of the colony. We numericallycalculate a critical value of the proportion of mites leaving the parent colony belowwhich the parent colony dies off and above which it survives.
2 Model Equations
The underlying model of honeybee-varroa-ABPV disease dynamics from [11] is:
dm y x D ˇ1 .M m/ ˇ2 m (1) dt xCy xCy Swarming in a Honeybee Colony 301
dx x D g.x/h.m/ ˇ3 m d1 x 1 Mx (2) dt xCy dy x D ˇ3 m d2 y 2 My (3) dt xCy dM M D rM 1 (4) dt ˛.x C y/
The parameters are assumed to be non-negative. Because the size of the beecolony and the life span of bees vary drastically with seasons, the parameters areassumed to be seasonally varying. In particular, we assume the parameters to beperiodic functions of time with a period T; in practice T D 1 year. The parameter in (2) is the maximum birth rate, specified as the number ofworker bees emerging as adults per day. The function g.x/ expresses that a sufficiently large number of healthy workerbees is required to care for the brood. We think of g.x/ as a switch function. If xfalls below a critical value, which may seasonally depend on time, essential workin the maintenance of the brood cannot be carried out anymore and no new bees areborn. If x is above this value, the birth of bees is not hampered. Thus g.0; / D 0,dg.0/ dx 0, limx!1 g.x/ D 1. A convenient formulation of such switch like behavioris given by the sigmoidal Hill function
xn g.x/ D (5) K n C xn
where the parameter K is the size of the bee colony at which the birth rate is halfof the maximum possible rate and the integer exponent n > 1. If K D 0 is chosen,then the bee birth terms of the original model of [13] is recovered. Then the brood isalways reared at maximum capacity, independent of the actual bee population size,because g.x/ 1. The function h.m/ in (2) indicates that the birth rate is affected by the presenceof mites that carry the virus. This is in particular important for viruses like ABPVthat kill infected pupae before they develop into bees. The function h.m/ is assumedto decrease as m increases, h.0/ D 1, dm dh .m/ < 0 and limm!1 h.m/ D 0; [13]suggests that this is an exponential function h.m/ emk , where k is non-negative.We will use this expression in the computer simulations later on. The parameter ˇ1 in (1) is the rate at which mites that do not carry the virusacquire it. The rate at which infected mites lose their virus to an uninfected hostis ˇ2 . The rate at which uninfected bees become infected is ˇ3 , in bees per viruscarrying mite and time. Finally, d1 and d2 are the death rates for uninfected and infected honeybees. Wecan assume that infected bees live shorter than healthy bees, thus d2 > d1 . Equation (4) is a logistic growth model for varroa mites. By r we denote themaximum mite birth rate. The carrying capacity for the mites changes with the hostpopulation site, x C y, and is characterized by the parameter ˛ which indicates how 302 V. Ratti et al.
many mites can be sustained per bee on average. The parameters 1;2 in Eqs. (2)and (3) represent the mortality rates of bees due to mites feeding on them.
3 Mathematical Formulation of Discrete Interventions
We formulate the process of swarming as discrete interventions in a continuoussystem in the form of a hybrid system. The discrete interventions (ti ’s) may dependon the time or the state variables. For instance, if we assume that swarming takesplace after fixed time intervals i.e., mimicking the natural life cycle of a queen bee,the time at which discrete interventions takes place will be given a priori. On theother hand, if we assume that swarming takes place due to overcrowding in thehive, the occurrence of discrete interventions will be state dependent. The mathematical formulation in terms of hybrid dynamical systems is
zP.t/ D f .t; z.t//; t 2 .ti ; tiC1 /; z.t0 / D z0 ; z.ti / D Xi ; (6) XiC1 D F.ziC1 /; where ziC1 D lim z.t/: (7) t%tiC1
Here ti are the discrete times at which events take place and t0 < : : : < ti < tiC1 . Also, z 2 R4 , t 2 RC ; i 2 N, X 2 R4 and f 2 K where
K D f.m; x; y; M/ 2 R4 W m 0; M 0; x 0; y 0; x C y > 0g:
In the case of swarming, we assume that F.X/ D DX, where D 2 R44 is adiagonal matrix. Also, z D .m; x; y; M/T , f is the RHS of (1), (2), (3) and (4), and, X and D aregiven by 3 2 2 3 m a0 00 6x7 60 b 0 07 XD6 7 4 y 5; DD6 40 0 7: 1 05 M 00 0a
We assume that the infected bees do not leave the parent colony, because thedisease progresses rapidly and sick bees are unable to fly. We also assume thatthe percentage of total mites and virus carrying mites leaving the parent colonyis the same. Thus, two new parameters a and b are introduced. The parameter ais the percentage of mites staying in the parent colony and the parameter b is thepercentage of uninfected bees staying in the parent colony after the swarm leaves. Swarming in a Honeybee Colony 303
4 Computer Simulations
In the simulation experiments, we study the process of swarming taking place due toovercrowding in the hive and due to events like queen supercedure. We investigatehow these two causes of swarming affect the strength and survival of the parentcolony infested with mites and virus. In the case where swarming takes place due toovercrowding in the hive, discrete interventions take place when
x.t/ D cx where 0 < c 1: (8)
The quantity x is calculated numerically by running the disease free model(without swarming) until the steady state is reached and then taking the maximumof the steady state solution over a 2 year period. For the simulation experiments, wefix c D 0:95. In the case where swarming takes place due to queen failure, the timeat which discrete interventions (ti ’s) take place are fixed a priori as ti D t0 C i. t/,where t D 2T, T D 365 days and i D 1; 2; : : : ;. We assume that t0 D 45 i.e.,swarming takes place in mid May [14]. The seasonal averages of the parameters ˇi ; r; d1 ; d2 ; k and are given in Table 1.Lacking more detailed information about the parameters, we use these valuesto construct piecewise constant time varying parameter functions, assuming fourequally long seasons of 91.25 days [11]. Two sets of seasonal averages of theparameter ˛, [0.4784, 0.5, 0.5, 0.4784] and [0.1 0.1 0.1 0.1], for the spring, summer,fall and winter respectively, are used depending upon the scenario under study(see [11] for the choice of these values). In order to better present the simulationexperiments, we introduce two new variables a1 D 1 a and b1 D 1 b. Here,a1 is the proportion of mites leaving the parent colony and b1 is the proportion ofuninfected bees leaving the parent colony during swarming.
Table 1 Seasonal averages of model parameters, derived from the data presented in the literature[6, 7, 11, 13]. The parameters included here are kept constant for all simulations; the values of theparameters that are varied are given in the textParameter Spring Summer Autumn Winter Sourceˇ1 0.1593 0.1460 0.1489 0.04226 [13]ˇ2 0.04959 0.03721 0.04750 0.008460 [13]ˇ3 0.1984 0.1460 0.1900 0.03384 [13]d1 0.02272 0.04 0.02272 0.005263 [13]d2 0.2 0.2 0.2 0.005300 [13] 500 1500 500 0 [13]k 0.000075 0.00003125 0.000075 N/A [13]K 8000 12,000 8000 6000 [11]r 0.0165 0.0165 0.0045 0.0045 [6, 7]1 D 2 107 107 107 107 [11] 304 V. Ratti et al.
4(a) x 10 4 (b) 3 x 10 Virus carrying mites(m) 3.5 Virus carrying mites(m) Uninfected bees(x) Uninfected bees(x) 2.5 Total mites(M) 3 Infected bees (y) 2.5 Total mites(M) 2
Population Population
2 1.5 1.5 1 1 0.5 0.5
0 0 0 500 1000 1500 2000 2500 3000 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (days) Time (days)
Fig. 1 (a) Bee-mite-virus system in the absence of swarming. (b) Bee-mite-virus system in thepresence of swarming due to overcrowding. Threshold bee population at which swarming takesplace is 31,342 and we assume a1 D 0:65; b1 D 0:5
Illustrative Simulation: The purpose of this simulation is to (i) show a typicalsimulation of the temporal dynamics of a honeybee colony infested with mites andvirus and, (ii) to compare the system with and without the process of swarming. InFig. 1a, we consider a honeybee-mite-virus system with no swarming taking place.We assume the lower value of the parameter ˛ i.e., [0.1 0.1 0.1 0.1] so that the colonyfights off the virus but not the mites [11]. The uninfected bee population increasesfrom 13,350 in the beginning of spring and reaches its maximum of 31,195 in thesummer, decreases in the fall and reaches its minimum of 13,350 in the winter.This pattern repeats annually. After an initial transient period of 2 years, the mitesestablish themselves in the colony and follow the same pattern as the bee population.In Fig. 1b, we consider a honeybee colony infested with mites and virus whereswarming takes place due to overcrowding i.e., swarming takes place if the beepopulation exceeds a certain threshold value x (see Eq. 8). We assume the highervalue of the parameter ˛ i.e., [0.5 0.4784 0.4784 0.5]. We assume that a1 D 0:65 andb1 D 0:5 which means 50 % of the uninfected bees and 65 % of the mites leave theparent colony during swarming. We choose the parameters, a1 ; b1 and ˛, in order topresent a reference case. The bee population follows the same pattern as in Fig. 1a inthe first year. In the second year, the population increases in the spring and summerand then swarming takes place when the bee population reaches the threshold valuex which in this case is 31,342. The population the drops from 29,670 to 15,207bees during swarming and then increases to 23,780 due to the model dynamics. Thepopulation starts decreasing again in the fall and winter and reaches its minimumof 13,180. This pattern repeats itself annually. After a transient period of 1 year, themite population also follows the same oscillatory behavior as the bee population.Simulation Experiment I: In this simulation experiment, we consider the casewhere swarming takes place due to overcrowding. As in the illustrative simulation,we assume that swarming takes place if the bee population exceeds a certainthreshold value, x . We use the value of the parameter ˛ to be [0.5 0.4784 0.47840.5] for the spring, summer, fall and winter. In Fig. 2a, we compare a disease freecolony, a mite infested colony in which only 5 % of the mites leave the parent colony(a1 D 0:05), and, a mite infested colony in which 70 % of the mites leave the parent Swarming in a Honeybee Colony 305
4 (a) 2 x 10
Average bee population at the finaltime Disease free Mite infested with tm = 5% Mite infested with tm = 70% 1.5
0.5
0 0.5 0.6 0.7 0.8 0.9 1 Percentage of the bees leaving the parent colony
(b)3 x 10 4 (c) x 10 4 3 Virus carrying mites(m) Virus carrying mites(m) Uninfected bees(x) Uninfected bees(x) 2.5 Total mites(M) 2.5 Total mites(M)
2 2
PopulationPopulation
1.5 1.5
1 1
0.5 0.5
0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (days) Time (days)
(d)3 x 10 4
Virus carrying mites(m) (e) 3 x 10 4
Virus carrying mites(m) Uninfected bees(x) Uninfected bees(x) 2.5 Total mites(M) 2.5 Total mites(M)
2 2Population
Population
1.5 1.5
1 1
0.5 0.5
0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (days) Time (days)
Fig. 2 Swarming due to overcrowding: (a) Effect of the percentage of the healthy bees leaving theparent colony on the average healthy bee population of the (i) disease free colony, (ii) mite infestedcolony with a1 D 0:05, and (iii) mite infested colony with a1 D 0:7. (b) The colony fights offthe virus when b1 D 0:5 and a1 D 0:65. (c) The colony dies off after 7000 days when b1 D 0:6,a1 D 0:65. (d) The colony dies off after 6600 days when a1 D 0:64, b1 D 0:5. (e) The colonyfights off the virus and survives as a properly working colony when a1 D 0:65, b1 D 0:5
colony (a1 D 0:7). We vary the percentage of healthy bees leaving the parent colonyover 50–100 % which covers the range (50–70 %) given by [3, 14] i.e., we vary b1from 0.5 to 1. In the case of the disease free colony, the average population startsfrom 19,000 and decreases gradually as the parameter b1 increases. The averagepopulation suddenly drops down when b1 D 0:87. In the case of the mite infestedcolony with a1 D 0:05, the average bee population starts at a lower level (i.e., at18,000) than in the disease free case and decreases to 11,000 bees as b1 increasesfollowed by a sudden drop to 0 when b1 D 0:87. When a1 D 0:7, the average beepopulation starts at 18,500 which is between the initial average population in thedisease free case and the case when a1 D 0:05. The critical value for the percentageof bees leaving the parent colony is the same in all three cases. Figure 2b, c showthe effect of the percentage of uninfected bees leaving the parent colony (b1 ) on the 306 V. Ratti et al.
dynamics of the colony; we fix the parameter a1 D 0:65. In Fig. 2b, the colony fightsof the virus and survives as a properly working colony when b1 D 0:5. In Fig. 2c,the colony dies off after 7000 days when b1 D 0:6. Figure 2d, e show how thecolony, that otherwise dies off due to virus, survives as a properly working colony ifthe percentage of mites leaving the parent colony is above a threshold value; we fixthe parameter b1 D 0:5. In Fig. 2d, the colony dies off due to virus after 6600 dayswhen a1 D 0:64. Figure 2e shows that when a1 D 0:65, the colony fights off thevirus and works as a properly working colony.Simulation Experiment II: In this simulation experiment, we assume that swarm-ing occurs at fixed time intervals of 2 years mimicking the natural death cycle ofthe queen bee. We assume that the swarm leaves the colony every 2 years in themid of May [14]. We use the value of the parameter ˛ to be [0.1 0.1 0.1 0.1] forthe spring, summer, fall and winter. In Fig. 3a, we compare a disease free colony, amite infested colony with a1 D 0:05, and, a mite infested colony with a1 D 0:7. Weinvestigate the effect of the percentage of bees leaving with the swarm (b1 ) on theaverage population of the parent colony. The parameter b1 is varied from 0.5 to 1.In case of the disease free colony, the average bee population starts from 21,000 andremains constant when the parameter b1 is varied from 50 % to 76 % and suddenlydrops down to 0 when b1 reaches 0:77. In case of the mite infested colony witha1 D 0:05, the average bee population starts below the disease free population andremains constant until the parameter b1 reaches 0:75 when it suddenly drops downto 0. In case of the mite infested colony where a1 D 0:7, the average bee populationstarts at the same level as in case of a1 D 0:05 and remains constant until a1 reaches0:76 and then it suddenly drops down to 0. It is interesting to note that the thresholdvalue for the parameter b1 is the maximum in case of the disease free colony whichis followed by the mite infested case with a1 D 0:7 which in turn is followed by themite infested case with a1 D 0:05. Figure 3b, c show the effect of the percentage of uninfected bees leaving theparent colony (b1 ) on the survival of the colony. The parameter a1 is fixed to be 0.7.In Fig. 3b, the colony fights off the virus and survives as a properly working colonywhen b1 D 0:76. In Fig. 3c, the colony dies off after 1000 days when b1 D 0:77.Figure 3d, e show how the colony, that otherwise dies off due to virus, survivesas a properly working colony if the parameter a1 is above a threshold value. Theparameter b1 is fixed to be 0.5. In Fig. 3d, the colony dies off due to virus after6000 days when a1 D 0:91. Figure 3e shows that when a1 D 0:92, the colony fightsoff the virus and survives as a properly working colony.
5 Summary and Conclusion
• A general framework is provided to incorporate the discrete interventions into the model using a discrete-continuous model. This framework is applied to the process of swarming which involves two types of discrete interventions: (i) due Swarming in a Honeybee Colony 307
4 x 10 (a) 2.5
Average bee population at the finaltime Disease free Mite infested with tm = 5% 2 Mite infested with tm = 70%
1.5
0.5
0 0.5 0.6 0.7 0.8 0.9 1 Percentage of the bees leaving the parent colony
(b) 3.5 x 10 4 (c) 3.5 x 10 4
Virus carrying mites(m) Virus carrying mites(m) 3 Uninfected bees(x) 3 Uninfected bees(x) Total mites(M) Total mites(M) 2.5 2.5 Population
Population 2 2
1.5 1.5
1 1
0.5 0.5
0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 500 1000 1500 2000 2500 3000 Time (days) Time (days)
(d) 3.5 x 10 4
Virus carrying mites(m) (e) x 10 4 3.5 Uninfected bees(x) Virus carrying mites(m) 3 Total mites(M) Uninfected bees(x) 3 Total mites(M) 2.5 2.5 Population
Population
2 2 1.5 1.5 1 1
0.5 0.5
0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (days) Time (days)
Fig. 3 When swarming takes place every 2 years: (a) Effect of the percentage of the healthy beesleaving the parent colony on the average healthy bee population of the colony that is (i) diseasefree, (ii) mite infested with a1 D 0:05, and (iii) mite infested with a1 D 0:7. (b) The colonyfights off the virus when b1 D 0:76 and a1 D 0:7. (c) The colony dies off after 1000 days whenb1 D 0:77 and a1 D 0:7. (d) The colony dies off after 6000 days when a1 D 0:91 and b1 D 0:5.(e) The colony fights off the virus and survives as a properly working colony when a1 D 0:92 andb1 D 0:5
to queen failure i.e., when discrete events occur at times that are a priori fixed (ii) due to overcrowding i.e., when occurrence of discrete events depend on the state of the system.• In case of a disease free colony and a mite-infested colony, a critical value of the percentage of bees leaving the colony plays an important role. The colony survives only if the percentage of bees is below this critical value.• In case of a colony infested with mites and virus, in addition to the percentage of the bees leaving the colony, the percentage of virus carrying mites and virus free mites leaving the colony during swarming also has a huge impact on the survival of the parent colony. Particularly, a colony, that otherwise dies off due to virus, can survive if the percentage of mites leaving the parent colony is above a critical value. 308 V. Ratti et al.
• The critical value of the percentage of bees leaving the colony is lower in case of swarming due to overcrowding as compared to the case where swarming takes place after fixed intervals. This difference could be due to the fact that swarming due to overcrowding takes place every year, however, swarming after fixed intervals is basically every 2 years. The parent colony is able to tolerate greater loss of bees because it gets longer time to establish itself before the next swarming event takes place.
References
1. Britton, N.F., Franks, N.R., Pratt, S. C., Seeley, T.D.: Deciding on a new home: how do honeybees agree? Proc. R. Soc. Biol. Sci. 269(1498), 1383–1388 (2002) 2. Eberl, H.J., Frederick, M.R., Kevan, P.G.: The importance of brood maintenance terms in simple models of the honeybee – Varroa destructor – acute bee paralysis virus complex. Electron. J. Differ. Equ. Conf. Ser. 19, 85–98 (2010) 3. Fries, I., Hansen, H., Imdorf, A., Rosenkranz, P.: Swarming in honey bees (Apis mellifera) and Varroa destructor population development in Sweden. Apidologie 34, 389–397 (2003) 4. Genersch, E., von der Ohe, W., Kaatz, H., Schroeder, A., Otten, C.,́ Büchler, R., Berg, S., Ritter, W., Mühlen, W., Gisder, S., Meixner, M., Liebig, G., Rosenkranz, P.: The German bee monitoring project: a long term study to understand periodically high winter losses of honey bee colonies. Apidologie 41, 332–352 (2010) 5. Kevan, P.G., Hannan, M., Ostiguy, N., Guzman-Novoa, E.: A summary of the varroa-virus disease complex in honey bees. Am. Bee J. 146(8), 694–697 (2006) 6. Martin, S.: A population dynamic model of the mite varroa jacobsoni. Ecol. Model. 109, 267– 281 (1998) 7. Martin, S.J.: Varroa destructor reproduction during the winter in apis mellifera colonies in UK. Exp. Appl. Acarol. 25(4), 321–325 (2001) 8. Myerscough, M.R.: Dancing for a decision: a matrix model for nest-site choice by honey- bees. Proc. R. Soc. Lond. B: Biol. Sci. 270(1515), 577–582 (2003) 9. Ostiguy, N.: Honey bee viruses: transmission routes and interactions with varroa mites. In: 11 Congreso Internacional De Actualizacion Apicola, vol. 9 al 11De Junio De 2004. Memorias., p. 47 (2004)10. Passino, K.M., Seeley, T.D.: Modeling and analysis of nest-site selection by honeybee swarms: the speed and accuracy trade-off. Behav. Ecol. Sociobiol. 59(3), 427–442 (2006)11. Ratti, V., Kevan, P.G., Eberl, H.J.: A mathematical model for population dynamics in honeybee colonies infested with varroa destructor and the acute bee paralysis virus. Can. Appl. Math. Q. 21(1), 63–93 (2013)12. Ratti, V., Kevan, P.G., Eberl, H.J.: A mathematical model for population dynamics in honeybee colonies infested with varroa destructor and the acute bee paralysis virus with seasonal effects. Bull. Math. Biol. 77(8), 1493–1520 (2015)13. Sumpter, D.J., Martin, S.J.: The dynamics of virus epidemics in varroa-infested honey bee colonies. J. Anim. Ecol. 73(1), 51–63 (2004)14. Winston, M.L.: The biology of the honey bee. Harvard University Press, Cambridge (1991)15. ZKBS (Zentralkommittee für biologiche Sicherheit des Bundesamts für Verbraucherschutz und Lebensmittelsicherheit); Empfehlung Az.: 45242.0087 - 45242.0094, 2012, (in German: Central Committee for Biological Safety of the Federal Agency for Consumer Protection and Food Safety, Recommendation 45242.0087-45242.0094, 2012) To a Predictive Model of Pathogen Die-off in SoilFollowing Manure Application
Andrew Skelton and Allan R. Willms
Abstract The application of manure is an important component of nutrient man-agement in the production of field crops. The regulations governing the safeapplication of manure are based on laboratory data, which may or may notaccurately reflect the environmental fluctuations seen in field conditions. This study aims to develop a predictive model for pathogen die-off in soilfollowing manure application. An ordinary differential equation model is presentedand fit to experimental data. The challenges of modelling field derived data,including detection thresholds, viable but nonculturable bacteria and difficulties inwinter data collection, are discussed. The capabilities of the model for predictivepurposes and the development of additional experimental trials and data collectionmethods are discussed.
1 Background
Field production of fruits and vegetables typically includes application of manureas a vital component of nutrient management. In organic farming, manure, compostand other organic materials are the only substances acceptable for use as fertilizer.The Standards Council of Canada 2008 regulations require that manure application shall be designed to ensure that manure application a. does not contribute to the contamination of crops by pathogenic bacteria, b. minimizes the potential for run-off into ponds, rivers and streams, c. does not significantly contribute to ground and surface water contamination. The non-composted solid or liquid manure shall be a. incorporated into the soil at least 90 days before the harvesting of crops for human consumption that do not come into contact with soil; b. incorporated into the soil at least 120 days before the harvesting of crops having an edible part that is directly in contact with the surface of the soil or with soil particles.
Pathogen die-off rates depend on environmental and biological conditions and,as such, will differ from season to season and year to year. A number of researchershave studied the effect such conditions have on pathogen die-off, such as the effect
A. Skelton () • A.R. WillmsDepartment of Mathematics and Statistics, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 309J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_29 310 A. Skelton and A.R. Willms
of temperature [7, 8, 14], soil and manure type [18], manure application methods[2], manure storage [9], intervention strategies [5] and genetic factors [11, 15]. Anice overview of studies conducted on survival of pathogens is given in [17]. Arecent study [6] presented results from a meta regression on a large number ofpotential influencing factors and concluded that the three most important factorsaffecting pathogen die-off rates were temperature, water and soil types, and whetheror not the study was conducted in the lab or in the field. Temperature and soilmoisture are environmental factors that are relatively easy to record and predict and,as such, are of use in a predictive model. The third factor is of particular interestas the waiting times required by the regulations are based on data obtained underlaboratory conditions. Data obtained from laboratory studies are subject to highly consistent andcontrolled environmental conditions and thus do not necessarily accurately reflectthe wide fluctuations seen in field conditions. The fixed waiting times required bythe regulations may either over- or under-estimate the time required to obtain safelevels of pathogen reduction. It is, therefore, of value to model pathogen die-offin local climatic conditions. Obtaining data in field conditions, however, brings itsown set of challenges and concerns. The aim of this study is to develop a predictivemodel of pathogen die-off and answer questions arising from the study of pathogendie-off in soil. Given measurements of pathogen levels in soil and some knowledgeof future environmental conditions, can future pathogen levels be predicted? Canfield derived data provide scientific information that can lead to informed policydevelopment which allows for sustainable food production while guarding againstfood contamination?
2 Experiment
A field study was conducted in 2011–2012 by a team led by Dr. Ann Huber from theSoil Resource Group and Dr. Keith Warriner from the Department of Food Scienceat the University of Guelph. The project was funded by the Food Safety ResearchProgram facilitated by the Ontario Ministry of Agriculture, Food and Rural Affairs(OMAFRA). A detailed description of the experimental procedure is available in[10] and illustrations and photographs of the experiment are presented in Fig. 1. Trials were conducted at a farm northwest of Belwood, Ontario at two sites,one with Perth Loam soil and the other with Hillsburg Fine Sandy Loam soil. Thetwo sites were adjacent to each other and thus subject to the same environmentalconditions. Manure (either dairy or swine) was spiked with either E.coli O157,Salmonella or Listeria, mixed with the soil, and placed in sentinel vials (seeFig. 1). These vials, which have a membrane impermeable to the contaminant, allowthe monitoring of pathogen populations in a confined environment and provide amechanism for separating the environmental effect on pathogen die-off from theeffect due to transport, such as run-off and movement through the soil profile.The sentinel vials were buried either directly under the surface or at a depth of To a Predictive Model of Pathogen Die-off in Soil Following Manure Application 311
Fig. 1 Illustrations and photographs of the experimental procedure (Images reproduced withpermission of Dr. Huber at the Soil Resource Group). Top: Introducing the inoculated sentinelvials into the test plot at different depths. Middle: Assembly of the sentinel vials into which thepathogen inoculated manure amended soil will be introduced. Bottom: Methodology to be appliedin the study 312 A. Skelton and A.R. Willms
approximately 15 cm and additional non-spiked manure was hand applied at a rateequivalent to within the vials. Three trials were conducted: the first from June 16,2011 to May 29, 2012, the second from September 22, 2011 to June 6, 2012 and thethird from November 2, 2011 to June 6, 2012. Each trial included pre-determinedsampling dates on which three sentinel vials were extracted and pathogen levelsreported. The three trials contained 13, 11 and 8 sampling dates respectively. Ourexperimental data therefore consists of 72 (3 trials 3 pathogens 2 depths 2 soiltypes 2 manures) time series data sets, with three data replicates at each samplingdate. A typical data set is found in Table 1. Note that there are no sampling datesover the winter months due to the soil becoming frozen and too snow covered tobe realistically sampled. When a sentinel vial is extracted from the ground, a soilsample of approximately 10 g is obtained. The sample is processed, diluted andbacteria are cultured and counted. If there are too many bacteria to count after beingcultured, the sample is repeatedly diluted by a factor of 10 until the number ofbacteria is countable. The data in row four of Table 1, for example, has been dilutedtwice, so the recordings of 7, 5 and 11, denote true counts of 700, 500 and 1100bacteria respectively. In each dilution, triplicate counts are obtained. For ease ofreading, only the median result is reported in Table 1, but averages are used inthe analysis. If there are no bacteria visible at a given dilution, we conclude onlythat there is less than 1 bacteria at that given dilution level. The data in the lastrow of Table 1, for example, displays counts of 0 bacteria at a dilution of 0.1. Theconclusion that we may draw from this observation is that there is less than 10bacteria in the sample. Our time series data will, therefore, contain censored datathat is known only to have a value beneath a detection threshold. The bacteria counts
Table 1 A typical data set. Data collected for over the September 22, 2011 to June 6, 2012 trial,using dairy manure, in sandy soil, at the surface, measuring levels of E.coli O157. To determinethe data values for model fitting, we take the bacterial count, divide by the dilution, scale by thesample weight and take the logarithm of the answer to determine the log bacterial count per 10 gof soil Sample weights (g) Bacterial countsDate Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Sample 3 DilutionSep 22/11 10:000 10:000 10:000 1 1 1 0:001Sep 25/11 10:063 10:019 10:253 75 73 101 0:01Oct 3/11 10:247 10:154 10:057 1 4 3 0:001Oct 13/11 10:297 10:024 10:054 7 5 11 0:01Oct 19/11 10:328 10:035 10:264 1 2 65 0:1Nov 10/11 10:254 10:106 10:012 3 0 3 0:1Nov 25/11 10:107 10:057 10:028 1 2 0 0:1Mar 27/12 10:039 10:041 10:048 0 0 0 0:1May 1/12 10:059 10:005 10:140 0 0 0 0:1May 17/12 10:152 10:128 10:020 0 0 0 0:1Jun 7/12 9:999 10:002 10:006 0 0 0 0:1 To a Predictive Model of Pathogen Die-off in Soil Following Manure Application 313
30 0.55Average Daily Air Temperature
0.5 20 0.45
% Soil Moisture 0.4 10 0.35
0.3 0 0.25
−10 0.2 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Days since June 10, 2011 Days since June 10, 2011
Fig. 2 Environmental data collected. Recordings from the weather station for average daily airtemperature and soil moisture over the course of the three trials. Missing data were interpolatedfrom records kept by nearby weather stations
are then scaled by the relative weight of the sample so if, for example, the sampleweight was 10:107 g, we would multiply the count by 10=10:107 to obtain a scaledcount. The data with which we will fit our model will be given as a log count ofbacteria per 10 g of soil. Environmental data were collected by a weather station installed at the site. Theaverage daily air temperature and percentage of soil moisture are shown in Fig. 2.There were two periods of missing environmental data. The first was due to strongwinds knocking over the weather station, and the second was due to a raccoonconsuming the power supply. In both cases, it was possible to obtain replacementweather data spatially averaged between nearby weather stations using data madefreely available by Environment Canada.
3 Model and Results
We now fit our experimental data using an ordinary differential equation model.Let x D x.t/ denote the logarithm of the bacteria count per 10 g of soil, and T DT.t/; M D M.t/ denote the average daily air temperature (degrees Celsius) and soilmoisture (%) linearly interpolated between adjacent data points from the data shownin Fig. 2. Pathogen die-off plots typically follow a characteristic logistic curve [2, 3, 7,11, 14, 18], so we set up a logistic differential equation in which the rate constantis dependent on both temperature and soil moisture. It is common to discusspathogen die-off in terms of numbers of log reductions, so an empirical model ofpathogen die-off in which the dependent variable is given on a logarithmic scale isappropriate. Using a quadratic dependence on temperature and linear dependenceon soil moisture, we obtain the following ordinary differential equation
x0 D .k1 C k2 T C k3 T 2 C k4 M/.x B/.C x/; (1) 314 A. Skelton and A.R. Willms
where the four rate constants k1 ; k2 ; k3 ; k4 , the two shape constants B; C (thebaseline and maximum values, respectively, which will depend on the trial, soil type,manure type and sample depth) and the initial condition x.t0 /, where t0 denotes thestart of the particular trial of interest and B < x.t0 / < C, will be obtained fromfinding the best model fit to the data. The initial condition must be estimated dueto the use of lab bacteria in field trials. When the lab bacteria are exposed to fieldconditions, a subset of the bacteria will experience shock due to the sudden changein environmental conditions and enter a viable but nonculturable (VBNC) state.Bacteria in a VBNC state cannot be cultured on growth media [13], but are stillviable and potentially dangerous, and thus must be accounted for in our analysis.This phenomenon can be seen in the data in Table 1. The first row of data, obtainedat the end of the first day of the trial, has an average of 1000 bacteria per 10 g ofsoil. This observed value is significantly less than the number of bacteria that wereoriginally placed in the spiked soil. The second row of data, obtained 3 days later,has an average of 8406 bacteria per 10 g of soil. This discrepancy reflects the numberof bacteria that entered a VBNC state initially, but had recovered after 3 days inthe field. In our analysis, we ignore the initial data value and instead estimate thenumber of bacteria present at the start of the experiment (Table 2). Owing to the fact that some of the data points are censored due to detectionthresholds, we will be required to use the Expectation-Maximization (EM) algo-rithm to fit our model to the data. The EM algorithm first sets all censored data toexactly half of the appropriate detection threshold (in our case, to either 5, 50, 500,or 5000 bacteria). The following steps are then repeated until convergence: a leastsquares procedure is used to find the best model fit given the current data valuesand then the censored data points are adjusted to fit the current parameter set. Themathematical details of the EM algorithm can be found in [4, 12] and the algorithmas applied to censored data at multiple detection thresholds can be found in [1].
Table 2 Best Fit Parameters for the plots shown in Fig. 3. Details of the four trials are as follows.(a) Dairy manure, loam soil, surface, E.Coli, (b) Dairy manure, sand soil, surface, E.Coli, (c) Dairymanure, sand soil, depth, Salmonella, (d) Swine manure, loam soil, depth, ListeriaParameter Figure 3a Figure 3b Figure 3c Figure 3dk1 5:15 104 3:46 103 1:62 103 7:48 106k2 6:51 105 3:87 104 4:39 104 4:53 103k3 2:00 105 2:89 105 1:21 104 1:85 104k4 1:60 104 2:33 104 9:95 104 3:17 103B 0:79 0:39 0:92 1:37C 9:71 5:25 6:56 6:32x.0/ 4:46 4:22 4:18 4:80 To a Predictive Model of Pathogen Die-off in Soil Following Manure Application 315
To find the best fit parameter set, we will be required to search in a seven-dimensional parameter space. Very little a priori information is available as to thevalues of many of the parameters. It is therefore useful to reduce the size of theparameter space we will be required to search. We used the method presentedin [16] to reduce the size of the search space and improve the speed of theminimization steps of the EM algorithm. This method uses interval analysis andlinear multistep discretizations to remove boxes of parameter space that are deemedto be inconsistent with the data. The method is able to quickly remove large regionsof parameter space to allow traditional minimization techniques to work moreeffectively. Results for representative data sets are presented in Fig. 3. We were, in general,able to obtain good fits to our experimental data. In plots (a) and (d), we can seethat there is an expected initial rapid decrease in the bacterial count, followed bya slower die-off (and in fact a small growth in the case of Listeria) over the winterseason, followed by another period of more rapid die-off during the spring months.Plots (b) and (c) illustrate the difficulties faced in modelling these data sets. Theonly viable data exists before the winter freeze, making any dynamics during orafter the winter break difficult to model. This lack of information in many data setshas made calibrating a model to the data very difficult. It has, however, directly ledto improved experimental design and data collection in a follow-up trial.
4 Conclusion and Future Work
This paper presents the first step in a process by which we hope to develop apredictive model for pathogen die-off in soil following manure application. We fitan ordinary differential equation model to field derived data. Our model was ableto reasonably describe the pathogen die-off using soil moisture and air temperature,both commonly available environmental data. To establish a predictive, rather thandescriptive, model, we will require additional winter and long-term data with whichwe can calibrate our model and then use for testing of predictive capabilities. Suchdata is currently being collected and analysis of the data is ongoing and will bereported elsewhere.
Acknowledgements This work was supported by an NSERC Discovery Grant. 316 A. Skelton and A.R. Willms
Fig. 3 Best fit plots. The (a) 6solution to the differentialequation model with best fit
Bacteria Log countparameters plotted for four 4representative data sets fromthe September 22/11 to June 27/12 trial. The open circlesrepresent data points whosevalues are known only to 0exist beneath the detectionthreshold. The closed dots −2represent the predicted data 100 150 200 250 300 350 400as outputted from the EM Days since June 10, 2011algorithm. (a) Dairy manure, (b) 6loam soil, surface, E.Coli, (b)Dairy manure, sand soil, Bacteria Log count
surface, E.Coli, (c) Dairy 4manure, sand soil, depth,Salmonella, (d) Swine 2manure, loam soil, depth,Listeria 0
−2 100 150 200 250 300 350 400 Days since June 10, 2011 (c) 5
4 Bacteria Log count
0 100 150 200 250 300 350 400 Days since June 10, 2011 (d) 5 Bacteria Log count
1 100 150 200 250 300 350 400 Days since June 10, 2011 To a Predictive Model of Pathogen Die-off in Soil Following Manure Application 317
References
1. Banks, H.T., Davidian, M., Hu, S., Kepler, G.M., Rosenberg, E.S.: Modelling HIV immune response and validation with clinical data. J. Biol. Dyn. 2(4), 357–385 (2008) 2. Bech, T., Dalsgaard, A., Jacobsen, O.S., Jacobsen, C.S.: Leaching of Salmonella enterica in clay columns comparing two manure application methods. Ground Water 49(1), 32–42 (2011) 3. Coroller, L., Leguerinel, I., Mettler, E., Savy, N., Mafart, P.: General model, based on two mixed Weibull distributions of bacterial resistance, for describing various shapes of inactivation curves. Appl. Environ. Microbiol. 72(10), 6493–6502 (2006) 4. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc.: Ser. B 39, 1–38 (1977) 5. Franz, E., sem*nov, A.V., van Bruggen, A.H.C.: Modelling the contamination of lettuce with Escherichia coli O157:H7 from manure-amended soil and the effect of intervention strategies. J. Appl. Microbiol. 105, 1569–1584 (2008) 6. Franz, E., Schijven, J., de Roda Husman, A.M., Blaak, H.: Meta-regression analysis of commensal and pathogenic Escherichia coli survival in soil and water. Environ. Sci. Technol. 48, 6763–6771 (2014) 7. Fremaux, B., Prigent-Combaret, C., Delignette-Muller, M.L., Mallen, B., Dothal, M., Gleizal, A., Vernozy-Rozand, C.: Persistence of Shiga toxin-producing Escherichia coli O26 in various manure-amended soil types. J. Appl. Microbiol. 104, 296–304 (2008) 8. Holley, R.A., Arrus, K.M., Ominski, K.H., Tenuta, M., Blank, G.: Salmonella survival in manure-treated soils during simulated seasonal temperature exposure. J. Environ. Q. 35, 1170– 1180 (2006) 9. Huber, A.: Survival of pathogens during storage of livestock manures. OMAFRA Final Report. SR9182 (2009)10. Huber, A., Warriner, K.: Pathogen die-off rates following manure application under Ontario field conditions – relationship to pre-harvest wait times. OMAFRA Final Report. SF6088 (2012)11. Ma, J., Ibekwe, A.M., Yi, X., Wang, H., Yamazaki, A., Crowley, D.E., Yang, C.-H.: Persistence of Escherichia coli O157:H7 and its mutants in soils. PLoS ONE 6(8), 1–9 (2011)12. McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, New York (1997)13. Oliver, J.D.: The viable but nonculturable state in bacteria. J. Microbiol. 43(S), 93–100 (2005)14. Ongeng, D., Muyanja, C., Ryckeboer, J., Springael, D., Geeraerd, A.H.: Kinetic model-based prediction of the persistence of Salmonella enterica serovar Typhimurium under tropical agricultural field conditions. J. Appl. Microbiol. 110, 995–1006 (2011)15. Saint-Ruf, C., Garfa-Traore, M., Collin, V., Cordier, C., Franceschi, C., Matica, I.: Massive diversification in aging colonies of Escherichia coli. J. Bacteriol. 196(17), 3059–3073 (2014)16. Skelton, A., Willms, A.R.: Parameter range reduction in ordinary differential equation models. J. Sci. Comput. 62(2), 517–531 (2015)17. van Elsas, J.D., sem*nov, A.V., Costa, R., Trevors, J.T.: Survival of Escherichia coli in the environment: fundamental and public health aspects. ISME J. 5, 173–183 (2011)18. You, Y., Rankin, S.C., Aceto, H.W., Benson, C.E., Toth, J.D., Dou, Z.: Survival of Salmonella enterica Serovar newport in manure and manure-amended soils. Appl. Environ. Microbiol. 72(9), 5777–5783 (2006) Mathematical Modeling of VEGF Binding,Production, and Release in Angiogenesis
Nicoleta Tarfulea
Abstract This paper presents a new mathematical model for the transductionof the extracellular vascular endothelial growth factor (VEGF) signal into theintracellular VEGF signal. It is based on a signal transduction pathway that accountsfor VEGF binding, production, and release. The resulting mathematical modelfor the evolution of the chemical species concentrations is analyzed analyticallyand numerically to address qualitative aspects (including positivity, stability androbustness with respect to variations of the secretion rate parameter). Varioussecretion rate functions are investigated as well.
1 Introduction
In recent years, tumor-induced angiogenesis has become an important field ofresearch because it represents a crucial step in the development of malignant tumors[18]. When a cancerous tumor initially forms, it acquires the nutrients neededfor survival from the surrounding tissue. When it grows to a large enough size[17], it needs another source of food and channel for depositing waste. As anearly response of tumor cells to hypoxia, they start releasing molecules calledtumor angiogenic factors, such as VEGF, into the surrounding tissue to initiate aprocess called angiogenesis (growth of new blood vessels from preexisting ones).Secreted VEGF then diffuses into the surrounding tissue and binds to specificendothelial cell (EC) receptors. As a response, ECs start to secrete proteolyticenzymes (such as PA-plasmin, MMPs) which dissolve the basal lamina of theparent vessel and the surrounding extracellular matrix (ECM). ECs respond tothe stimulus chemotactically and begin to migrate toward it, and so small sproutswhich grow toward the tumor are formed [1, 23, 32, 35]. Over the last years it hasbecome evident that an effective way of treating cancer is inhibition of angiogenesis[15, 19, 20, 29, 41], which could be done by targeting endothelial cells rather thantumor cells since: (i) ECs are more accessible than tumor cells; (ii) death of a single
N. Tarfulea ()Purdue University Calumet, 2200 169th Street, Hammond, IN, 46323, USAe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 319J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_30 320 N. Tarfulea
cell will cause death of approximate 500 tumor cells; (iii) ECs are relatively stableand will not mutate easily into drug resistant variants [50]. The existing mathematical models in the literature [35] have been able to repro-duce some characteristics of angiogenesis. These include continuous approaches[2–5, 10–12, 24, 27, 28, 31, 38–40, 44, 52–54], random walk models [2, 6, 13, 22,38, 39, 46, 47, 51], and cell-based formulations [7–9, 25, 30, 33, 36, 43, 50]. Reviewsof such models can be found in [35, 37, 42]. However, it has been discovered that theactivated ECs secrete detectable amounts of VEGF [16, 45]. Thus, in this paper wewill develop a new mathematical model that account for VEGF binding, production,and release.
2 Description of the Signal Transduction Pathway
The dynamic interaction between angiogenic factors, the uPA-plasmin system, andthe presumed activation of matrix-degrading metalloproteinases during angiogene-sis is shown in Fig. 1. It can be described as follows [50]: (1) hypoxic tumor cellssecrete VEGF; (2) VEGF is sequestered in the ECM by binding to ECM heparinbinding sites; (3) VEGF binds to VEGFR (VEGF receptor) on EC cell (VEGFR-2 is considered the main signaling receptor in ECs); (4) inactive uPA (urokinaseplasminogen activator) pro-uPA, binds to specific cell surface receptor uPAR;(5) plasminogen (pgn) binds to pgnR; (6) uPAR-bound uPA reacts with bound-plasminogen to generate Plasmin; (7) uPAR occupied by uPA forms complexes withPAI-1 (plasminogen activator inhibitor); (8) Plasmin degrades ECM; (9) Plasminmobilizes VEGF from ECM reservoir; (10) Plasmin induces the activation of LTGF-
(1) (3) Cell Endothelial cell receptors VEGF
(2)
pro−MMPs pgnR uPAR (9) Plasmin pgn uPA pro−uPAR (5) (4) (6) (14) (15) (10) ECM (13) MMPs (8) LTGF−β TGF− β PAIs (12) (16) (17) TIMPs
Matrix degradation
Fig. 1 The biochemical pathway of angiogenesis Mathematical Modeling of VEGF Binding, Production, and Release in Angiogenesis 321
ˇ (latent type beta transforming growth factor) from ECM; (11) Plasmin can convertLTGF-ˇ into TGF-ˇ (type beta transforming growth factor); (12) active TGF-ˇinduces synthesis of PAI-1; (13) PAI-1 blocks the conversion of plasminogen intoPlasmin; (14) Plasmin, uPA activate some pro-MMPs (matrix metalloproteinase);(15) some MMPs can activate other MMPs; (16) all MMPs are inhibited by TIMPs(tissue inhibitor of MMP enzyme) by binding to pro-MMPs or MMPs; (17) MMPsdegrades ECM. The signal transduction part incorporates known biology morecompletely than any other model [35] and involves 18 variables and 14 biochemicalreactions. Using the law of mass action, these lead to a system of ten differentialequations and auxiliary algebraic equations for the time evolution of the intracellularspecies.
3 The Simplified Pathway and the Corresponding Model
3.1 Assumptions for the Simplified Model Signal Pathway
Since biological and biochemical processes in angiogenesis are very complex, weseek reductions in the number of variables that do not alter the dynamics of themodel. After VEGF is been secreted into the surrounding tissue by tumor cells andhas encountered endothelial cells, it binds to specific proteins (called receptors)that are sitting on the outer surface of the cell. This binding activates a series ofrelay proteins which transmit a signal that starts a series of cascades inside the cell[50]. As a result the motile machinery of the endothelial cell is activated, and alsosome VEGF is secreted [26, 45]. Using these facts, we have reduced the schemeto four primary species for the intracellular dynamics. A schematic diagram of theinteractions in the proposed model is shown in Fig. 2. The definition of the symbolsused in these reactions as well as the mathematical symbol used later are given inTable 1. Here ki represent the reaction rates, hi represent the degradation rates, andbsr is the basal secretion rate; all of them are given in Table 2. Two classes of species are represented in this model: volumetric (growth factor– VEGF) and surface (receptors and ligand-receptor complexes). No factors in the
Fig. 2 Simplified pathway Vefor angiogenesis
R Y*
Vi Y 322 N. Tarfulea
Table 1 The variables, their symbols, and non-dimensional formSpecies Conc. Definition Dimensionless formVe y1 .t/ Extracellular VEGF u1 D kk15 y1 y2R y2 .t/ VEGFR u2 D ŒR T y3Ve R y3 .t/ VEGFR bound with VEGF u3 D ŒR T k2 Ck3Ve RY y4 .t/ Ve RY complex bound with Y u4 D y k2 ŒR T ŒY T 4 y5Y y5 .t/ Activated enzyme u5 D ŒY T k4S y6 .t/ Substrate u6 D k3 6 yYS y7 .t/ Activated enzyme Y bound with S u7 D y7 ŒY TVi y8 .t/ Intracellular VEGF u8 D k1 k5 8 yY z.t/ Free Y –t – Time D k5 t
Table 2 Estimates for the kinetic parameters – for numerical simulationsPar. Estimated value Source Par. Estimated value Sourcek1 3:8 106 M1 s1 [14, 21] k4 104 m2 /.mol s/ [48, 49]k1 95 106 s1 [14, 21] k4 0 s1 [48, 49]k2 103 m2 =.mol s/ [14, 21] k5 6 102 s1 [48, 49]k2 0 s1 [48, 49] k6 10s1 [48, 49]k3 103 s1 [48, 49] bsr 3 102 s1 [49]s 3:8 109 mol=.m2 s/ [48, 49] l3 2:8 104 s1 [48, 49]l2 105 s1 [21, 49] l4 2:8 104 s1 [48, 49]
fluid that could bind the ligands, such as secreted ECM, are included. Also, weassume that receptor concentration is uniform over the cell surface (with no receptorclusters being formed). The detailed biochemical reactions in this mechanism are asfollows.
k1 Ve C R ! Ve R (1) k1 k2 Ve R C Y ! Ve RY (2) k2 k3 Ve RY Y + Ve R (3) k4 Y C S ! Y S (4) k4 k5 Y S Vi + Y (5) Mathematical Modeling of VEGF Binding, Production, and Release in Angiogenesis 323
k6 Y Y (6) bsr Vi Ve (7)
3.2 The Mathematical Model for the Intracellular Dynamics
We consider the time evolution of species concentrations deduced according to themechanism described above. A rectangular box was used having dimensions 1 mm1 mm 1:5R, where R D 10m is the radius of EC when spherical geometry isassumed. Furthermore, N represents the number of cells, Ac and Vc represent thesurface area and the volume of each cell. The reactions that are considered in themechanism then lead to the following system of differential equations:
dy1 V0 D N Ac k1 y3 N Ac k1 y1 y2 C N Vc sr.y8 / N Vc h1 y1 (8) dt dy2 D k1 y3 k1 y1 y2 C s l2 y2 (9) dt dy3 D k1 y1 y2 k1 y3 k2 y3 z C .k2 C k3 /y4 l3 y3 (10) dt dy4 D k2 y3 z .k2 C k3 /y4 l4 y4 (11) dt dy5 D k4 y5 y6 C .k4 C k5 /y7 C k3 y4 k6 y5 (12) dt dy6 D k4 y5 y6 C k4 y7 (13) dt dy7 D k4 y5 y6 .k4 C k5 /y7 (14) dt dy8 Vc D Ac k5 y7 Vc sr.y8 / Vc h2 y8 : (15) dtHere s is the insertion rate of surface species into endothelial cell membrane andl2 , l3 , and l4 represent the internalization rates of surface receptors and complexes.Their values are listed in Table 2. We introduce the secretion function sr.y8 / whichoccurs in the secretion step (7). First, we consider it to be a linear function of theintracellular VEGF (Vi ) of the form sr.y8 / D bsry8 : This does not affect the validityof our model since the true secretion rate function is not known. Also, we considerthe case where bsr is piecewise and periodic function. Moreover, the secretionfunction can be easily modified and the model amended as new experimental databecomes available. 324 N. Tarfulea
We proved the nonnegativity and boundedness of the solution to the initial-valueproblem corresponding to the system (8), (9), (10), (11), (12), (13), (14) and (15).The initial data is nonnegative.
3.3 Insertion of Surface Species into EC Membrane
In this section, we consider two separated sub-cases, namely, when insertionof surface species into endothelial cell membrane and internalization of surfacereceptors and complexes are not taken into consideration (Case A), and when theyare considered in the system (Case B). A comparison between the two systems willbe done. In Case A, we consider the system of differential equations (8), (9), (10), (11),(12), (13), (14) and (15), with the values of the parameters s, l2 , l3 , and l4 beingzero. The conservation relations are ŒVe R C ŒR C ŒVe RY D ŒR T ; ŒY C ŒY S CŒY D ŒY T ; where subscript T denotes the initial value of a quantity. When theseconditions are solved for y2 .t/ and z.t/ z.t/ D ŒY T y5 .t/ y7 .t/ y2 .t/ DŒR T y3 .t/ y4 .t/; and the result is used in Eqs. (8), (9), (10), (11), (12), (13),(14) and (15), we obtain the system of differential equations for y1 ; y3 ; : : : ; y8 . Inthe interest of space, this will not be included here. The values used for the kineticparameters are in Table 2 and were found from experimentally based estimates[14, 21, 48, 49]. We also use Ve .0/ D 2:22 nM, RT D 3:3 105 mol/m2 , andYT D 2:50 104 mol/m2 . In Case B we take into consideration insertion ofsurface species into the endothelial cell membrane and internalization of surfacereceptors and complexes. The reactions that are considered in the mechanism thenlead to system of differential equations (8), (9), (10), (11), (12), (13), (14) and(15), which describes the time evolution of species concentrations. The conservationrelation is ŒY C ŒY S C ŒY D ŒY T and the quantity that is determined by it is z.t/(corresponding to Y – free enzyme) is z.t/ D ŒY T y5 .t/ y7 .t/; where subscript Tdenotes the initial value of a quantity. As before, using this conservation condition,we obtain a system of differential equations for y1 ; : : : ; y8 .
3.4 Nondimensionalization and Analysis of the Systems
We introduce dimensionless variables for the independent concentrations (Table 1).We find that they are of two categories: those who reach steady-state rapidly, whichare called fast variables and the remaining ones which are called slow variables.In our case the fast variables are denoted by u4 and u7 . The nondimensionalization k5 k5gives rise to two small parameters 1 D and 2 D . On the time scale, k2 C k3 k3the singular equations, corresponding to the fast variables u4 and u7 , can be reduced Mathematical Modeling of VEGF Binding, Production, and Release in Angiogenesis 325
du4 du7to the algebraic system of equations D 0 and D 0. By replacing u4 and d d u7 and by dropping the O./ terms, after some simplifications, we obtain a systemof differential equations for the species concentrations. In case A, the variables areu1 ; u3 ; u5 ; u6 , and u8 . In addition, in Case B we have u2 . Having investigated thedynamics of species concentrations in the described cases, several observations canbe made. The analysis in each case was made using the same set of parameters andinitial conditions. The reduced system in Case A agrees with the full system only onshort time, however we see that the reduced system in Case B and the full systemagree on both short and long time period.
3.5 Further Assumptions for Intracellular Dynamics Model
After VEGF is been secreted and released into nearby tissues by the hypoxic tumor,it binds to endothelial cell receptors, process that activates a series of relay proteinswhich in turn transmit a signal that starts a series of cascades inside the cell. So farthe mechanism was reduced to four primary species for the intracellular dynamics.Next, we make the biologically realistic assumption that the substrate is not onlyconsumed, but also produced during the entire process Thus, we can consider thatthe rate of change of substrate concentration is approximately zero. Moreover, weinvestigate Case A and Case B too. A similar analysis as well as a comparisonbetween the two systems is performed and we obtain that Case B agrees with thefull system on both short and long time period. Thus, we consider the followingsystem to represent the governing equations for the intracellular dynamics.
du1 D N1 ˛1 u3 N1 u1 u2 C ˇ1 u8 ˇ2 u1 (16) d du2 D ˛1 u3 u1 u2 C ı1 ı2 u2 (17) d du3 D u1 u2 .˛1 C ı3 /u3 (18) d du5 D ˛4 ˛2 2 .1 .1 C ˛3 u6 /u5 /u3 ˛5 u5 (19) d du8 N 1 D ˛3 u5 u6 .ˇ1 C ˇ3 /u8 ; (20) d 2
where u6 is constant and initial condition
.u1 .0/; u2 .0/; u3 .0/; u5 .0/; u8 .0//T D .u10 ; u20 ; 0; 0; 0/T : 326 N. Tarfulea
We performed a stability and bifurcation analysis for the above system consideringit to be on the form du D f .u; p/; u.0/ D u0 (21) dt
where u 2 M R5 and p 2 I Rm [34]. We consider m D 1 and the p D ˇ1 whichis the nondimensional form of the VEGF secretion rate. Furthermore, the subsetI Œ0; 1 , according with data from biological experiments. We showed that thesystem has a steady state .0; 1; 0; 0; 0/ and a unique steady state in the interior of R5C .Moreover, the unique positive steady state is stable for any ˇ1 2 Œ0; 1 and, since theJacobian is nonsingular for any ˇ1 2 Œ0; 1 , there are no possible bifurcation points.
4 Conclusions
We have reduced the complicated biochemical pathway of angiogenesis (Fig. 1) toa simplified one (Fig. 2) involving only four primary species for the intracellulardynamics (namely, VEGF, Receptor, Enzyme, and Substrate). Two separated caseshave been considered for describing this mechanism (rate of change of substrateconcentration is not zero or approximately zero). The behavior of time evolutionof species concentrations have been investigated in each case and compared withthe expected biological phenomenon. As a conclusion, we have been able todevelop a new mathematical model that describes the intracellular dynamics whichincorporates a realistic model for signal transduction, VEGF production and release. Considering the fact that it has only recently been established that the activatedendothelial cells secrete detectable amounts of VEGF, quantitative data relative tothis fact is almost entirely absent. Therefore, the model was tested for different setsof parameters. A qualitative analysis revealed the fact that the system has a uniquepositive, asymptotically stable steady state solution. Finally, a bifurcation analysis ofthe ODE kinetics of the model showed that there are no possible bifurcation pointswith respect to the coefficient of the secretion rate function, and that the system isrobust to variations of this parameter. First, we considered that the VEGF secretion rate function has a linear depen-dency on the intracellular VEGF. Next, we investigated the cases when bsr is apiecewise function and a periodic function in time, with the same average as theconstant value. Interestingly, we obtained the same behavior for short and long timeperiods for our system (Fig. 3). We note that this analysis could be extended furtherby investigating different parameter variations, for example the ratio of the totalcell volume to the extracellular volume. Additionally, the secretion rate functioncan be further modified and the model amended as new experimental data becomesavailable. Mathematical Modeling of VEGF Binding, Production, and Release in Angiogenesis 327
Fig. 3 Piecewise (left) and periodic (right) secretion rate-bsr
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Marianne Waito, Scott R. Walsh, Alexandra Rasiuk, Byram W. Bridle,and Allan R. Willms
Abstract Cytokine storms are a potentially fatal exaggerated immune responseconsisting of an uncontrolled positive feedback loop between immune cells andcytokines. The dynamics of cytokines are highly complex and little is known aboutspecific interactions. Researchers at the Ontario Veterinary College have encoun-tered cytokine storms during virotherapy. Multiple mouse trials were conductedwhere a virus was injected into mice whose leukocytes lacked expression of the typeI interferon receptor. In each case a rapid, fatal cytokine storm occurred. A nonlineardifferential equation model of the recorded cytokine amounts was produced toobtain some information on their mutual interactions. Results provide insight intothe complex mechanism that drives the storm and possible ways to prevent suchimmune responses.
1 Introduction
Overly robust cytokine responses are responsible for a broad array of very chal-lenging and often fatal clinical conditions. These include infectious diseases suchas influenza [16], severe acute respiratory syndrome (SARS) [11], bacteria-inducedtoxic shock syndrome [20] and sepsis [9], as well as multiple sclerosis [13], graft-versus-host disease [7] and sometime adverse effects of therapies [22]. Cytokinesare released by cells to coordinate an immune response to help protect againstforeign and/or dangerous matter. They are produced in response to infection andinflammation [5]. There are five major classifications of cytokines; interferons(IFN), interleukins (IL), chemokines, colony-stimulating factors (CSF), and tumournecrosis factors (TNF) [23]. Together they function in order to stimulate a responsethat will control cellular stress as well as minimize the amount of damage to aparticular cell or group of cells [5].
M. Waito () • A.R. WillmsDepartment of Mathematics and Statistics, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]. Walsh • A. Rasiuk • B.W. BridleDepartment of Pathobiology, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 331J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_31 332 M. Waito et al.
Although the goal of each cytokine is to ultimately protect the host againstdangerous insults, it is not uncommon for this process to become unbalanced.For reasons unknown, positive feedback loops can become up-regulated during animmune response involving complex interaction between cytokines and immunecells. The fine balance that is typical of healthy individuals is lost and gives rise to acascade which is termed a cytokine storm. The dynamics of cytokines during normalimmune responses and even more so during storms are highly complex and littleis known about specific interactions [5, 27]. Cytokine storms not only have verynegative effects on the immune response, they often have life threatening effectsincluding a decrease in blood pressure, increase in heart rate and can often lead todeath of the host [27]. As more biological data is collected, the immense complexities of the immunesystem is becoming increasingly apparent [26]. It is clear that mathematical mod-elling has become an essential tool to complement experimental techniques both invitro and in vivo. Combining these approaches can result in major advancements inboth the understanding of cancer and immune system dynamics including cytokineinteractions, with the potential to identify strategies to control toxic cytokine storms[26]. Previous mathematical models include at most two cytokines and do not focusthe research on the interactions and function of the cytokines. Typical cytokinesincluded in models are IL-2, IL-10, IL-12 and TNF. Unfortunately, even the modelsthat contain cytokines do not investigate the specific interactions that we areinterested in. These models exclude many of the cytokines that are of greatestinterest to us [6, 12, 15, 19]. Since our focus is to model only the dynamics of cytokines, we turn to ordinarydifferential equation (ODE) models to provide a good framework for exploringinteractions between different cytokine populations. To keep the model relativelysimple, we ignore the effects of effector cells (activated immune cells) due to thecomplexities it brings to the model. To date, the only model that is of a similarnature is produced by [27] called Dynamics of a Cytokine Storm. In this paper, theinteractions of nine cytokines were modelled using data from a human clinical trialwhere six volunteers took part in a study that accidentally led them to undergo theeffects of a cytokine storm [27]. This model was primarily constructed to look at theeffects of the antibody responsible for the cytokine storm using a linear ODE modelwhere parameter estimation determined coupling parameters between cytokines.The coupling parameters identified which cytokine was responsible for enhancingor inhibiting each other cytokine as well as self-regulation [27]. In this paper, we determine how cytokines interact with one another based on aset of time series data provided by Dr. Byram Bridle and his lab members Dr. ScottWalsh and Alexandra Rasiuk at the Ontario Veterinary College (OVC). Followingadministration of a highly attenuated virus to mice with leukocytes lacking thetype I IFN receptor, a deadly cytokine storm developed leading to death in only24 h. Down-regulation of anti-viral IFN signaling is a common mechanism usedby viruses during infection, including those associated with cytokine storms (e.g.influenza virus [14], SARS-coronavirus [3] and Ebola virus [2]). Thus, a natural A Mathematical Model of Cytokine Dynamics During a Cytokine Storm 333
question arises: what are the specific dynamics of cytokines with respect to bothinitiating and exacerbating a cytokine storm when type I interferon signalling isimpaired?
2 Mathematical Model
Most biological systems are innately non-linear and thus a non-linear ODE modelis essential to obtain the proper dynamics [26]. The model we use is given by:
Mi xP i D i xi C for i; j D 1; 2; : : : ; 13: (1) 1 C eyi
The first and most common assumption throughout the literature is that theconcentration of a particular cytokine, xi , continues to decrease linearly when thereis no outside stimulus [10, 17]. Since cytokines are secreted by other cells it makessense that in the absence of these producers, the number of cytokines will rapidlydecline at a rate i (>0) [10, 17, 25]. The second assumption is that the rate ofproduction of cytokines is dependent on interactions with other cytokines and issigmoidal in shape of the form
Mi ; 1 C eyi
where Mi (>0) is the maximum production rate. The interaction with othercytokines, yi , determines the slope of the function, and thus how much of a cytokineis produced, potentially offsetting some or all of the decay [10, 17, 25]. Theinteraction factor yi is given by
X n yi D ˛0;i C ˛i;j xj C Si ; (2) jD1
meaning that yi is composed of effects from the presence of other cytokines, ˛i;j ;effects from components for which there is no data, ˛0;i ; and the stimulus, Si , by thevirus. The stimulus is of the form
Si D bet= ;
where there is an initial dose of the drug, b, which decays exponentially withcharacteristic time . 334 M. Waito et al.
3 Data
Data was collected and provided by Dr. Byram Bridle, Dr. Scott Walsh andAlexandra Rasiuk who study the role of type I IFN signalling in the regulationof cytokine responses at the OVC. Chimeric mice were made by lethal irradiationof the bone marrow of C57BL/6 mice (Charles River Laboratories) followed byreconstitution with bone marrow from either wild-type or type I IFN receptor(IFNAR)-knockout donors (the latter provided by Laurel Lenz, University ofColorado School of Medicine). These mouse-based experiments were approvedby the institutional Animal Care Committee and complied with the standard ofthe Canadian Council of Animal Care. These mice were infected intravenouslywith recombinant Vesicular Stomatitis Virus with a deletion of methionine atposition 51 of the matrix protein (VSV m51) [21]. The matrix protein of VSVsuppresses antiviral type I IFN responses. Therefore, this mutant virus renders thealready attenuated laboratory strain of VSV even safer and it is being developedas an oncolytic virus for the treatment of cancers via intravenous infusion [24].Surprisingly, mice lacking the IFNAR on their leukocytes experienced a profoundcytokine storm, ultimately leading to death in only 24 h. The resulting time seriesdata provided concentrations of 13 different cytokines in plasma, measured using amultiplex array (BioRad). Concentrations of 13 cytokines were recorded at times 0, 2.5, 5, 10 and24 h for both the wild-type mice and mice lacking the IFNAR on 20% of theirleukocytes. Raw data was then normalized to account for the vast variability in theconcentrations. This was done by dividing the concentration at each time point bythe sum of the concentrations across all time points for a particular cytokine. In orderto produce an accurate fit to the model it was essential to group cytokines to reducethe number of parameters. Grouping was based on the inflammatory classificationof each cytokine as well as similarity of cytokine profile. The time at which the peakconcentration occurred was recorded along with either the pro- or anti-inflammatoryclassification. Groupings can be seen in Table 1. Parameter estimation techniques,specifically fmincon in MATLAB, were used to fit the model to the data. The cost
Table 1 Grouping of 13 cytokines based on raw data and inflammatory classification Inflammatory WT data peak KO data peakGroup Cytokines classification (h) (h)1 IL-13, IL-5 Anti 2:5 242 MIP-1ˇ, TNF-˛, IL-12 Pro 2:5 243 IL-1ˇ, eotaxin, IFN- Pro 5 244 IL-4, IL-10 Anti 5 245 IL-6 Pro/Anti 2:5 106 MIP-1˛ Pro 5 247 Rantes Pro 10 24 A Mathematical Model of Cytokine Dynamics During a Cytokine Storm 335
function is the typical least squares function with a normalization matrix to offsetdiscrepancies between groups.
4 Results
Mathematical analysis is an important tool for biological predictions and forproducing more questions and further areas of research both in biology andmathematics [1]. Model results from parameter optimization of IFNAR-knockoutmice data are shown in Fig. 1. The model provides an accurate fit to the cytokinetime series data for both IFNAR-knockout and wild-type mice (not shown). Themodel parameters that provide the most interesting information are the ˛i;j values, asthey relate how each cytokine group interacts with each of the other groups. The signand magnitude of these values indicate the type of effect a particular cytokine has; anegative parameter value indicates an inhibitory effect, a positive value an enhancingeffect and the largest magnitude produces the most significant coupling. The threelargest ˛i;j values for each group of the IFNAR-knockout mice are displayed inFig. 2 where the lines indicate either enhancing (solid line with an arrowhead) orinhibiting (dotted line with a ‘T’) and the line thickness displays the magnitude (thethicker the line, the greater the magnitude). Preliminary results from Fig. 2 show that Groups 2, 4, 5 and 6 are the mostsignificant. Each of them have a large number of interactions, multiple two-waypaths, exhibit definite enhancement or inhibition, and fit well with what is knownbiologically. The most significant group, Group 2, consists of three pro-inflammatorycytokines: MIP-1ˇ, TNF-˛ and IL-12. Of the three, TNF-˛ is assumed to be thedominating cytokine. TNF has been thoroughly studied and is one of the most wellknown cytokines as it plays an important role in the outcome of immune function
GROUP 1 GROUP 2 GROUP 3 GROUP 4 0.4 0.4 0.5 0.8
0.4 0.3 0.3 0.6 Concentration
Concentration ConcentrationConcentration
0.3 0.2 0.2 0.4 0.2
0.1 0.1 0.2 0.1
0 0 0 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time (hours) Time (hours) Time (hours) Time (hours)
GROUP 5 GROUP 6 GROUP 7 0.8 0.35 0.8
0.6 0.3 0.6Concentration
Concentration
Concentration
0.4 0.25 0.4
0.2 0.2 0.2
0 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 Time (hours) Time (hours) Time (hours)
Fig. 1 Model results from parameter estimation of IFNAR-knockout mice. Dots represent rawtime series data, lines show the model prediction 336 M. Waito et al.
GROUP 3
GROUP CYTOKINES GROUP GROUP 1 IL-13, IL-5 2 4 2 MIP- , TNF- , IL-12
3 IL-1 , eotaxin, IFN- GROUP 1 4 IL-4, IL-10
5 IL-6
6 MIP-1 GROUP GROUP 7 5 7 Rantes
GROUP 6
Fig. 2 Largest three interactions for each cytokine group of IFNAR-knockout mice are repre-sented with a line. Solid lines with an arrow indicate enhancing effects while dotted lines with a‘T’ indicate inhibitory effects
[18, 23]. Model results for Group 2 show that it is an integral part of six significantinteractions and has multiple two-way paths, one with Group 5 and another withGroup 7. All three interactions that affect Group 2 are enhancing, indicating that asthe storm progresses, the amount of cytokine will continue to rise. TNF-˛, alongwith IL-1ˇ (found in Group 3), are considered early-response cytokines, occurringsoon after an immune response is triggered. It is known biologically that TNF-˛promotes the generation of IL-1ˇ [23]. Although the results are not shown for thewild-type case, the model results indicate that in fact Group 2 does enhance Group3. In the case of IFNAR-knockout mice however, Group 2 inhibits Group 3 (therelatively small interaction is not shown in Fig. 2). This difference begins to shedlight on why a storm occurred in mice lacking IFNAR on leukocytes. Another central group is Group 4 which is made up of two anti-inflammatorycytokines, IL-4 and IL-10, with the latter being a very prominent inhibitor. IL-10 isoften produced once a cytokine storm has begun in an attempt to return the balancethat has been lost, termed immunoparalysis. Although overproduction can oftenallow the host to survive the cytokine storm longer, it is not likely it will survive longterm [8, 18, 23]. Model results for Group 4 show that there are six interactions thatare significant as well as two primarily enhancing two-way interactions with Groups3 and 5. The three interactions that affect Group 4 are all enhancing, meaning thatthe amount of Group 4 will likely increase as the storm continues. Biologically it isknown that IL-10 plays a role in the down regulation of both TNF-˛ (Group 2) andIL-1ˇ (Group 3) [18]. Referring to Fig. 2, the model verifies that in fact Group 4does inhibit Group 2, however Group 4 enhances Group 3. This could be due to thegrouping of cytokines, since Group 2 contains MIP-1ˇ and IL-12 as well. On the A Mathematical Model of Cytokine Dynamics During a Cytokine Storm 337
contrary, in the wild-type case, Group 4 inhibits Group 3 while it enhances Group2. A well-studied cytokine that is known to be a key component of a cytokine stormis IL-6, Group 5, which has both pro- and anti-inflammatory properties and is acentral cytokine used to assess cytokine responses in the host [23]. Figure 2 displaysthe importance of this group with the eight interactions and two primarily enhancingtwo-way paths. It is known biologically that the production of IL-6 is stimulated byTNF-˛ and IL-1ˇ [23]. Results from the model using wild-type data verify thatindeed both Groups 2 and 3 enhance Group 5, however the interaction betweenGroup 3 and 5 is relatively small. IFNAR-knockout results shown in Fig. 2 implythat Groups 2 and 3 inhibit Group 5 and that those interactions are significant. Again,this can provide some insight into how a storm was able to occur. Group 6, MIP-1˛, is required for a typical inflammatory response to viruses [4].It is a pro-inflammatory chemokine that inhibits proliferation of hematopoietic stemcells in vitro and in vivo [4]. Model results for MIP-1˛ show that six interactionsare significant as well as a two-way interaction with Group 3. The three interactionsthat affect MIP-1˛ are both inhibiting and enhancing whereas in the wild-type casethe interactions are purely enhancing, causing an increase in the amount of MIP-1˛as the storm continues. It has been noted that the fine balance of pro- and anti-inflammatory mechanismsis critical in maintaining stability, and if these mechanisms become unbalanced,the outcome may contribute to a cytokine storm [23]. Groups 1 and 4 are anti-inflammatory cytokines, while the remainder act primarily as pro-inflammatorycytokines. For wild-type populations, of the three most significant groups (2, 4, and6), the anti-inflammatory cytokines are being inhibited, while the pro-inflammatorycytokines are being enhanced. This balance in this system becomes lost in micelacking the IFNAR on their leukocytes, as shown in Fig. 2. Instead of the anti-inflammatory cytokines (Group 4) being inhibited, they are instead enhanced.Immediately it becomes apparent that the fine balance that is typical of healthyindividuals has become unstable. Future work including sensitivity analysis of cytokine groupings could providefurther information on the significance of the groupings and individual cytokines.In conclusion, cytokines belonging to Groups 2, 4, 5 and 6, particularly TNF-˛,IL-10, IL-6 and MIP-1ˇ, have the largest effects on the dynamics of this particularcytokine storm. Changes introduced into the system by knocking out IFNAR causekey interactions to swap from enhancing to inhibiting and vice versa. It is possiblethat reducing the alterations in the effects of Groups 2, 4, 5 and 6 could lead to thereduction in severity and possibly even the entire storm.
Acknowledgements This research was supported in part by the Government of Ontario and theUniversity of Guelph. 338 M. Waito et al.
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Michael Yodzis, Chris T. Bauch, and Madhur Anand
Abstract Pollution-induced illnesses are caused by toxicants that result fromhuman activity and should be entirely preventable. However, social pressuresand misperceptions can undermine the efforts to limit pollution, and vulnerablepopulations can remain exposed for decades. We present a human-environmentsystem model for the effects of water pollution on the health and livelihood of afishing community. The model includes dynamic social feedbacks that determinehow effectively the population recognizes the injured and acts to reduce its pollutionexposure. Our work is motivated by a historical incident from 1949 to 1968 inMinamata, Japan where methylmercury effluent from a local factory poisonedfish populations and humans who ate them. We will discuss the conditions thatallow for the outbreak of a pollution-induced epidemic, and explore the role thatmisperception plays in allowing it to persist.
1 Introduction
A human-environment system (HES) is characterized by the interaction of humanactivities and natural processes [12, 13]. The HES model that we present in thispaper is developed in [20] in the context of the methylmercury-poisoning incidentin Minamata, Japan. It represents the transmission of pollution to humans throughfish-eating, and the social feedbacks to limit exposure through the boycott of fish-eating and the demand for pollution control. This model is motivated by several key observations. First is that the epidemio-logical data from Minamata show that the community was unwilling to recognizenew pollution victims from 1960 to 1971 [7, 8]. Historical accounts describe thedependence of the community on the polluting industry, and how this dependence
M. Yodzis () • M. AnandUniversity of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]. BauchUniversity of Waterloo, Waterloo, ON, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 341J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_32 342 M. Yodzis et al.
fed a widespread stigmatization of the pollution victims and underestimation of thehealth damages [5, 9, 17]. Our model depicts the voluntary actions that citizens must take to protect them-selves when proactive leadership from the government or industry is lacking. Legaltheorists who study the regulatory compliance of polluting industries observe thatgovernment policy may not be enforced effectively without pressure from citizens[19]. Social misperception can be the single largest obstacle to resolving pollutionproblems. Sometimes this is fuelled by an inconclusive scientific understanding,given that the symptoms of pollution exposure can be slow to appear, and difficultto separate from other causes or confounding variables [6, 7, 19]. Misperceptionmay also be fuelled by economic expediency and deliberate foul play. Therefore,citizen action and changing social perception play a large role in helping to resolvepollution problems. Our objectives are to examine the ecological and social conditions that cause theoutbreak of a pollution-induced epidemic, and to study the role of social feedbacksand misperception effects that allow the epidemic to persist.
2 Model
To focus on numerical simplicity and qualitative behaviour, in [20] we build asystem of normalized equations whose state variables are proportions rather thanabsolute quantities. The pollution is driven by an input n.t/, which is the baselineemissions loading rate from an industry into a water source. Whereas n.t/ is aprescribed function, the net emissions loading E.t/ is affected by feedback fromthe social demand for abatement. The fishery consists of fish F.t/ and boats B.t/,whose interactions are governed by Lotka-Volterra type equations and emissionsmortality. Human injuries are driven by the fish-borne pollution exposure. This exposuredepends on the net emissions level E.t/, the fish catch HF.t/B.t/ with harvest rateH, and a factor of pollution availability to humans, "I . To model human perception, we introduce P.t/ for the perceived injured and I.t/for the actual injured. The parameter s is the stigma rate, or the rate at which thecommunity denies new victims as the perceived injured P.t/ increases. This stigmaeffect causes a disparity between the actual and perceived number of injured. Ifthere is no stigma, s D 0, then I.t/ and P.t/ grow at the same rate. However, whens > 0, then a distinct negative feedback is introduced in P0 : the rate at which thecommunity is willing to recognize new victims decreases as the number of perceivedvictims increases. Then P.t/ is an underestimate of the actual health damages I.t/. Observe that in the model, all social feedbacks and decisions are driven by P.t/but are blind to I.t/. P.t/ affects the level of fish-eating, 1 bP.t/, where b isthe boycott rate. P.t/ also influences the public demand for abatement X.t/, whichgrows through social learning in the form of an imitation game [10]. We assumethat citizens sample the preferences of others and aim to minimize a cost function, Social Feedbacks in a Model of Fish-Borne Pollution Illness 343
hP.t/ ˇX.t/, by choosing either to favour abatement or non-abatement. is asampling and imitation rate, h measures the health costs from pollution, and ˇreflects costs to productivity and job security that is claimed by the industry and itssupporters in response to growing X. In turn, the population preference X.t/ feedsback to influence the net emissions E.t/, thus closing the loop. Altogether, the equations for the human-environment system are: 9 F 0 .t/ D rF.t/.1 F.t// HF.t/B.t/ "F E.t/F.t/ > > > > B0 .t/ D .1 bP.t//HF.t/B.t/ cB.t/ > = I 0 .t/ D "I E.t/HF.t/B.t/ Œ1 bP.t/ I.t/ (1) > > P0 .t/ D "I E.t/HF.t/B.t/ Œ1 bP.t/ sP.t/ P.t/ > > > ; X 0 .t/ D X.t/.1 X.t// ŒhP.t/ ˇX.t/
where the net emissions loading E.t/ is given by:
E.t/ D n.t/ .1 f .X.t/// (2)
Note that the form of the abatement function f .X/ is important. It should bemonotone increasing in X, and have a social concern threshold xM above whichthere is complete abatement: i.e. whenever X xM , then f .X/ D 1 and E.t/ D 0.Table 1 gives a summary of the model variables and parameters, including the valuesselected in [20] to simulate the historic pollution epidemic in Minamata. In thesesimulations, f is chosen to be a nonlinear function with xM D 0:4, With the appropriate choice of initial conditions, we can constrain the solutionsof (1) to an invariant region ˝ in the non-negative orthant of the FBIPX state space, ( ) .F; B; I; P; X/ W F; X 2 Œ0; 1 ; B 0; ˝ WD 1Cs 1 (3) I 2 0; 1CbCs ; P 2 0; 1CbCs
This result is proved in the supplementary appendix of [20] using standardinvariance theorems for systems of ordinary differential equations.Pollution Epidemic The system (1) has nine equilibria, with a single nontrivialequilibrium: 0 1 0 c 1 F H.1bP / B B C B 1 .r .1 F / " .1 f .X /// C B C BH F C B C B 1Cs C B I C D B 1CbCs C (4) B C B 1 C @ P A @ 1CbCs A X h ˇ.1CbCs/
This equilibrium corresponds to a pollution epidemic steady-state, as it occurswhen the emissions and fish catch coexist. 344 M. Yodzis et al.
Table 1 Model variables and parameters Variable Meaning Range I.C.a E Net emissions loading Œ0; 1 0.0 F Fish Œ0; 1 1.0 B Boats 0 0.1 I Cumulative injured Œ0; 1 0.0 P Cumulative perceived injured Œ0; 1 0.0 X Demand for pollution abatement Œ0; 1 0.01 f .X/ Abatement level Œ0; 1 0.0 Parameter Meaning Range Baselineb H Harvesting rate Œ0; 1 0.7 r Fecundity .0; 1 1.0 c Normalized boat costs .0; 1 0.35 "F Fish pollution mortality Œ0; 1 0.7 "I Pollution availability to humans Œ0; 1 0.009 b Rate of fish boycott per unit injury 0 100 s Rate of stigmatization per unit injury 0 600 Sampling and imitation rate Œ0; 1 0.5 h Rate of health concern per unit injury 0 1000 ˇ Pushback to demand for abatement 0 4.0 xM Social concern threshold Œ0; 1 0.4a Initial Condition values selected for simulation in [20]b Baseline parameter values selected for simulation in [20]
Existence and Stability Conditions This equilibrium cannot occur in ˝ unless thefollowing three inequality conditions hold: the social concern is below the thresholdneeded to abate the emissions,
h 0 < xM (5) ˇ.1 C b C s/
the fish reproduction rate exceeds the pollution mortality,
r > "F .1 f .X // (6)
and the returns from the harvest exceed the cost of fishing, "F .1 f .X // c < H.1 bP / 1 : (7) r
Using these same inequalities, it is shown in [20] that when the nontrivialequilibrium exists in ˝ it is locally attractive there.Convergence of Solutions to the Pollution Epidemic Equilibrium Even if a pollutionepidemic equilibrium exists and is locally attractive in ˝, it will not necessarily Social Feedbacks in a Model of Fish-Borne Pollution Illness 345
be reached. The existence and local stability conditions alone are not enough toguarantee convergence to steady-state, since the system (1) is highly coupled andnonlinear. The convergence behaviour of this system is sensitive to the relativegrowth rates of each variable, particularly those that affect the viability of the fishery. Solutions converge to the pollution epidemic equilibrium if the injuries P.t/ andsocial concern X.t/ grow fast enough that the fish catch F.t/B.t/ attains its steady-state. Whereas when they grow too slowly, the fish catch F.t/B.t/ will collapse.These observations are illustrated in the following schematic:
P.t/ ! P P.t/ ! P FAST SLOW X.t/ ! X X.t/ ! X FAST SLOW F.t/B.t/ ! F B F.t/B.t/ ! 0 converges collapses
Due to the nonlinearity of the system, we are not able to characterize theconvergence behaviour more precisely in terms of the model parameters, as wewere in (5), (6) and (7). However, there are insights to be gained from writing eachvariable in terms of its associated integral equation, evaluated up to some time T:
ZTP.T/ D P.t0 / C "I H F.t/ B.t/ .1 f .X.t/// Œ1 .1 C b C s/ P.t/ dt t0 ZTX.T/ D X.t0 / C X.t/ .1 X.t// Œh P.t/ ˇ X.t/ dt t0 ZT ZTF.T/ D F.t0 / C r F.t/ .1 F.t// H F.t/ B.t/ dt "F F.t/ .1 f .X.t/// dt t0 t0 ZTB.T/ D B.t0 / C .1 b P.t//H F.t/ B.t/ c B.t/ dt t0
Observe that these integral equations depend on the initial conditions and theparameter values "I and , whereas the inequalities (5), (6) and (7) do not. We seethat the fishery is sensitive to the time evolution of P.t/ and X.t/, because P.t/affects both the demand for fish 1 bP.t/ and the demand for abatement X.t/, whileX.t/ affects the emissions. In general, during the time that P.t/ and X.t/ are small,both the number of boats and the level of emissions are high. If the fish populationis driven below the break-even cost required for the boats, the fish catch collapsesand an epidemic is averted. 346 M. Yodzis et al.
3 Analysis
In [20] our model is applied to simulate the pollution epidemic that occurred from1949 to 1968 in Minamata, Japan. Given that some of the datasets from Minamataare incomplete or sampled at irregular time intervals, the simulations are meant toagree qualitatively with the available data, and to identify a set of baseline parametervalues that are physically plausible. These baseline values displayed in Table 1. Thesimulations run from 1945 to 1976. We are interested in examining how the social feedbacks, driven by the parame-ters s, b and h, affect the resulting steady-state. The system (1) has nine equilibriathat we class into four types: (I) No fish catch, no emissions; (II) Fish catch, noemissions; (III) No fish catch, emissions; (IV) Fish catch and emissions. Type IV isa nontrivial equilibrium, a pollution epidemic steady-state. Figure 1 depicts the steady-states of the system solved in s-b parameter space,with all other parameters held at their baseline values. We numerically solve F, Band X for large time t D 1;000;000 to approximate the steady-state, and then plot itin s-b space coloured according to its equilibrium type. We are also able to test our theoretical expectations against the numerical results.In Fig. 1, the hatched region demarcates where there is potential for a pollutionepidemic according to the inequalities (5), (6) and (7). We find that this regionoverlaps exactly with the numerically solved steady-state region in dark grey, whichindicates that a pollution epidemic has occurred. As the parameter plane in Fig. 1 shows us, an epidemic does not occur unlessthe stigma s is sufficiently large. It is very important to recognize how unlikelythe conditions for a pollution epidemic (type-IV) equilibrium are without socialstigma/misperception.
Fig. 1 Changing values for the social feedbacks s (stigma) and b (boycott) yield alternativequalitative outcomes. The parameter plane shows the dynamical outcomes defined by theequilibrium-types I, II, III, and IV that result for various values of s and b. The hatched regionsignifies where a potential epidemic is possible, and the solid point represents the baseline valuesused in the simulation of Minamata; s D 600, b D 100 Social Feedbacks in a Model of Fish-Borne Pollution Illness 347
We want to know what parameters are required for us to have a pollutionepidemic when s D 0, and how physically plausible they are compared to thebaseline parameter values used in the simulation of Minamata. Figure 1 depicts theaxis s D 0 in black. To find when a pollution epidemic becomes possible withoutstigma, we must vary our parameters so that the inequalities (5), (6) and (7) aresatisfied on the axis s D 0. To find when this happens, we can consider the boundarycase of (5) with X D xM ,
h D xM (8) ˇ.1 C b/
so that (6) reduces to r > 0 and (7) reduces to 1 c<H (9) 1Cb
Combining (8) and (9) yields
h HMx < (10) ˇ c
Varying each of these parameters individually while holding the others at theirbaseline values from Table 1, we find that the potential for a pollution epidemicexists for some b if either the health concern is decreased to 0 < h < 3:2, the boatcosts are decreased to 0 < c < 0:00112, or the fish harvesting rate is increased toH > 218:75. Note that in the ratio h=ˇ, it suffices to consider changing h only. Asthe numerical results in Fig. 2 show, the time evolution of the solutions is sensitiveto the collapse of the fishery, and they do not necessarily converge to the pollutionepidemic equilibrium. When the health concern is decreased to h D 3 a pollution epidemic becomespossible for 0:25 < b < 0:55. However, as Fig. 2a demonstrates, small values of hand b make the social response to the pollution negligible. The fish catch collapsesbefore the injuries grow large enough to trigger any abatement, and a pollutionepidemic is averted. Decreasing the boat costs to c D 0:001, a pollution epidemic becomes possiblefor values of b > 624. The numerical results in Fig. 2b indicate that the pollutionepidemic occurs. However, is this plausible? The accompanying plot in Fig. 2bsuggests that this operating cost for the boats is unrealistically low. The boats growto an unrealistically high level and push the fish population toward zero. For theshort time-scale we are interested in, the potential for a pollution epidemic is avertedby the decline of the fish catch to near-zero levels. If we solve numerically over avery large time scale as t ! 1, it turns out that the fish and boats exhibit dampedoscillations, and the injuries grow. 348 M. Yodzis et al.
Fig. 2 We seek the conditions for which a pollution epidemic is possible with no stigma. Thepanels depict the s-b parameter planes that result when (a) h D 3, (b) c D 0:001 and (c) H D 220.Each is coloured according to the legend in Fig. 1. The solid dot in each plane represents the valuesof s and b used to generate each of the plots below, which show the time evolution of the fish (indark grey) and boats (in light grey)
Another option is to increase the harvest rate to H D 220 with b > 624. However,as the plots in Fig. 2c show, this unsustainable harvest pressure causes the fish catchto collapse before the variables are able to reach a pollution epidemic steady-state. To examine the effects of the parameters r and "F on the occurrence of a pollutionepidemic, we return to the inequalities (5), (6) and (7). Allowing r, "F and b tobe free while holding s D 0 and all other parameters at baseline, we find thatthe inequalities are satisfied when r > 0 and "F < 0. However, negative valuesfor the pollution mortality rate of fish violate our parameter requirements and arebiologically implausible. Instead, we explore the case "F D 0 so that the fish do not die from pollution,while freeing r and another parameter, h. Holding s D 0 and all other parametersat baseline, we find that the inequalities (5), (6) and (7) are satisfied if r > 0, 0 <h < 3:2, and 0 < b < 1. Figure 3a shows that an epidemic does occur for the valuesr D 1, h D 1 and b D 0:9. To determine whether this is realistic, we look to the timeseries plots in Figs. 3b–e. The emissions and fish catch coexist and the injuries growsteadily. Since s D 0, all of these injuries are fully perceived. However, the growthof the social concern X and the reduction of fish-eating bP remain negligible,because the social feedbacks h and b are very small. Assuming that the nature of thepollution injuries is serious, like those experienced from methylmercury-poisoningin Minamata, this situation of full perception without response is socially unrealistic. Social Feedbacks in a Model of Fish-Borne Pollution Illness 349
Fig. 3 We seek the conditions for which a pollution epidemic is possible with no stigma. Panel(a) depicts the s-b parameter plane that results when "F D 0 and h D 1. The plane is colouredaccording to the legend in Fig. 1, and the black point gives the values s D 0 and b D 1 used togenerate the accompanying time series plots: (b) emissions, (c) social concern, (d) fish (in darkgrey), boats (in light grey), (e) perceived and actual injuries
4 Discussion and Conclusion
The human-environment system model studied in this paper allows us to understandhow social feedbacks that protect people from pollution rely on accurate informa-tion. Our model is capable of reaching a pollution epidemic steady-state where thepollution exposure continues indefinitely. We conclude that the pollution epidemicequilibrium is unlikely to occur in the absence of stigma/misperception, exceptunder conditions that are socially or biologically implausible, or which violate ourparameter constraints. This highlights the importance of including misperception asa factor in the model. Our HES model is based on interactions that are endogenous and localized;there are no external shocks, and no corrective mechanisms to move the systemaway from a pollution equilibrium once it settles there. The very existence of apollution epidemic equilibrium, and the role played by misperception in augmentingits occurrence, reveals that our model can fail to be self-regulating. This reflectsreal-world circ*mstances in which citizen stakeholders have information that isinaccurate or different to that held by industry, rendering citizens’ voluntaryprotective actions inadequate to resolve the problem. Our HES model representsthe conditions that might have persisted for decades in Minamata if a secondmethylmercury-poisoning incident had not occurred in Niigata, Japan, to increasecitizens’ awareness and stimulate relief for the victims [1, 17]. In models where human activity interacts with natural processes, we believethere is more room to study the extent to which human decisions can be madefrom inaccurate information, and to study how this affects the qualitative outcomeof the model. For example, although medical researchers recognize the role ofstigma in prolonging epidemics, especially when it shames people from seeking 350 M. Yodzis et al.
medical attention [4, 18], it is not clear that these insights have been readily takenup in epidemiological models. Risk perception and behavioural strategies to avoidinfectious disease are more widely modelled [14], and some agent-based modelsexplicitly distinguish between injured and perceived injured [15]. Social learningmodels are increasingly used in mathematical epidemiology to study vaccine scares[3], and in ecology to study resource management and conservation [2, 11, 16]. Up to now, these methods have not been widely applied to understand pollutionillnesses, in spite of the fact that pollution is intimately bound to social activity.We have worked to incorporate these insights into our model, and the feedbacksinvolving stigma and fish-eating boycott represent a break from existing approaches.
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Mohamad S. Alwan, Humeyra Kiyak, and Xinzhi Liu
Abstract This paper addresses impulsive switched singular systems with nonlinearperturbation term. The main theme is to establish exponential stability of thesystems where the impulses are of fixed time type and treated as perturbation.We first establish the exponential stability of a single-mode impulsive systemsusing the Lyapunov method. We have observed that if the underlying continuoussystem is stable and the impulses are applied slowly, then it is guaranteed thatthe impulsive system maintains the stability property. Later, a switched systemwith impulsive effects is considered. The method of multiple Lyapunov functionand average dwell time switching signal are used. We have noticed that if allsubsystems are exponentially stable and the average dwell time is sufficiently large,then the impulsive switched system is exponentially stable. Numerical exampleswith simulations are given to illustrate the effectiveness of the proposed theoreticalresults.
1 Introduction
Singular systems (or differential algebraic systems) have received considerableattentions due their extensive applications in the areas of control systems, electricalpower systems, mechanical systems, and robotics. Readers may refer to [4, 4–6, 9, 12]. The theory of singular system is well studied in the literature; see forinstance [3, 4, 6, 7, 12, 15], and [5]. On the other hand, the dynamics of many natural or engineering systems aresubject to abrupt changes, where the durations of these changes are sufficientlysmall that can be reasonably approximated by instantaneous changes of states orimpulses. The resulting systems can be suitably modeled as an impulsive systems.Nowadays, the applications of impulsive system have been found in many differentreal world or man-made systems, such as mechanical and engineering systemsincluding mass-spring systems, robotics, or electrical circuits. It can also be found inbiological systems including the function of the heart and biological neural network,
M.S. Alwan () • H. Kiyak • X. LiuDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 355J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_33 356 M.S. Alwan et al.
and epidemic disease models including pulse vaccination. The theory of impulsivesystem is well presented and more application can be found in [1, 2, 11, 14, 16]. When applying the impulsive effects to the singular system, the resulting systemis called impulsive singular system. Many singular systems exhibit impulsive andswitching behaviors, which are characterized by switches of states and abruptchanges at switching time; that is, the systems switch with impulse effects [1, 2,11, 13]. The objective of the paper is to establish the exponential stability of a nonlinearimpulsive switched singular system. The nonlinearity is linearly bounded, and boththe impulsive effects and switching are applied to the system at fixed times. Usingthe method of Lyapunov function for impulsive singular system, some sufficientconditions are given to establish the stability property. On the other hand, multipleLyapunov function and the average dwell-time approach are used to determine somesufficient conditions for impulsive switched singular system. The rest of the paper is organized as follows: In Sect. 2, the problem formulationand some background are given. In Sect. 3, we state and prove the main results.Numerical examples are presented in Sect. 4 to clarify the proposed approach.Finally, a conclusion of these results is given in Sect. 5.
2 Problem Formulation
Denote by Rn the n-dimensional Euclidean space with the norm k k, Rnxn the set ofall nxn square matrices and N the set of natural numbers. If A 2 Rnxn , denote by minand max the p minimum and maximum eigenvalues of A; respectively, and its normis kAk D max .AT A/; where the superscript T represents the matrix transpose. IfA is a positive definite matrix, then we write A > 0. Consider the impulsive switched singular systems of the form
EPx.t/ D A .t/ x.t/ C g .t/ .t; x/; t ¤ tk ; x.t/ D Bk x.t/; t D tk ; (1) x.t0 / D x0 ;
where x 2 Rn is the system state variable, and A .t/ ; Bk ; E 2 Rnxn are systemcoefficient matrices with E being singular with rank.E/ D r < n, the matrixpairs .E; A .t/ / being regular, and Bk being constant matrices. The switching signal W Œt0 ; 1/ ! S is a piecewise constant function taking values in a finite compactset S D f1; 2; : : : ; Ng for some N 2 N. ftk g1kD1 are the impulsive times that form anincreasing sequence satisfying tk1 < tk and limk!1 tk D 1. x D x.tC / x.t /where x.t / (and x.tC /) is the state just before (and just after) the impulsive actionwith x.tC / D lims!tC x.s/. The solution x is assumed to be left-continuous, i.e.,x.tk / D x.tk /. For i 2 S , gi .t; x/ W RC Rn ! Rn is piecewise continuous vector- Stability of Singular Impulsive Systems 357
valued functions with gi .t; 0/ 0; t 2 RC to ensure the existence of solutions forsystem (1). When .t/ D for all t > t0 , system (1) is reduced to the following single-modeimpulsive singular system:
EPx.t/ D Ax.t/ C g.t; x/; t ¤ tk x.t/ D Bk x.t/; t D tk ; k2N (2) x.t0 / D x0 ;
where the subscript is dropped for simplicity of notation. In the following, some definitions and theorems are introduced.Definition 1 ([7, 8]) Matrix pair .E; A/ is regular if there exists a constant scalar 2 C such that det. E A/ ¤ 0: The matrix pair .E; A/ is said to be impulse freeif deg.det. E A// D rank.E/:Definition 2 ([7, 17]) Regular system (1) is said to be exponentially stable if thereexist ˛; ˇ > 0 such that its state x.t/ satisfies kx.t/k ˛kx.t0 /keˇt ; for all t > t0 :Definition 3 ([17]) System (1) is said to be E-exponentially stable if there existconstants ˛; ˇ > 0 such that kEx.t/k ˛kEx.t0 /keˇt ; for all t > t0 :Definition 4 ([7]) System (1) is admissible if it is stable and impulse-free.Theorem 1 ([8]) Matrix pair .E; A/ is regular if and only if there exist two Q1 1 2 1 2 and P D P P where Q 2 R rnnonsingular matrices Q; P such that Q D , QQ2 2 R.nr/n , P1 2 Rnr , P2 2 Rn.nr/ and the following standard decompositionholds: g .t; x1 ; x2 / QEP D diag .Ir ; N/ ; QAP D diag .A1 ; Inr / ; Qg.t; x/ D 1 ; g2 .t; x1 ; x2 / (3)
where r D rank.E/, A1 2 Rrr , N 2 R.nr/.nr/ is a nilpotent matrix with nilpotentindex h.Theorem 2 ([7]) The singular system EPx.t/ D Ax.t/ is stable if and only if .E; A/has finite eigenvalues with negative real parts.Theorem 3 ([7]) If system (1) is admissible, then for any Y > 0 there exists X > 0satisfying ET XA C AT XE D ET YE: 358 M.S. Alwan et al.
3 Main Results
In this section, exponential stability of system (1) and (2) are discussed.
3.1 Singular Systems Subject to Impulsive Effects
Theorem 4 Assume that system (2) is impulse free, the eigenvalues of the matrixpair .E; A/ have negative real parts, the singular matrix E and the matrix I C Bk forI 2 Rnxn identity matrix are commutative, and kg.t; x/k kExk for < 2kXk min .Y/
positive constant where X > 0 satisfying
AT XE C ET XA D ET YE (4)
for any Y > 0. Then, the trivial solution of the nonlinear singular impulsive system(2) is exponentially stable if the following inequality holds:
ln ˛k .tk tk1 / 0; k D 1; 2; : : : (5)
max Œ.ICBk /T X.ICBk / min .Y/2kXk where ˛k D min .X/ , 0 < < , and D max .X/ .Proof Let x .t/ D x.t; t0 ; x0 / be the solution of the system (2). Define
$.t/ D V.x.t// D xT .t/ET XEx.t/; t ¤ tk
as a Lyapunov function candidate. Then, along the trajectory of (2), $.t/ P D xT .t/ AT XE C ET XA x.t/ C gT .t; x/XEx.t/ C xT .t/ET Xg.t; x/ D xT .t/ET YEx.t/ C 2xT .t/ET Xg.t; x/ (6)
where ET YE D AT XE C ET XA: Since X > 0 and Y > 0, we have [10]
min .X/kExk2 xT ET XEx max .X/kExk2 (7) min .Y/kExk2 xT ET YEx max .Y/kExk2 : (8)
Thus, using (7), (8), and kg.t; x/k kEx.t/k in (6) lead to P min .Y/ 2kXk kEx.t/k2 $.t/; $.t/ t 2 .tk1 ; tk ;
min .Y/2kXk min .Y/where D max .X/ is a positive constant if < 2kXk . Then,
C $.t/ $.tk1 /e .ttk1 / ; t 2 .tk1 ; tk : (9) Stability of Singular Impulsive Systems 359
At t D tkC ; we have
$.tkC / D xT .tk /.I C Bk /T ET XE.I C Bk /x.tk / ˛k $.tk /; (10)
max Œ.ICBk /T X.ICBk / where ˛k D min .X/ : Using (9) and (10), one may get
$.t/ ˛1 ˛2 : : : ˛k $.t0C /e .tt0 / D $.t0C /˛1 e.t1 t0 / ˛2 e.t2 t1 / : : : ˛k e.tk tk1 / e.ttk / e./.tt0 / :
By assumption (5), we have
$.t/ $.t0C /e./.tt0 / ; t t0 ;
which implies that
min .X/kEx.t/k2 $.t/ $.t0C /e./.tt0 / max .X/kEx.t0C /k2 e./.tt0 / ) kEx.t/k KkEx.t0C /ke./.tt0 /=2 ; t t0 ; (11) pwhere K D max .X/=min .X/. Thus, the trivial solution of the system (2) isE-exponentially stable. Let 1 x1 .t/ P x.t/ D : (12) x2 .t/
Then, it follows from the standard decomposition form that system (2) is equiva-lent to
xP 1 D A1 x1 C Q1 g.t; x/ (13) 0 D x2 C Q2 g.t; x/ (14)
where x1 2 Rr and x2 2 Rnr . By (12), we have x1 .t/ Ir 0 x1 .t/ x1 .t/ QEx.t/ D QEP D D : (15) x2 .t/ 0 0 x2 .t/ 0
From (11) and (15), we have
kx1 k D kQEx.t/k KkQkkEx.t0C /ke./.tt0 /=2 (16)
which implies that x1 is exponentially stable. 360 M.S. Alwan et al.
We need to show that x2 is also exponentially stable. It follows from (14) andkg.t; x/k kEx.t/k that
kx2 k D kQ2 g.t; x/k KkQ2 kkEx.t0C /ke./.tt0 /=2 : (17)
This means x2 is exponentially stable. From (16) and (17), the trivial solution of (2)is exponentially stable. This completes the proof.Remark 1 The validity of the matrix equation (4) guarantees that the Lyapunovfunction is decreasing for all t 6D tk . While the inequality in (5) is made to ensure thatthe impulses are applied slowly in order to maintain the stability of the impulsivesystem in (2).
3.2 Switched Singular Systems Subject to Impulsive Effects
Theorem 5 For any i 2 S , assume that system (1) is impulse free, the eigenvaluesof the matrix pairs .E; Ai / have negative real parts, and kgi .t; x/k i kExk for i <min .Yi / 2kXi k positive constant where Xi > 0 satisfying Ai Xi E C E Xi Ai D E Yi E for T T T
any Yi > 0. Then, the trivial solution of (1) is exponentially stable if the followingassumptions hold:(i) For any i; j 2 S there exists ˛k > 1 such that
.I C Bk /T ET Xj E.I C Bk / ˛k ET Xi E: (18)
(ii) For any t0 , the switching law satisfies
t t0 N.t; t0 / N0 C Ta
where N.t; t0 / represents the number of switchings in .t; t0 /, and N0 and Ta are the chatter bound and average dwell time to be defined, respectively.Proof Let x .t/ D x.t; t0 ; x0 / be the solution of the system (1). Define
$i .t/ D Vi .x.t// D xT .t/ET Xi Ex.t/; t ¤ tk ;
as a Lyapunov function candidate for ith subsystem. Then, derivative of $i along thetrajectory of (1) is given by
$P i .t/ D xT .t/ET Yi Ex.t/ C 2xT .t/ET Xi gi .t; x/ min .Yi /kEx.t/k2 C 2kxT .t/ET kkXi kkgi .t; x/k .min .Yi / 2kXi ki / kEx.t/k2 i $i .t/; t 2 .tk1 ; tk ; (19) Stability of Singular Impulsive Systems 361
min .Yi /2kXi ki min .Yi /where i D max .Xi / > 0 if i < 2kXi k . Then,
C $i .t/ $i .tk1 /ei .ttk1 / ; t 2 .tk1 ; tk : (20)
At t D tkC , k 2 N, suppose .tkC / D j, it follows from (1) and (18) that
$j .tkC / D xT .tk /.I C Bk /T ET Xj E.I C Bk /x.tk / ˛k $i .tk /: (21)
Using (20) and (21) leads to
$i .t/ ˛1 ˛2 : : : ˛i1 $1 .t0C /e1 .t1 t0 / e2 .t2 t1 / : : : ei .ttk / ; t t0 :
Let D minfi I i 2 S g. Then, we have
$i .t/ ˛1 ˛2 : : : ˛i1 $1 .t0C /e .tt0 / : (22)
To use assumption (ii), let ˛ D maxf˛k I k D 1; 2; : : : ; i 1g. Then, (22) becomes
$i .t/ ˛ i1 $1 .t0C /e .tt0 / D $1 .t0C /e.i1/ ln ˛ .tt0 / :
ln ˛Applying (ii) with N0 D ln ˛ , where is an arbitrary constant, and Ta D ,. > / leads to .tt / $i .t/ $1 .t0C /e 0 ;
which implies that .tt //=2 kEx.t/k KkEx.t0C /ke. 0 ; t t0 ; (23) pwhere K D max .X1 /=min .Xi /. Thus, the trivial solution of the system (1) isE-exponentially stable. As done in Theorem 4, we can show that .tt //=2 .tt //=2kx1 k KkQi kkEx.t0C /ke. 0 and kx2 k i KkQ2i kkEx.t0C /ke. 0 :
That is, the trivial solution of (1) is exponentially stable.
4 Numerical Example
In this section, we present some examples to illustrate the effectiveness of thetheoretical results. 362 M.S. Alwan et al.
Example 1 Consider the singular impulsive system in (2) where 2 3 2 3 2 3 1000 0 100 0 60 0 1 07 6 1 0 0 07 6 0 7 ED6 7 6 7 6 40 0 0 05 ; A D 41 0 0 15 ; g.t; x/ D 4 7; 5 0 0000 0 111 x1 .t/ sin.t/ 2 3 0:01 0 0 0 6 0 0:01 0 0 7 Bk D 6 4 0 7 0 0:01 0 5 0 0 0 0:01
and the admissible initial value is x.0/ D Œ2 1 0 0 T : Since deg.det.sE A// D rank.E/ D 2, the system ispimpulse free. Also, thesystem is stable because the eigenvalues of .E; A/ are 1˙i 2 3 . Given 2 3 2 0 00 60 2 0 07 Y D6 4 0 7; 0 1 05 0 0 0 1
the equation ET XA C AT XE D ET YE is satisfied where 2 3 2 1 2 2 61 3 1 17 XD6 42 1 7: 3 05 2 1 0 8
We also get D 0:1112, D 0:000001 and ˛k D 84:5806, and from (5) tk tk1 39:91. The simulation result is shown in Fig. 1, where tk tk1 D 40:Example 2 Consider the impulsive switched singular system given by (1) where x1 .t/ 40xD , .t/ 2 S D f1; 2g, E D with rank.E/ D 1, Bk D 0:1I and x2 .t/ 20 2 1 1 1 T A1 D ; g1 .t; x/ D 15 tanh.x1 .t// 15 tanh.x2 .t// ; 1 2 4 1 T A2 D ; g2 .t; x/ D 12 tanh.x1 .t// 1 2 tanh.x2 .t// : 1 4 Stability of Singular Impulsive Systems 363
x1(t) 0
–2 0 20 40 60 80 100 120 t 2 x2(t)
–2 0 20 40 60 80 100 120 t 2 x3(t)
–2 0 20 40 60 80 100 120 t 2 x4(t)
–2 0 20 40 60 80 100 120 t
Fig. 1 Exponential stability result of Example 1
From the Jordan canonical form of . E Ai /1 E for 2 C, we find that 1 1 41 1 1 0 0 Q1 D 5 10 2 4 ; P1 D 4 ; Q2 D 99 ; P2 D 2 ; 5 5 5 12 1 1 12 6 1 3 43 3 10 10 0such that Q1 EP1 D Q2 EP2 D ; Q1 A 1 P 1 D Q2 A 2 P 2 D ; 00 0 1 5 6 0 : From the decomposition form it is clear that the systems are impulse free. 0 1Moreover,the eigenvalues of .E; A1 / and .E; A2 / are negative. 5 0 X1 D 3 5 > 0 satisfies AT1 X1 E C ET X1 A1 D ET Y1 E for any Y1 D I > 0. 0 3Using X1 and Y1 ,we can 2 find that 1 < 0:3 where 1 satisfies kg1 .t; x/k 1 kExk. 0Similarly, X2 D 3 1 > 0 satisfying AT2 X2 E C ET X2 A2 D ET Y2 E for any Y2 D 0 3I > 0. Thus, by X2 and Y2 , 2 < 0:75 is found where 2 satisfies kg2 .t; x/k 2 kExk. We also get 1 D 0:62 and 2 D 1:5020. Thus, D 0:62. The inequality.I CBk /T ET Xj E.I CBk / ˛k ET Xi E holds where ˛k D 1:21: If we choose D 0 and D 0:02, then the chatter bound N0 D 0, and the average dwell time Ta D 0:3177.The simulation result is shown in Fig. 2, where tk tk1 D 1. 364 M.S. Alwan et al.
6 5 4
3 x1(t)
1 0
–1 0 5 10 15 20 25 t 6 5 4
3 x2(t)
1 0
–1 0 5 10 15 20 25 t
Fig. 2 Exponential stability result of Example 2
5 Conclusion
A nonlinear impulsive singular system has been studied. In Sect. 3.1, the exponentialstability has been established for the impulsive system when we have used theLyapunov method. It has been noticed that if the continuous system is stableand the time between every successive impulses is sufficiently large, then theimpulsive system is also exponentially stable. In Sect. 3.2, we have addressed theimpulsive switched system and developed new sufficient condition to guarantee theexponential stability using multiple Lyapunov function method and average dwelltime switching signal to organize the jumps among the subsystems. It has beennoticed that when all subsystems are stable, if the average dwell time is sufficientlylarge, then the entire system is also exponentially stable.
References
1. Bainov, D.D.: Systems with Impulse Effect: Stability, Theory, and Applications. Ellis Hor- wood, Chichester/Halsted Press, New York, Toronto (1989) 2. Bainov, D.D., Simeonov, P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific & Technical, Harlow/Wiley, New York/Burnt Mill, Harlow (1993) 3. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, New York (1989) Stability of Singular Impulsive Systems 365
4. Campbell, S.L.: Singular Systems of Differential Equations. Pitman Advanced Pub. Program, San Francisco (1980) 5. Campbell, S.L.: Singular Systems of Differential Equations II. Pitman, San Francisco (1982) 6. Dai, L.: Singular Control Systems. Springer-Verlag, Berlin (1989) 7. Duan, G.: Analysis and Design of Descriptor Linear Systems. Springer, New York (2010). 8. Feng, G., Cao, J.: Stability analysis of impulsive switched singular systems. IET Control Theory Appl. 9(6), 863–870 (2015) 9. Hill, D.J., Mareels, I.M.Y.: Stability theory for differential/algebraic systems with application to power systems. IEEE Trans. Circuits Syst. 37(11), 1416–1423 (1990)10. Khalil, H.K.: Nonlinear Systems. Prentice Hall, Upper Saddle River (2002)11. Lakshmikantham, V.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)12. Lewis, F.: A survey of linear singular systems. Circuits Syst. Signal Process 5(1), 3–36 (1986)13. Li, Z.: Switched and Impulsive Systems: Analysis, Design, and Applications. Springer, Berlin/New York (2005)14. Sun, J.T., Zhang, Y.P.: Stability analysis of impulsive control systems. IEE Proc. Control Theory Appl. 150(4), 331–334 (2003)15. Wang, C.J.: Controllability and observability of linear time varying singular systems. IEEE Trans. Autom. Control 44(10), 1901–1905 (1999)16. Yang, T.: Impulsive control. IEEE Trans. Autom. Control 44(5), 1081–1083 (1999)17. Yao, J., Guan, Z.H., Chen, G., Ho, D.W.C.: Stability, robust stabilization and control of singular-impulsive systems via switching control. Syst. Control Lett. 55(11), 879–886 (2006) Input-to-State Stability and H1 Performancefor Stochastic Control Systems with PiecewiseConstant Arguments
Mohamad S. Alwan and Xinzhi Liu
Abstract This paper addresses stochastic control system of differential equationswith piecewise constant arguments (SEPCA). The piecewise constant argumentsare of delay type. The system is viewed as a hybrid (or particularly switched)system. This approach motivates the applicability of the classical theory of ordinarydifferential equations, but not of functional differential equations, and the designof a switching law. The main theme of this work is to establish the problems ofinput-to-state stabilization (ISS), and H1 performance for a class of an uncertaincontrol SEPCA. To analyze these result, a common Lyapunov function togetherwith the techniques of differential inequalities and Razumikhin condition is used.A numerical example with simulations is presented to clarify the validity of theproposed theoretical approaches.
1 Introduction
A nonlinear deterministic differential equations with piecewise constant arguments(EPCA) may have the form
xP .t/ D f .t; x.t/; x..t///; (1)
where the argument is a piecewise constant function defined on intervals with acertain length, and it may be defined by .t/ D Œt ; Œt n ; t nŒt , or Œt C 1 , for allt and a positive integer n, where Œ is the greatest-integer function [8, 22]. EPCA appeared in some sequential-continuous disease models [6], controlsystems [7], population growth models [1, 4, 10], chaotic synchronization–includingdelayed secure communication [3], and in approximating the solutions of somedelay differential equations [9, 11, 22]. The dynamics of these equations containboth continuous and discrete components, which enable this type of systems ofequations to be adequately embedded under the hybrid system umbrella [4]. EPCA
M.S. Alwan () • X. LiuDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 367J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_34 368 M.S. Alwan and X. Liu
can also be a special type of functional differential equations, where the state historyis given at certain individual points, rather than intervals. The theory of EPCA is well discussed in [8, 22]. Further properties andapplications of these equations were considered in many other works; readers mayrefer to [2, 5, 7, 14, 15, 21, 23]. The notion of ISS deals with the response of asymptotically stable systems tobounded, small input or disturbance regardless of the magnitude of system initialstate. During the last two decays, ISS has become a central foundation of modernnonlinear feedback and design. It has been now playing a key role in systems withrecursive design and co-prime factorizations, and a connection between the input-output (or external) stability and state (or internal) stability. For more properties,applications, and implications of the ISS, readers may consult [13] and [16–20]. In this paper, stochastic EPCA (or SEPCA) is studied. As stated earlier, thesystem will be viewed as a switched system. The main objective is to establishthe problem of input-to-state stabilization and H1 performance. To analyze theseresults, the Lyapunov function method and Razumikhin condition are used. We alsopresent exponential stability as a special case of the aforementioned results. Finally,to enhance the findings of this paper, some numerical examples and simulationsare presented. We believe that these results will have a great contribution to manyresearch fields in sciences and engineering, particularly in the fields of modernnonlinear control design and recursive system design and in dynamical systemssubject to external disturbances. The paper is organized as follows: In Sect. 2, the system is described, andsome notations and definitions that will be used in the sequel are given. The maincontributions of the paper are presented in Sect. 3, and a conclusion is given inSect. 4.
2 Problem Formulation and Background
Denote by RC the set of all non-negative qP real numbers, R the n-dimensional n
n 2 iD1 xi for every x 2 R , R n nmEuclidean space and its norm kxk D the setof all n m real matrices, and ˝ a sample space. Let C .Œa; b ; D/ be the space ofcontinuous functions mapping Œa; b , with a < b for any a; b 2 RC , into D, forsome open set D Rn , and C 1;2 .RC Rn I RC / be the space of all functions V.t; x/defined on RC Rn such that they are continuously differentiable once in t andtwice in x. For instance, if V.t; x/ 2 C 1;2 .RC Rn I RC /, then we have
@V @V @V @2 V Vt D ; Vx D ; ; ; Vxx D : @t @x1 @xn @xi @xj nn
Let .˝; F ; fF t gt0 ; P/ be a complete probability space with filtration fF t gt0satisfying the usual condition (i.e., it is right continuous and F 0 contains all P- ISS and H1 for Stochastic Control EPCA 369
null set of F ). For all t 2 RC and ! 2 ˝, let W.t; !/ 2 Rm be an m-dimensionalWiener (or Brownian motion ) process. We also assume that, for a given W andfiltration F t for all t 2 Œa; b RC , the process W is F t -measurable and, forall s t, the random variable W.t; !/ W.s; !/ is independent of the -algebraF s . Denote by L ad .˝; Lp Œa; b / the class of all random processes f which are F t -adapted and almost all their sample paths are integrable in the Riemann sense. Forsimplicity of notation, we will drop ˝ and ! when writing random processes. A nonlinear system with SEPCA may have the following form dx.t/ D f t; x.t/; x..t// dt C g t; x.t/; x..t// dW.t/; x.t0 / D x0 ; (2)
where x 2 Rn is the system state, and for all t t0 , the function .t/ is a piecewiseconstant function taking values in the set % D fk g1 kD0 . So that one may redefine thisfunction as follows: for a non-negative integer k, W Œtk ; tkC1 / ! %. This piecewiseconstant function represents the switching signal of the system which has the rolesof switching between the values of system state argument x. Accordingly, one maydefine system (2) as follows: for all t 2 Œtk ; tkC1 /, dx.t/ D f t; x.t/; x.k / dt C g t; x.t/; x.k / dW.t/; x.t0 / D x0 ; (3)
or, equivalently, Z Z t t x.t/ D x0 C f s; x.s/; x.k / ds C g s; x.s/; x.k / dW.s/; (4) t0 t0
where the first integral is a Riemann integral almost surely (a.s.), and the second oneis an Itô integral satisfying hZ t i E g s; x.s/; x.k / dW.s/ D 0; and t0 Z t 2 Z t 2 E g s; x.s/; x.k / dW.s/ D Eg s; x.s/; x.k / ds: t0 t0
The following definitions and assumption will be needed throughout this paper.Definition 1 For any ˛; ˇ 2 R, an Rn -valued stochastic process x W .˛; ˇ/ ! R issaid to be a solution of (2) (or (3)) if the following hold:1. x.t/ is continuous and F t -adapted for all t 2 .˛; ˇ/;2. f .t; x.t/; x.k // 2 L ad .˝; L1 .˛; ˇ// and g.t; x.t/; x.k // 2 L ad .˝; L2 .˛; ˇ//;3. the stochastic integral equation (4) holds (a.s.). 370 M.S. Alwan and X. Liu
Itô Lemma. For any t0 2 RC and t t0 , let x.t/ be an Rn -dimensional Itôstochastic process, i.e., it is F t -adapted and satisfying, Z t Z t x.t/ D x.t0 / C f .s; x.s//ds C .s; x.s//dW.s/; (a.s.): t0 t0
Suppose that V 2 C 1;2 .RC Rn I R/. Then, for any t t0 , V.t; x.t// is an Itôstochastic process satisfying Z t Z tV.t; x.t// D V.t0 ; x.t0 // C L V.s; x.s//f .s; x.s//ds C Vx .s; x.s// .s; x.s//dW.s/; t0 t0
(a.s.), where
1L V.t; x.t// D Vt .t; x.t// C Vx .t; x.t//f .t; x.t// C trΠT .t; x.t//Vxx .t; x.t//.t; x.t// 2
is the infinitesimal operator acting on the process V.t; x.t// with Vt .t; x.t//,Vx .t; x.t//, and Vxx .t; x.t// being the partial differentials of the process V.t; x.t//with respect to t, x, and twice with respect to x, respectively. Having defined SEPCA, we consider the following uncertain control SEPCA dx.t/ D .A C A/x.t/ C Bu.t/ C Gw.t/ C f .x / dt C g.x / dW.t/; (5a)
z.t/ D Cx.t/ C Fu.t/; (5b)
x.t0 / D x0 ; (5c)
where x 2 Rn is the system state, u 2 Rp is the control input of the form Kx,where K 2 Rpn is a control gain matrix, R 1w 2 R 2is a disturbance ninput, whichnm q is 2assumed to be in L2 Œ0; 1/ (i.e., kwk2 D 0 kw.t/k dt < 1), f 2 R and g 2 Rrepresent lumped uncertainties, z 2 Rr is the controlled measured output, A, B,G, C, and F are constant matrices that describe the nominal system, and A./ isreal-valued matrix, which is piecewise continuous function representing parameteruncertainties. We also assume that f .0/ D 0 2 Rn and g.0/ D 0 2 Rnm to ensurethat the system admits a trivial solution. A symmetric matrix P is said to be positivedefinite if the scalar xT Px > 0 for all nonzero x 2 Rn and xT Px D 0 for x D 0.Definition 2 A function a 2 C .RC I RC / is said to belong to class K if a.0/ D 0and it is strictly increasing [12]; it said to belong to class K 1 if a 2 K and it isconvex; it said to belong to class K 2 if a 2 K and it is concave.Definition 3 For any t0 2 RC , t t0 , and x0 2 Rn , let x.t/ D x.t; t0 ; x0 / be asolution of uncertain SEPCA (5). Then, the system is said to be robustly globallyinput-to-state stable (ISS) in the m.s. if there exist functions ˇ 2 KL and 2 K ISS and H1 for Stochastic Control EPCA 371
such that, for any input disturbance w, the solution satisfies EŒkx.t/k2 ˇ EŒkx0 k2 ; t t0 C .kw.t/k/; 8 t t0 ;
for any solution x.t/ D x.t; t0 ; x0 / of (5), and any x0 2 Rn with EŒkx0 k2 < 1.Particularly, if ˇ.s; t/ D set , for some positive , then the system is said to beinput-to-state exponentially stable in the m.s.Definition 4 Given a constant N > 0, uncertain SEPCA (5) is said to be input-to-state stabilizable with an H1 -norm bound N if there exists a state feedback law
u.t/ D Kx.t/;
where K D 12 "BT P for some a constant " > 0 and positive-definite matrix P, suchthat, for any admissible parameter uncertainty A./, the corresponding closed-loop system is uniformly asymptotically (or exponentially) ISS in the (m.s.) andthe controlled output z satisfies hZ 1 i kzk2E WD E kz.t/k2 dt N 2 kwk22 C m0 ; 0
for some positive constant m0 .Assumption A For all t 2 RC , the admissible parameter uncertainty of the systemis defined by A D DU .t/H, where D and H are known real constant matriceswith appropriate dimensions that give the structure of the uncertainty, and U is anunknown real time-varying matrix containing the uncertain parameters in the linearpart and satisfying kU .t/k 1.
3 Main Results
Theorem 1 For all t 2 Œtk ; tkC1 /, let the controller gain K and disturbance levelN > 0 be given. Assume that Assumption A holds, and there exist positive constants"1 ; "2 , and a positive-definite matrix P such that the following inequality T 1 A C BK P C P A C BK C P "1 DDT C I C N 2 GGT P C qN P "2 1 T C H H C "2 qN kUk2 I C CcT Cc C ˛P < 0 (6) "1
holds, where qN > 1, ˛ > 0, > 0 is such that
trŒgT .y/Pg.y/ 2 qN xT Px; (a.s.); (7) 372 M.S. Alwan and X. Liu
U is an n n matrix such that k f .y/k2 qN kUk2 kxk2 , and Cc D C C FK. Supposefurther that for any k 2 N, the following dwell-time-type condition
1 adk tkC1 tk ln (8) ˛N bdkC1
holds, where ˛N > 0 and dk is a constant satisfying dkC1 < dk < 1 and limk!1 dk D0. Then, system (5) is m.s. robustly globally exponentially input-to-state stabilizedby the feedback control u D Kx with an H1 -norm bound N > 0.Proof For all t 2 Œtk ; tkC1 /, let x.t/ D x.t; t0 ; x0 / be the solution of (5) and defineV.x/ D xT Px as Lyapunov function candidate. Then,
L V.x/ xT ŒAT P C PA C 2K T BT P x C 2xT P. A/x C 2xT Pf .x / C qN xT Px C 2xT PGw; (a.s.);
where qN is provided by using the Razumikhin technique. 1 Claim There exits 2 > 0 such that 2xT Pf .x / xT "2 qN kUk2 I C P2 x. "2Proof of the Claim Since p 1 p 1 T 0 "2 f T .x / p xT P "2 f T .x / p xT P "2 "2 1 T 2 D "2 f T .x /f .x / C x P x 2xT Pf .x / "2 1 T 2 "2 kUk2 kx k2 C x P x 2xT Pf .x / "2
which implies the desired result. Where qN D q=min .P/ > 1, with q > 1. By the facts that 2xT P. A/x xT "1 PDDT P C "11 H T H x, we get
1 L V.x/ xT .A C BK/T P C P.A C BK/ x C xT "1 PDDT P C H T H x "1 1 1 C xT "2 qN kUk2 I C P2 x C qN xT Px C "3 xT PGGT Px C wT w; (a.s.): "2 "3
Making use of inequality (6) with "3 D N 2 , we get, for all t 2 Œtk ; tkC1 /,
1 T L V.x/ .˛ /xT Px xT Px C w w "3 D ˛V.x/; N ˛N D ˛ ; (a.s.); ISS and H1 for Stochastic Control EPCA 373
qprovided that V.x/ 1"3 kwk2 or kxk a1"3 kwk, for some positive constant <˛. Applying the Itô formula to process V.x/ and taking the mathematical expectationgive DC m.t/ ˛m.t/, N which implies that s ˛.tt N k/ 1 m.t/ m.tk /e ; whenever kxk kwk; a"3
where m.t/ D EŒV.x.t// ; 8t 2 Œtk ; tkC1 /. By the dwell-time condition (8), we get
EŒkx.tkC1 /k2 dkC1 EŒkx.tk /k2 ;
where we have used the fact bkxk2 xT Px akxk2 . Since limk!1 q dk D 0, thelimit of x.tkC1 / will eventually converge to the limit set of radius a1"3 kwk in them.s.; that is the solution x is robustly globally input-to-state stabilized in the m.s. To prove the upper bound on the output magnitude kzk, we introduce theperformance function Z 1 JDE .zT z N 2 wT w/dt: t0
It follows that Z Z 1 1JDE zT z N 2 wT w dt C E dV.x/dt EŒV.x.1// C EŒV.x0 / t0 t0 Z h Z 1 T 1 1 E zT z N 2 wT w dt C E xT A C BK P C P A C BK C P "1 DDT C I P t0 t0 "2 1 i C qN "2 kUk2 I C H T H C qN P N 2 PGGT P C N 2 PGGT P x C 2xT PGw dt C EŒV.x0 / "1 Z 1
T 1 DE xT A C BK P C P A C BK C P "1 DDT C I C N 2 GGT P t0 "2 1 C qN "2 kUk2 I C H T H C qN P C CcT Cc x dt "1 Z 1 T E w N 2 GT Px N 2 w N 2 GT Px dt C EŒV.x0 / : t0
By the theorem assumption in (6) and strict negativeness of the second term, we get
kzk2E N 2 kwk22 C m0 ; where m0 D EŒV.x0 / t u
Remarks1. The conditions on f and g mean that the system perturbations are bounded by linear growth bounds. 374 M.S. Alwan and X. Liu
2. The dwell-time-type condition in (8) is made to generate a sequence of solution trajectories evaluated at the switching times that is convergent in the m.s. to a limit set with a radius depending on the system disturbance.3. If the controlled measured output is available at the time instances , that is to say z D Cc x , where Cc is as defined before, then assumption (6) becomes T 1 A C BK P C P A C BK C P "1 DDT C I C N 2 GGT P C qN P "2 1 T C H H C "2 qN kUk2 I C qN CcT Cc C ˛P < 0; "1 Z 1 where in this case J D E .zT z N 2 wT w/dt. t0
Corollary In Theorem 1, if w.t/ 0 for all t t0 , and the inequality T 1 A C BK P C P A C BK C P "1 DDT C I P "2 1 T C qN P C H H C "2 qN kUk2 I C CcT Cc C ˛P < 0 "1
holds, then x 0 is robustly globally exponentially stable in the m.s.Example Consider uncertain control SEPCA (5), where " # " # " # " # 7 0 0:1 0:4 01 1 AD ; BD ; GD ; w.t/ D sin.t/ ; C D I22 ; 0 0:1 0 9 10 1
" # " # " # 0:1 0:1 1 h i 1 x1 FD ; DD ; HD 01 ; U .t/ D sin.t/; f .x / D ; 2 0 0 2 x2
" # x1 0g.x / D ; "1 D 1; "2 D 0:1; N D 0:1; ˛ D 2; qN D 2; and D 0:6386: 0 x2
By the theorem assumptions, we have, after some algebraic manipulations, U D0:5I22 , and from the algebraic Riccati-like inequality (6), we have " # 0:0593 0:0125PD ; with b D min .P/ D 0:0579 and a D max .P/ D 0:1762; 0:0125 0:1748 ISS and H1 for Stochastic Control EPCA 375
2.5
2 E||x||2 & β(||w||)
1.5
0.5
0 0 2 4 6 8 10 12 14 16 t
Fig. 1 Input-to-state stability in the m.s.
2.5
1.5 E[||(x,y)T||2]
0.5
0 0 1 2 3 4 5 6 7 t
Fig. 2 Exponential stability in the m.s. (w.t/ 0)
0:0009 0:0002and the control gain matrix is K D . The simulation results of 0:0204 0:2308EŒkx.t/k2 (above) and ˇ.kw.t/k/ D a1"3 sin2 .t/ (below) is shown in Fig. 1, where D 1:3, "3 D N 2 D 1=0:01, and, by the dwell-time condition in (8), we havechosen tkC1 tk D 1, where dk D 1=2k , and k D tk , for all k D 0; 1; . Ifw.t/ 0, x.t/ 0 is exponentially stable in the m.s., as shown in Fig. 2. 376 M.S. Alwan and X. Liu
4 Conclusions
In this paper, we have considered a control system with uncertain SEPCA, whichhas been viewed as a switched system. The focus was on establishing input-to-statestabilization and H1 performance, where the method of Lyapunov function andRazumikhin techniques have been used to write some sufficient conditions.
Acknowledgements This research was financially supported by Natural Sciences and Engineer-ing Research Council of Canada (NSERC).
References
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21. Ozturk, I., Bozkurt, F.: Stability analysis of a population model with piecewise constant arguments. Nonlinear Anal.: Real World Appl. 12, 1532–1545 (2011)19. Sontag, E.D.: Comments on integral variants of ISS. Syst. Control Lett. 34, 93–100 (1998)16. Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34(4), 435–443 (1989)18. Sontag, E.D., Wang, Y.: New characterization of input-to-state stability property. IEEE Trans. Autom. Control 34(41) 1283–1294 (1996)17. Sontag, E.D., Wang, Y.: On characterization of input-to-state stability property. Syst. Control Lett. 24, 351–359 (1995)20. Teel, A.R., Moreau, L., Nešic, D.: A unified framework for input-to-state stability in systems with two time scales. IEEE Trans. Autom. Control 48(9), 1526–1544 (2003)22. Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)23. Wu, Z.-G., Park, J.H., Su, H., Chu, J.: Stochastic stability analysis of piecewise hom*ogeneous Markovian jump neural networks with mixed time-delays. JFI 349, 2136–2150 (2012) Switched Singularly Perturbed Systemswith Reliable Controllers
Mohamad S. Alwan, Xinzhi Liu, and Taghreed G. Sugati
Abstract This paper addresses the problem of exponential stability for a classof switched control singularly perturbed systems (SCSPS) not only when all thecontrol actuators are operational, but also when some of them experience failures.Multiple Lyapunov functions and average dwell-time switching signal approachare used to establish the stability criteria for the proposed systems. In this paper,we assume that a full access to all the system modes is available, though themode-dependent, slow-state feedback controllers experience faulty actuators of anoutage type. In the stability analysis, the system under study is viewed as aninterconnected system that has been decomposed into isolated, lower order, slow andfast subsystems, and the interconnection between them. It has been observed that ifthe degree of stability of each isolated mode is greater than the interconnectionbetween them, the interconnected mode is exponentially stable, and, then, the fullorder SCSPS is also exponentially stable for all admissible switching signals withaverage dwell-time. A numerical example with simulations is introduced to illustratethe validity of the proposed theoretical results.
1 Introduction
A switched system is a special class of hybrid systems that consists of a familyof continuous- or discrete-time dynamical subsystems, and a switching rule thatcontrols the switchings amongst them. In reality, many applications in differentfields such as aircraft, automotive industry, robotics, control systems, biological,epidemic disease models, etc., are operating based on switchings between multipledynamical modes. The stability of such systems has received great attention in thelast few decades. For more readings, one may refer to [3, 9, 10, 14, 17] and thereferences therein. It is well known that even if the individual modes are stable, the switched systemmay not be stable [9]. However, it has been shown that the stability of the switchedsystem can be achieved if the dwell-time, the time between any two consecutive
M.S. Alwan () • X. Liu • T.G. SugatiDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 379J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_35 380 M.S. Alwan et al.
switchings, is sufficiently large [12]. Practically, in some situations, such as agingsystems or systems with finite escape time, the dwell-time condition may not hold,and yet one can get the same stability result if the average dwell-time is satisfied [5]. The reliable control is the controller that tolerates actuator and/or sensor failures.The failure of control components is frequently encountered in practice, yet theimmediate repair may be impossible in some critical cases. Therefore, designinga reliable controller to guarantee stability is necessary. This problem has drawnresearcher attention for many years; see for instance [4, 13, 15, 16]. Systems involving multiple time-scale dynamics (known as singularly perturbedsystems) arise in a large class of applications in science and engineering suchas celestial mechanics, many-particle dynamics, and climate systems, etc. Theyare characterized by small parameters multiplied by the highest derivatives. Thestability problem of these systems has attracted many researchers; see [1, 2, 6–8, 11]and some references therein. In this paper, we address SCSPS where the controllers are subject to faultyactuators. The continuous states are viewed as an interconnected system with two-time scale (slow and fast) subsystems. Moreover, due to dominant behaviour ofthe reduced systems, the stabilization of the full order systems is achieved throughthe controller of the slow reduced order subsystem. This in turn results in lesseningsome unnecessary sufficient conditions imposed on the fast subsystem. The stabilityanalysis is obtained by multiple Lyapunov functions method after decomposing thesystem into isolated, lower order, slow and fast subsystems, and the interconnectionbetween them. It has been observed that if the degree of stability of each isolated mode is greaterthan the interconnection between them, the underlying interconnected mode of theswitched system is exponentially stable. Moreover, if switching among the systemmodes follows the average dwell-time rule, then the SCSPS is also exponentiallystable. The relationship between the stability degrees of the isolated subsystemsand the interconnection strength is usually formulated by the so-called M-matrix.Finally, a numerical example and simulations are provided to justify the proposedtheoretical results. The rest of this paper is organized as follows. Section 2 involves the problemdescription and some definitions. The main results and proofs are stated in Sect. 3.A numerical example with simulations is presented in Sect. 4. The conclusion isgiven in Sect. 5.
2 Problem Formulation and Preliminaries
Throughout this paper, Rn denotes the n-dimensional Euclidean space; RC refersto the nonnegative real numbers; Rnm is the class of all n m real matrices. Asymmetric matrix P is said to be positive definite if all its eigenvalues are positive.Moreover, If P 2 Rnn , denote by max .P/(min .P/) the maximum (minimum)eigenvalue of P. If V.x/ D xT Px, the inequalities min .P/jjxjj2 V.x/ Switched Singularly Perturbed Systems with Reliable Controllers 381
max .P/jjxjj2 are true. If x 2 Rn , then jjxjj refers to the Euclidean vector normof x. Consider the following system
xP D A11.t/ x C A12.t/ z C B1.t/ u; (1a) ".t/ zP D A21.t/ x C A22.t/ z C B2.t/ u; (1b) x.t0 / D x0 ; z.t0 / D z0 ; (1c)
where x 2 Rm ; z 2 Rn are the system slow and fast states respectively, u 2 Rl is thecontrol input of the form u D Kx for some control gain K 2 Rlm , W Œt0 ; 1/ !S D f1; 2; ; Ng is a piecewise constant function known as the switching signal(or law). For each i 2 S ; A11i 2 Rmm ; A12i 2 Rmn ; A21i 2 Rnm ; A22i 2Rnn ; are known real constant matrices with A22i is a nonsingular Hurwitz matrix,B1i 2 Rml ; B2i 2 Rnl ; and 0 < "i 1. Setting "i D 0 implies that z D hi .x/ DA122i ŒA21i x C B2i u : Plug z into (1a) gives the slow reduced subsystem xPs D A0i xs CB0i u where A0i D A11i A12i A1 1 22i A21i , and B0i D B1i A12i A22i B2i . Choose u D Kxssuch that .A0i ; B0i / is stabilizable. For simplicity of notation, we use x instead of xs to refer to the slow reducedsystem.Definition 1 The trivial solution of system (1) is said to be globally exponentiallystable (g.e.s.) if there exist positive constants L and such that
jjx.t/jj C jjz.t/jj L.jjx.t0 /jj C jjz.t0 /jj/e.tt0 / ; t t0 2 RC ;
for all x.t/ and z.t/, the solutions of system (1), and any x0 2 Rm ; z0 2 Rn .Definition 2 An n n matrix M D Œmij with mij 0, for all i 6D j, is said to be anM-matrix if all its leading successive principle minors are positive, i.e., 2 3 m11 m12 m1k 6m21 m22 m2k 7 det 6 4 7 > 0; k D 1; 2; ; n: 5 mk1 mk2 mkk
Average dwell-time Condition (ADTC) [5]. The number of switchings N.t0 ; t/ inthe interval .t0 ; t/ for a finite t satisfies N.t0 ; t/ N0 C tt a ; where N0 is the chatter 0
bound, and a is the average dwell-time.
3 The Main Results
In this section, we present our main results. 382 M.S. Alwan et al.
3.1 Normal Case
For any i 2 S , the closed-loop system becomes 8 < xP D .A11i C B1i Ki /x C A12i z; " zP D .A21i C B2i Ki /x C A22i z; (2) : i x.t0 / D x0 ; z.t0 / D z0 :
Theorem 1 The trivial solution of system (1) is g.e.s. if ADTC holds, and thefollowing assumptions hold (i) ReŒ.A22i / < 0, and .A0i ; B0i / is stabilizable;(ii) there exist positive constants aji ; j D 1; ; 6 such that
2xT P1i A12i hi .x/ a1i xT x; (3) 2xT P1i A12i .z hi .x// a2i xT x C a3i .z hi .x//T .z hi .x//; (4) 2.z hi .x//T P2i R 1i x a4i xT x C a5i .z hi .x//T .z hi .x//; (5) .z hi .x//T R 2i .z hi .x// a6i .z hi .x//T .z hi .x// (6)
where hi .x/ D A1 22i .A21i C B2i Ki /x, P2i is the solution of the Lyapunov equa- tion AT22i P2i CP2i A22i D Iin ; where Iin is an identity matrix, R 1i D A1 22i .A21i C B2i Ki /ŒA11i C B1i Ki A12i A122i .A 21i C B K 2i i / ; and R 2i D 2P A 1 2i 22i .A21i C B2i Ki /A12i ;(iii) there exist a positive constant "i such that AN i is an M-matrix where " max .Ni / a3i # max .P1i / min .P2i / AN i D a4i a6i .1a5i "i / ; min .P1i / min .P2i / "i max .P2i /
where Ni D Qi C .a1i C a2i /I C M T Pi C Pi M T such that M D A12i A1 22i .A21i C B2i Ki / and .A0i C B0i Ki /T P1i C P1i .A0i C B0i Ki / D Qi for a given Ki .Proof Let Vi .x/ D xT P1i x and Wi ..z hi .x//.t// D .z hi .x//T P2i .z hi .x// beLyapunov function candidates for the slow and the fast subsystem, respectively.Then,
VP i .x/ xT .Qi C a1i I C a2i I/x C a3i .z hi .x//T .z hi .x// max .Ni / a3i Vi .x/ C Wi ..z hi .x//.t//; (7) max .P1i / min .P2i / Switched Singularly Perturbed Systems with Reliable Controllers 383
where Ni D Qi C.a1i Ca2i /I CM T Pi CPi M T such that M D A12i A1 22i .A21i CB2i Ki /is negative definite. We also have
1WP i ..z hi .x//.t// D .z hi .x//T .z hi .x// 2.z hi .x//T P2i hP i .x/ "i 1 .a5i /.z hi .x//T .z hi .x// C .z hi .x//T R 2i .z hi .x// "i C a4i xT x a4i h a .1 a5i "i / i 6i Vi .x/ C Wi ..z hi .x//.t//: min .P1i / min .P2i / "i max .P2i / (8)
where R 2i D 2P2i A1 22i .A21i C B2i Ki /A12i . Combining (7) and (8), we get
" max .Ni / a3i # P i .x/ V max .P1i / min .P2i / Vi .x/ : P i .z hi .x//.t/ W a4i a6i .1a5i "i / Wi .z hi .x//.t/ min .P1 / i min .P2i / "i max .P2i /
Then, we have " max .Ni / a3i # max .P1i / min .P2i / AN i D : a4i a6i min .P1i / min .P2i / ".1a 5i "i / i max .P2i /
Then there exists i D max .AN i / > 0 such that for t 2 Œtk1 ; tk /; Vi .x/ Vi .x.tk1 // C Wi .z hi .x//.tk1 / ei .ttk1 / ;
and Wi .z hi .x//.t/ Vi .x.tk1 // C Wi .z hi .x//.tk1 / ei .ttk1 / ;
For any i; j 2 S ; M > 1, we have
Vj .x.t// Vi .x.t//; Wj .z hj .x//.t/ Wi .z hi .x//.t/ :
If the system switches among its modes, one may get for all t t0 ,
Vi .x.t// 2e1 .t1 t0 / 2e2 .t2 t1 / 2ek1 .tk1 tk2 / h i V1 .x.t0 // C W1 .z h1 .x//.t0 / ek .ttk1 / : 384 M.S. Alwan et al.
Let D minfj W j D 1; 2; ; kg: Then h i Vi .x.t// .2/k1 V1 .x.t0 // C W1 .z h1 .x//.t0 / e.tt0 / h i V1 .x.t0 // C W1 .z h1 .x//.t0 / e.k1/ ln .tt0 / ;
where D 2. Applying the ADTC with N0 D ln , is an arbitrary constant, ln a D . / with > leads to h i Vi .x.t// V1 .x.t0 // C W1 .z h1 .x//.t0 / e.tt0 /
and h i Wi .z hi .x//.t/ V1 .x.t0 // C W1 .z h1 .x//.t0 / e.tt0 /
This implies that there exists L > 0 such that .tt /=2 jjx.t/jj C jjz.t/jj L.jjx.t0 /jj C jjz.t0 /jj/e 0 : t u
3.2 Faulty Case
To analyze the reliable stabilization with respect to actuator failures, for any i 2 S ,consider the decomposition of the control matrix Bi D Bi˙ C Bi˙N ; where ˙ theset of actuators that are susceptible to failure, and ˙N f1; 2; : : : ; lg ˙ theset of actuators which are robust to failures and essential to stabilize the givensystem, moreover, the matrices Bi˙ ; Bi˙N are the control matrices associated with˙; ˙N respectively, and are generated by zeroing out the columns correspondingto ˙N and ˙, respectively. The pair .Ai ; Bi˙N / is assumed to be stabilizable. For afixed i, let ˙ corresponds to some of the actuators that experience failure,and assume that the output of faulty actuators is zero. Then, the decompositionbecomes Bi D Bi CBiN ; where Bi and BiN have the same definition of Bi˙ and Bi˙N ,respectively. Since the control input u is applied to the system through the normalactuators, the closed-loop system becomes
xP D .A11i C B1N i KiN /x C A12i z; (9a) "i zP D .A21i C B2N i KiN /x C A22i z; (9b) x.t0 / D x0 ; z.t0 / D z0 : (9c)
where KiN D 12 ˇi BT0iN PiN , with B0iN D B1N i A12i A1 22i B2N i , and PiN is a positivedefinite matrix such that .A0i C B0iN KiN / PiN C PiN .A0i C B0iN KiN / D I. Setting T Switched Singularly Perturbed Systems with Reliable Controllers 385
"i D 0, one may get z D hiN .x/ D A1 22i .A21i C B2N i KiN /x. In the following theorem,we assume that N i D ˙i . N
Theorem 2 The trivial solution of system (9) is g.e.s. if ADTC and the followingassumptions hold for any i 2 S (i) ReŒ.A22i / < 0, and AT11i P1i C P1i A11i C ˇi P1i A12i A1 T 22i B2˙N i B1˙N i B1˙N i B1˙N P1i C ˛i I D 0; T i (ii) there exist positive constants aji ; j D 1; ; 6 such that
2xT P1i A12i hi˙N .x/ a1i xT x; (10) 2x P1i A12i .z hi˙N .x// a2i x x C a3i .z hi˙N .x// .z hi˙N .x//; T T T (11) 2.z hi˙N .x// P2i R 1i˙N x a4i x x C a5i .z hi˙N .x// .z hi˙N .x//; T T T (12) .z hi˙N .x//T R 2˙N i .z hi˙N .x// a6i .z hi˙N .x//T .z hi˙N .x//; (13)
where hi˙N .x/ D A1 22i .A21i C B2˙N i Ki˙N /x, P2i is the solution of A22i P2i C T 1 1 P2i A22i D Iin , R 1i Ṅ D A22i .A21i C B2˙N i Ki˙N /ŒA11i C B1i Ki A12i A22i .A21i C B2˙N i Ki˙N / where Ki˙N D 12 ˇi BT0i˙N Pi˙N , and R 2˙N i D 2P2i A1 22i .A21i C 1 1 T 1 ˇ B B .A12i A22i / P1i /A12i 2 ˇi B2˙N i B1˙N P1i ; T 2 i 2˙N i 2˙N i T i(iii) there exist a positive constant "i such that AN i˙N is an M-matrix where " max .N # i Ṅ / a3i max .P1i / min .P2i / AN i˙N D a4i "i .a5i Ca6i /1 : min .P1i / "i max .P2i /
Proof Let Vi .x/ D xT P1i x and Wi .z h˙N i .x//.t/ D .z h˙N i .x//T P2i .z h˙N i .x//be Lyapunov function candidates. Then, we have
VP i .x/ xT .˛i C a1i C a2i /Ix C a3i .z hi˙N .x//T .z hi˙N .x// ˛i C a1i C a2i a3i Vi .x/ C Wi .z h˙N i .x//.t/ (14) max .P1i / min .P2i /
We also have 1 WP i .z h˙N i .x//.t/ D .z hi˙N .x//T .z hi˙N .x// 2.z hi˙N .x//T P2i hP i˙N .x/ "i 1 .a5i /.z hi˙N .x//T .z hi˙N .x// C a4i xT x "i C .z hi˙N .x//T R 2˙N i .z hi˙N .x// a4i h " .a C a / 1 i i 5i 6i Vi .x/ C Wi ..z hi˙N .x//.t//; min .P1i / "i max .P2i / (15) 386 M.S. Alwan et al.
1 1where R 2˙N i D 2P2i A1 1 T 22i .A21i 2 ˇi B2˙N i B1˙N i P1i C 2 ˇi B2˙N i B2˙N i .A12i A22i / P1i /A12i . T T
Combining (14) and (15), we get the M-matrix AN i˙N with " max .N # i Ṅ / a3i max .P1i / min .P2i / AN i˙N D a4i "i .a5i Ca6i /1 : min .P1i / "i max .P2i /
Proceeding as done in the proof of Theorem 1, we get the desired result. t u
4 Numerical Example
Example 1 Consider system (1) with S D f1; 2g; 5 0 0:1 2 13 1 2 A111 D ; A121 D ; A211 D ; A221 D ; 0 10 0:1 0 21 3 2 3 1 1 0 23 2 1 A112 D ; A122 D ; A212 D ; A222 D ; 0 6 0:1 0:3 11 1 1 5 0:5 3 1 4 5 2 2 B11 D ; B21 D ; B12 D ; B22 D ; 0:1 0:15 1 4 0:5 1 1 3
"1 D 0:01; ˇ1 D 0:5; a11 D 0:1; a21 D 0:15; a31 D 0:02; a41 D 0:01; a51 D70; Q1 D 4I; "2 D 0:02; ˇ2 D 0:25; a12 D 0:3; a22 D 0:2; a32 D 0:2; a42 D 0:02; a52 D30; and Q2 D I: 0:1025 0:0274 Case 1. When all actuators are operational, we have P11 D ; 0:0274 0:0615 0:1697 0:0616 1:5 1 0:5 0:5 P12 D ; P21 D ; P21 D ; and K1 D 0:0616 0:1322 1 1:75 0:5 1 0:0217 0:0031 0:1638 0:0869 ; K2 D : Thus, the matrices A0i C B0i Ki 0:0840 0:0238 0:1449 0:0842 (i D 1; 2) are Hurwitz and a D ˛ln D 1:8330. Case 2. When there are failures in the first actuator of B1i , and the second actuator of B2i for both modes, i.e., 0 0:5 30 05 20 B1˙N 1 D ; B2˙N 1 D ; B1˙N 2 D ; B2˙N 2 D ; 0 0:15 10 01 10 Switched Singularly Perturbed Systems with Reliable Controllers 387
a4 b 3.5 3.5 3
3 2.5
2.5 2||x||&||z||
||x||&||z|| 2
1.5 1.5
1 1
0.5 0.5
0 0 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 9 t t
Fig. 1 Singularly perturbed switched system. (a) Operational actuators. (b) Faulty actuators
0:0993 0:0257 0:2828 0:1570 we have P11 D ; P12 D ; P21 and P22 0:0257 0:0606 0:1570 0:2145 0:1024 0:0278 are the same as for the normal case, and K1 D ; K2 D 0:0134 0:0055 0:1355 0:0991 : Thus, the matrices A0i C B0i Ki (i D 1; 2) are Hurwitz and 0:1964 0:1249 a D ˛ln D 4:1498.Figure 1a,b show the simulation results of jjxjj (top) and jjzjj (bottom) for the normaland the faulty cases respectively.
5 Conclusion
This paper has established new sufficient conditions that guaranteed the globalexponential stability of SCSPS. The output of the faulty actuators has been treatedas an outage. So that, as a future work, one may consider nonzero output whichcan be viewed as an external disturbance to the system. We have shown that, usingADTC with multiple Lyapunov functions, the full order switched system has beenexponentially stabilized by u D Ki x where in the faulty case, KiN D 12 ˇi BT0iN PiN .A numerical example has been introduced to clarify the proposed results.
Acknowledgements This work was partially supported by NSERC Canada. The third authoracknowledges the sponsorship of King Abdulaziz University, Saudi Arabia. 388 M.S. Alwan et al.
References
1. Alwan, M.S., Liu, X.Z.: Stability of singularly perturbed switched systems with time delay and impulsive effects. Nonlinear Anal. 17, 4297–4308 (2009) 2. Alwan, M.S., Liu, X.Z., Ingalls, B.: Exponential stability of singularly perturbed switched systems with time delay. Nonlinear Anal.: Hybrid Syst. 2(3), 913–921 (2008) 3. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998) 4. Cheng, X.M, Gui, W.H., Gan, Z.J.: Robust reliable control for a class of time-varying uncertain impulsive systems. J. Cent. S. Univ. Technol. 12(1), 199–202 (2005) 5. Hespanha, J.P., Morse, A.S.: Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 3, Phoenix, pp. 2655– 2660 (1999) 6. Kang, K.-I., Park, K.-S., Lim, J.-T.: Exponential stability of singularly perturbed systems with time delay and uncertainties. Int. J. Syst. Sci. 46(1), 170–178 (2015) 7. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2002) 8. Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic, London (1986) 9. Liberzon, D.: Switching in Systems and Control. Birkhäuser, Boston (2003)10. Liberzon, D., Morse, A.S.: Basic Problems is Stability and Design of Switched Systems. IEEE Control Syst. Mag. 19(5), 59–70 (1999)11. Liu, X., Shen, X., Zhang, Y.: Exponential stability of singularly perturbed systems with time delay. Appl. Anal. 82(2), 117–130 (2003)12. Morse, S.: Supervisory control of families of linear set-point controllers, part 1: exact matching. IEEE Trans. Autom. Control 41(10), 1413–1431 (1996)13. Seo, C.J., Kim, B.K.: Robust and reliable H1 control for linear systems with parameter uncertainty and actuator failure. Automatica 32(3), 465–467 (1996)14. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49(4), 545–592 (2007)15. Veillette, R.J.: Reliable state feedback and reliable observers. In: Proceedings of the 31st Conference on Decision and Control, Tucson, pp. 2898–2903 (1992)16. Veillette, R.J., Medanic, J.V., Perkins, W.R.: Design of reliable control systems. IEEE Trans. Autom. Control 37(3), 290–304 (1992)17. Zhai, G., Hu, B., Yasuda, K., Michel, A.N.: Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Proc. Am. Control Conf. 1(6), 200– 204 (2000) Application of an Optimized SLW Model in CFDSimulation of a Furnace
Masoud Darbandi, Bagher Abrar, and Gerry E. Schneider
Abstract Radiation is the most important part of heat transfer in combustionprocesses in applications such as furnaces. A proper radiation model for CFDsimulations is the one, which provides the required accuracy with minimumcomputational cost. In this work, we focus on radiation modeling in CFD simulationof a laboratory scaled furnace. We use an optimized spectral line-based weighted-sum-of-gray-gases (SLW) model, which only needs four radiation transfer equations(RTE) solution for accurate prediction of radiative heat transfer in non-gray combus-tion fields. This is while the classic non-optimized SLW model needs at least 10–20RTEs solution for the same case. We apply both the optimized and non-optimizedSLW model to CFD simulation of the furnace. To evaluate the achieved results,we compare them with available measured data. We further compare the results ofoptimized SLW model with those of non-optimized SLW model. The comparisonsdemonstrate that the optimized SLW model provides the same accuracy of non-optimized SLW mode, while it requires less than 80% computational time.
1 Introduction
The fully coupled computational fluid dynamic (CFD) simulation is widely used fordesign and optimization proposes in different industrial furnaces, boilers, and fire-heaters. Radiation is the most important part of heat transfer in such combustiondevices. The process of turbulent combustion and heat transfer strongly depend oneach other and the small influence of thermal radiation may be magnified by thenon-linear processes of turbulent combustion. Moreover, radiation can affect flametemperature and species concentration calculations. The inclusion of radiative heattransfer reduces the size of flame region, where the maximum temperatures occur.
M. Darbandi () • B. AbrarDepartment of Aerospace Engineering, Center of Excellence in Aerospace Systems,Sharif University of Technology, Tehran, P.O. Box 11365-8639, Irane-mail: [emailprotected]. SchneiderDepartment of Mechanical and Mechatronics Engineering, University of Waterloo,Waterloo, ON, N2L 3GI, Canada
© Springer International Publishing Switzerland 2016 389J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_36 390 M. Darbandi et al.
Therefore, the effects of thermal radiation should take into account even in case ofnon-luminous flames [1–3]. This requires the solution of radiative transfer equation(RTE), which determines the variation of radiation intensity in spatial and angularspace. However, accurate solution of RTE in combustion fields is a very complicatedtask. It is mainly because of non-gray behavior of combustion gases. The absorptioncoefficient of non-gray gases, which appears in RTE, varies very rapidly andstrongly with the wave number. Cumber and Fairweather [4] recommended thatnon-gray radiation models should be used for accurate simulations of combustionprocesses. Among different non-gray radiation models, the spectral line-based weighted-sum-of-gray-gases (SLW) model has been the subject of many researches in the lasttwo decades [5]. The SLW is categorized in the modern class of global models. Theglobal models can be considered as improved versions of the weighted-sum-of-gray-gases (WSGG) model, which benefit form definition of absorption-line-black-body-distribution-function (ALBDF) to obtain the radiation parameters. Goutiere, et al.[6] evaluated different radiation models to calculate the radiative heat transfer in twodimensional enclosures. They showed that the SLW model can accurately predict theradiation heat transfer in their test cases. Similar investigations were also conductedby Coelho [7] in three-dimensional enclosures. He also confirmed the accuracy ofSLW model. Johansson, et al. [8] evaluated WSGG and SLW models in several 1Dtests mimic the conditions in oxy-fired boilers. Their results showed that the SLWmodel usually yields more accurate predictions than the WSGG model. Recently,Modest [9] reviewed historical development of nongray models. He proposed toinclude global models in simulations of complicated combustion systems. However,one should consider the computational cost of global models, such as the SLWmodel, before applying it in CFD simulations. The classic SLW model approximatesthe non-gray gases by sum-up of 10–20 gray gases. Therefore, it requires 10–20RTEs solution for these gray gases, which could be very time consuming in realscale problems. It is well known that reducing the number of SLW’s gray gases andtherefore reducing the computational cost of RTEs solution is not readily possible,because it could introduce some errors in the predicted results [5]. In this work, we conduct a CFD simulation for turbulent reacting flow in alaboratory scaled gas fired furnace. We apply an optimized SLW model for efficientcalculation of radiation heat transfer. The optimized SLW model approximatesnon-gray combustion gas mixture by sum-up of only 3 gray gases plus 1 cleargas. However, the accuracy of the optimized SLW model is preserved through theuse of an optimization procedure. This would make the optimized SLW modelcomputationally more efficient than the classic non-optimized SLW model. Application of an Optimized SLW Model in CFD Simulation of a Furnace 391
2 Governing Equations and Physical Models
2.1 Turbulent Flow Field and Energy Transport Equations
The mass and momentum governing equations for a turbulent flow can be expressedas
r .V/ D 0 (1)
r .VV/ D pI C r N (2)
in which, is the density, V is the velocity vector, p is the pressure, and N is thestress tensor. Here, we use the -" model along with the standard wall functions[10], which is commonly accepted for turbulence modeling in industrial applicationssuch as furnaces and boilers [11, 12]. The governing equation for turbulent energy transport is given by
keff r .VH/ D r . r H/ C SH (3) cp
in which, H is enthalpy. The viscous heating terms are neglected here. The effectivethermal conductivity is given by keff D k C .cp t /=Prt , with Prt D 0:85. k, cp ,t are thermal conductivity coefficient, specific heat at constant pressure, and fluidviscosity coefficient, respectively. SH also represents source term due to radiationheat transfer.
2.2 Combustion Model
Here, we use the mixture fraction theory for combustion modeling. The mixturefraction is defined as f D .Yi Yi;o /=.Yi;f Yi;o /, in which Yi is the mass fraction ofspecies i. The subscripts f and o also refer to fuel and oxidizer at inlet streams.In this theory, a transport equation is solved for each of mean mixture fractionand mixture fraction variance. One advantage of using mixture fraction theoryfor turbulent non-premixed combustion modeling is that, under the assumption ofchemical equilibrium, instantaneous values of species concentration, density, andtemperature can uniquely relate to mixture fraction via 'i D 'i . fi ; Hi /. However,only averaged values of these scalars, i.e. ', f , and H are predicted in turbulentflow modeling. The relationship between the averaged values and the instantaneousvalues can be achieved through the use of presumed ˇ-shaped probability densityfunction (PDF)[13]. Here, we perform the chemical equilibrium calculations by means of Gibbs’ freeenergy minimization. We assume there are 20 species and radicals in the equilibriummixture including CH4 , C2 H6 , C3 H8 , C4 H10 , CO2 , N2 , O2 , H2 O, CO, H2 , OH, O, 392 M. Darbandi et al.
O3 , HOCO, HCO, CHO, H2 O2 , HONO, HO2 , and H. More details on governingequations and combustion modeling in turbulent reacting flow can be found inRef. [14].
2.3 Radiation Model
Radiation effects are taken into account in the CFD simulation of combustionprocesses via addition of radiation source term in the right hand side of energytransport equation, i.e., Eq. (3). In the SLW model, this source term is given by
X J Z SH D j .4aj Ib Ij d˝/ (4) jD1 4
The calculation of above source term requires the solution of j RTE s as follows:
dIj D j .aj Ib Ij / (5) dsin which, I is the radiation intensity, j is the absorption coefficient, and aj isthe emissivity weighting factor. j is related to the absorption cross section asdescribed by j D NCj . To achieve Cj , the entire range of absorption crosssection between Cmax and Cmin is divided into J logarithmically spaced intervals.The interval borders named as supplemental absorption cross section q are obtained .j=J/using CQ j D Cmin .Cmax =Cmin / , where Cj is defined as Cj D CQ j CQ jC1 . Theemissivity weighting factor aj is also calculated by employing the ALBDF definitionas follows: Z 1 F.C ; Tg ; Tb ; Y/ D Ib .Tb ; /d (6) Ib .Tb / fWC .Tg ;Y/Cg
Using the above definition, aj is written as aj D F.CQ jC1 / F.CQ j /. To obtain thevalues of ALBDFs, here we use the tabulated values, which are presented by Pearsonand Webb [15] for H2 O, CO2 , and CO gases based on the recent HITEMP2010spectroscopic database [16].
2.4 The Optimized SLW Model
As outlined in the previous sections, the SLW model is involved with J C 1 RTEssolutions selections. About J D 10 20 gray gases are usually required to achievethe required accuracy in the SLW model. However, 10–20 RTEs solutions may needmuch computational efforts in real scale combustion application problems. This is Application of an Optimized SLW Model in CFD Simulation of a Furnace 393
a disadvantage, which makes the use of classic SLW model prohibitive in CFDsimulations. On the other hand, choosing a value of J bellow 10 may deteriorate theaccuracy of SLW model [5]. Here, we propose the use of optimized SLW modelto reduce the number of gray gases. Instead of logarithmic discretization in theclassic non-optimized SLW model, we benefit from the optimization procedure toobtain gray gas parameters aj and j in the optimized SLW model. The optimizationprocedure minimizes the difference between the calculated values of total emissivitybased on only three gray gases parameters and the true values of total emissivityover the path lengths Li . Mathematically, it is required to minimize the followingobjective function: X error D Œ"optimized SLW .Li / "true .Li / 2 (7) i
where the values of total emissivity, based on the three gray gases parameters, arecalculated from
X JD3 "optimized SLW .Li / D aj .1 ej Li / (8) jD1
and the true values of total emissivity "true can be obtained using the detail spectro-scopic data in the HITEMP2010 database and taking the following integration: Z 1 "true .Li / D Ib .1 e Li /d (9) Ib
The path lengths used in Eqs. (7), (8) and (9) should cover a range of typical lengthsconsistent with the problem in hand. Here, we use a typical range from 0.01 to 10times of the characteristic length of the problem including about 10 path lengths ineach decade. The minimization of the objective function in Eq. (7) is in fact a non-linear curve-fit problem, in which the values of aj and j parameters are obtained foreach of the three gray gases as of the results. We use the trust region algorithm forsolving this problem. In this way, we can accurately approximate the non-gray gasesmixture by sum-up of only 3 gray and 1 clear gases. In the optimization procedure,the true values of " are first calculated over a range of distances L typical of problemunder investigation. Then, the optimized parameters aj and j for each of 3 graygases are directly achieved using the least square curve fit to these " data.
3 Numerical Methods and Computational Procedure
We use the finite-volume method to treat the flow governing equations and toderive the sets of linear algebraic equations. In this method, the solution domain isdiscretized to a number of control volumes. The governing equations are integrated 394 M. Darbandi et al.
over the entire faces of each control volume. We also use the finite-volume methodto derive a conservative statement for the RTE. For this purpose, each octant of theangular space 4 is discretized into N N' non-overlapping control angles ˝l . Inthis work, we use the SIMPLE algorithm to solve the flow governing equations.Details of the employed finite volume method for the flow and RTE governingequations and the SIMPLE algorithm are given in our previous works [17, 18].
4 Problem Description
We consider a laboratory scale furnace with available measured data in this study[19]. The furnace is a circumferentially symmetric 300 kW BERL combustor, withproperly insulated octagonal cross section, conical hood and cylindrical exhaustduct. It is equipped with a vertically fired burner in the bottom. The burner has 24radial fuel injection holes and a bluff center body. Natural gas is radially introducedthrough these holes. Air is introduced through an annular inlet equipped with aproper swirler. Therefore, a non-premixed turbulent swirl stabilized flame forms inthe furnace. We model this problem as axisymmetric by appropriate adjustment withthe real 3D furnace. Figure 1 presents a schematic of the furnace and a close up ofburner’s head along with main dimensions. The natural gas is assumed to be composed of 96.5 % CH4 , 1.7 % C2 H6 , 0.1 %C3 H8 , 0.1 % C4 H10 , 0.3 % CO2 , and 1.3 % N2 . Fuel jet is considered to have amean radial velocity of v D 157:77 m/s at fuel inlet boundary. Air stream isalso considered to have a mean axial velocity of u D 31:35 m/s and a meanswirl velocity of w D 20:97 m/s. The fuel and air inlet temperatures are 312 and
Fig. 1 Geometry of the furnace and close-up of the burner, Sayre et al. [19] Application of an Optimized SLW Model in CFD Simulation of a Furnace 395
308 K, respectively. The temperature of the bottom wall, side wall, conical wall,and exhaust duct wall are set to be constant at 1100, 1220, 1305, and 1370 K,respectively. All walls are treated to be diffuse with an internal emissivity of 0.5.The gauge pressure is also set to be zero at the furnace outlet.
5 Results and Discussion
Here we solve the problem using both the optimized and non-optimized SLWmodels. In the non-optimized SLW model, j D 20 gray gases are used. In thissection, we first provide the results, which are achieved using the optimized SLWmodel and compare them with the available measured data of furnace. Then wefurther compare the achieved radiative quantities of the optimized SLW model withthose of the non-optimized SLW model. We obtain our results using a non-uniformstructured grid of about 10,000 control volumes and N N' D 4 4 control angles. Figure 2 gives qualitative representations of the achieved flame and using theoptimized SLW model. Figure 2a displays the temperature contour, Fig. 2b and cdisplays the H2 O and CO2 mole fraction contours, and Fig. 2d displays the CO molefraction contour. All the presented contours in Fig. 2 have important role in radiationcalculations. Figure 3 presents radial temperature profiles in different axial locations of thefurnace. We compare the results of optimized SLW model with the availablemeasured data of Sayre, et al. [19]. It seems that our results present a thinner flamethan the measured data. This may be partially because of the axisymmetric modelingof a three-dimensional furnace. Also the sharp gradients of temperature variationsin our results can be partially due to ignoring the finite rate of reactions in ourequilibrium chemistry calculations. Moving downstream in the furnace, these arebetter agreement between our predictions and the measured data. Figure 4 compares the results of the optimized and non-optimized SLW modelswith each other. Figure 4a presents the contours of radiation source term. For a bettercomparison, we limit the range of depicted source term. Comparison in this figurereveals that the results of optimized SLW model are in complete agreement withthose of the non-optimized SLW model. Figure 4b displays the variation of incidentradiation heat flux along the furnace walls and the variation of temperature alongthe furnace centerline. Again, there are complete agreement between the resultsof optimized and non-optimized SLW models. These comparisons reveal that theoptimized SLW model can provide the same accuracy as it is provided by the non-optimized SLW model. Table 1 compares the number of equations and the required CPU times for1 iteration of coupled CFD solution using different SLW models. The optimizedSLW model only needs 4 RTEs to be solved, while the non-optimized SLW modelneeds 21 RTE solutions. This would indicate a great computational advantage forthe optimized SLW model. Here, all the computations are conducted on a laptopequipped with an Intel CORE i5 CPU and 4 GB RAM. As is seen in Table 1, 396 M. Darbandi et al.
0.6 Temperature (K): 400 600 800 1000 1200 1400 1600 1800
r (m) 0.4 0.2 0 0 0.5 1 x (m) 1.5 2 2.5 3 (a) Temperature Contour 0.8 mole fraction H2O (%): 1 3 5 7 9 11 13 15 17 0.6 r (m)
0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 x (m) (b) H2O mole fraction Contour 0.8 0.6 mole fraction CO2 (%): 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 r (m)
0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 x (m) (c) CO2 mole fraction Contour 0.6 mole fraction CO (%): 0.2 0.6 1 1.4 1.8 2.2 2.6 r (m)
0.4 0.2 0
0 0.5 1 x (m) 1.5 2 2.5 3 (d) CO mole fraction Contour
Fig. 2 Contours of a: temperature and b: H2 O, c: CO2 , and d: CO mole fractions in the furnace
each solution iteration only lasts 0.8 s with the optimized SLW model. However, therequired time for each solution iteration is 4.1 s with the non-optimized SLW model.In other words, using the optimized SLW model makes over 80 % reductions inthe required computational time in comparison with using the non-optimized SLWmodel. This is where both models provide the same level of accuracy, see Fig. 4. Application of an Optimized SLW Model in CFD Simulation of a Furnace 397
0.5
0.4
x = 27 mm
x = 109 mm
x = 177 mm
x = 343 mm
x = 432 mm 0.3 r (m)
0.2
0.1
0 1000
2000
1000
2000
1000
2000
1000
2000
1000
2000 T (K)
Fig. 3 Radial temperature profiles at different axial distances in furnace and comparison with themeasured data of Sayre, et al. [19]
–∇.q (kW/m2): –200 –150–100 –50 0 50 100 0.6
0.4 optimized SLW 0.2 r(m)
–0.2 non-optimized SLW –0.4
–0.6
0 0.5 1 1.5 2 2.5 x(m) (a) contour of radiation source term 2100 optimized SLW 300 non-optimized SLW qr, inc (kW/m ) 2 T (K)
1800 200
1500 100 0 0.5 1.5 1 2 2.5 x (m) (b) variations of incident radiation heat flux along the furnace wall and temperature along the furnace centerline
Fig. 4 Comparison between the results of optimized and non-optimized SLW models, a: contoursof radiation source term, b: variations of incident radiation heat flux along the furnace wall andtemperature along the furnace centerline 398 M. Darbandi et al.
Table 1 Number of equations and required CPU times for each coupled CFD solution iterationusing different SLW models Model Number of RTEs CPU time (s) Optimized SLW 4 0.8 Non-optimized SLW 21 4.1
6 Conclusion
In this work we introduced the optimized SLW model, which used only 3 gray gasesplus 1 clear gas to approximate the radiation in non-gray gas mixtures. Therefore, itreduces the number of radiative transfer equatous to only 4 RTEs solution. This iswhile the classic non-optimized SLW model requires about 10–20 RTEs solution tosupport the same accuracy. We applied both the optimized and non-optimized SLWmodels in the CFD simulation of turbulent reacting flow in a furnace. The achievedresults were in good agreements with available measured data in furnace. We furthercompared the results of optimized SLW model with those of non-optimized SLWmodel. The comparisons demonstrated that the accuracy of the optimized SLWmodel was as good as the non-optimized SLW model. This is while the use ofoptimized SLW model required 80 % less computational time. This improvement isof great value for intensive CFD simulations. Therefore, we suggest the optimizedSLW model for radiation heat transfer calculation in CFD simulation of combustionprocesses.
Acknowledgements The authors would like to thank the research deputy of Sharif University ofTechnology for the financial support received during this research work.
References
1. Keramida, E.P., Liakos, H.H., Founti, M.A., Boudouvis, A.G., Markatos, N.C.: Radiative heat transfer in natural gas-fired furnaces. IJHMT 43(10), 1801–1809 (2000) 2. Xu, X., Chen, Y., Wang, H.: Detailed numerical simulation of thermal radiation influence in Sandia flame D. IJHMT 49(13), 2347–2355 (2006) 3. Kontogeorgos, D.A., Keramida, E.P., Founti, M.A.: Assessment of simplified thermal radiation models for engineering calculations in natural gas-fired furnace. IJHMT 50(25), 5260–5268 (2007) 4. Cumber, P.S., Fairweather, M.: Evaluation of flame emission models combined with the discrete transfer method for combustion system simulation. IJHMT 48(25), 5221–5239 (2005) 5. Denison, M.K., Webb, B.W.: A spectral line-based weighted-sum-of-gray-gases model for arbitrary RTE solvers. J. Heat Transf. 115(4), 1004–1012 (1993) 6. Goutiere, V., Liu, F., Charrette, A.: An assessment of real-gas modelling in 2D enclosures. JQSRT 64(3), 299–326 (2000) 7. Coelho, P.J.: Numerical simulation of radiative heat transfer from non-gray gases in three- dimensional enclosures. JQSRT 74(3), 307–328 (2002) 8. Johansson, R., Andersson, K., Leckner, B., Thunman, H.: Models for gaseous radiative heat transfer applied to oxy-fuel conditions in boilers. IJHMT 53(1), 220–230 (2010) Application of an Optimized SLW Model in CFD Simulation of a Furnace 399
9. Modest, M.F.: The treatment of nongray properties in radiative heat transfer: from past to present. J. Heat Transf. 135(6), 061801 (2013)10. Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3(2), 269–289 (1974)11. Belosevic, S., Sijercic, M., Oka, S., Tucakovic, D.: Three-dimensional modeling of utility boiler pulverized coal tangentially fired furnace. IJHMT 49(19), 3371–3378 (2006)12. Crnomarkovic, N., Sijercic, M., Belosevic, S., Tucakovic, D., Zivanovic, T.: Numerical investigation of processes in the lignite-fired furnace when simple gray gas and weighted sum of gray gases models are used. IJHMT 56(1), 197–205 (2013)13. Cumber, P.S., Onokpe, O.: Turbulent radiation interaction in jet flames: sensitivity to the PDF. IJHMT 57(1), 250–264 (2013)14. Kuo, K.K.: Principles of Combustion. Wiley, Hoboken (2005)15. Pearson, J.T., Webb, B.W., Solovjov, V.P., Ma, J.: Updated correlation of the absorption line blackbody distribution function for H2O based on the HITEMP2010 database. JQSRT 128:10– 17 (2013)16. Rothman, L.S., Gordon, I.E., Barber, R.J., Dothe, H., Gamache, R.R., Goldman, A., Tennyson, J.: HITEMP, the high-temperature molecular spectroscopic database. JQSRT 111(15), 2139– 2150 (2010)17. Darbandi, M., Abrar, B., Schneider, G.E.: Solving combined natural convection-radiation in participating media considering the compressibility effects. In: Proceeding of the 52nd Aerospace Sciences Meeting, AIAA, Maryland (2014)18. Darbandi, M., Abrar, B.: A compressible approach to solve combined natural convection- radiation heat transfer in participating media. Numer. Heat Transf. B 66(5), 446–469 (2014)19. Sayre, A., Lallemant, N.D.J., Weber, R.: Scaling Characteristics of Aerodynamics and Low- NOx Properties of Industrial Natural Gas Burners, The SCALING 400 Study, Part IV: The 300 kW BERL Test Results. International Flame Research Foundation (1994) Numerical Investigation on Periodic Simulationof Flow Through Ducted Axial Fan
Seyedali Sabzpoushan, Masoud Darbandi, Mohsen Mohammadi,and Gerry E. Schneider
Abstract In this paper, the flow through an axial fan is suitably simulatedconsidering relatively low mesh sizes and benefiting from the periodic boundarycondition. The current periodic boundary condition implementation has majordifferences with the past classical ones, which were routinely used in literature.In this regard, we first discuss the ambiguities behind the current periodic geometryand the proposed mesh generation and possible boundary condition choices. Then,proper remedies are proposed to resolve them. One remedy returns to the properchoice of pitch ratio magnitude. After practicing various pitch ratio magnitudes, weeventually arrive to an optimum one, which provides suitable numerical accuracydespite using sufficiently low number of mesh elements. In order to validate theachieved numerical solutions, they are compared with the experiments and otheravailable numerical results. The proposed approach can be also used in simulationsof other turbomachinary cases, where there are serious computer memory andcomputational time limitations. For instance, using the current periodic boundarycondition approach, one can readily simulate very huge wind tunnels considering afull 3D model of its fans, i.e., including its rotor and stator instead of consideringsimple fan pressure jump or actuator disc models.
S. Sabzpoushan • M. MohammadiDepartment of Aerospace Engineering, Sharif University of Technology, Tehran, P.O. Box11365-8639, Irane-mail: [emailprotected]; [emailprotected]. Darbandi ()Department of Aerospace Engineering, Center of Excellence in Aerospace Systems,Sharif University of Technology, Tehran, P.O. Box 11365-8639, Irane-mail: [emailprotected]. SchneiderDepartment of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo,ON, N2L 3G1, Canada
© Springer International Publishing Switzerland 2016 401J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_37 402 S. Sabzpoushan et al.
1 Introduction
Fan has various applications in different industrial equipment, e.g., heat exchangers,air conditioning systems, and wind tunnels. In case of applications with largevolume of air, the industries may have to consider using parallel fans. However,this may lead to a difficulty that the flow at upstream and/or downstream of fansmay not perform axisymmetric pattern. In such cases, one may not simply applythe periodic boundary condition at the boundaries of a slice of flow passage. It isbecause if one wishes to simulate such a flow field, this would cause pitch ratio atthe interfaces between the moving rotor domain and the stationary ducts domains,which in turn will cause critical regions at such interfaces. We should be careful ontwo key points at these interfaces. First, one should know that the face mesh at theseinterfaces must be as similar as possible to avoid missing data from one zone to thenext one [1]. Second, there is an important constraint indicating that the pitch ratiocannot be further up than a certain limit [2]. As a rule of thumb, the pitch ratio mustnot exceed the upper limit of about 10 if one expects an almost reliable simulation[3]. Literature shows that there have been several efforts to simulate axial fansassuming both full rotor disc consideration and periodic blade boundary conditionimplementation. Fidalgo [1] simulated a three-dimensional unsteady flow through afull-annulus rotor disc with distortion in its inlet total pressure. Additionally, Denton[2] raised some limitations in turbomachinary CFD indicating that if the engineersdo not care those limitations, they may not get solutions with sufficient accuracies.He specially focused on specific geometries such as the tip clearance and leadingedge shapes. Cevik [4] performed a simulation on the full geometry of an axial fanwith a diameter of 3.13 m placed in a duct with 12.5 m length. Using the k-epsilonturbulence model, the total number of mesh elements were about 2,500,000. Thereare some other publications, which use the true periodic boundary condition. Forexample, Le Roax [5] simulated the rotor disc and its upstream/downstream ductscompletely and all together using the periodic boundary condition. At this stage, we need to clarify the “pitch ratio” importance in axial fanmodelling by introducing a simple example (see Fig. 1). Prior to any explanation,we should mention that this simple example is so helpful to examine the validityand reliability of current periodic simulation at next stage. This can be done formore complicated cases by comparing its solution with the fan experimental dataand full blade rotor simulation results. As it is shown in Fig. 2(a), the target fanhas 12 blades located in a long duct. For this geometry, the pitch ratio is 360ı /360ıD 1 at the interface of rotational and stationary zones. Now, suppose a periodicdomain in which, we consider only two blades of the fan with (2/12)360ı D 60ıannulus slice cross section. This slice would interact with the full 360ı domainsat its downstream and upstream zones. For this geometry, the pitch ratio becomes360ı/60ı D 6. In this article, we focus on mass flow rate and flow uniformity ratherthan other parameters. In case of using parallel set of fans, interactions between theinflow and outflow streams would also become important. Numerical Investigation on Periodic Simulation of : : : 403
Fig. 1 Two types of rotor disc geometry. (a) Full rotor disc. (b) Sliced rotor disc
(a) (b)
Fig. 2 The fan and its rotor blade geometries. (a) The assembled fan. (b) The result of rotor bladeCMM
2 Geometry and the Solution Domain
The length of rotor blade shown in Fig. 2 is 0.5 m from root to tip and the diameterof its casing is 2 m. The maximum allowable rotational speed is about 900 rpm. Inorder to eliminate the swirl of flow through the rotor, a stator has been embeddeddownstream of the rotor, if this swirl can have considerable effects. Fortunately, incase of using periodic rotor disc, the flow passage in sliced rotor domain does notneed to be coincident with the streamlines. It is because any amount of air exitingfrom one of the two periodic sides, would enter from the other side. Before applying the current approach in complicated geometries with complexconditions, it is better to examine it in a simple geometry test case. Figure 3 presentsthis simple test case, which is constructed based on a complex wind tunnel air drivergeometry. According to the standards related to the methods of fan testing [4, 6, 7],the minimum lengths for the upstream and downstream ducts of an axial fan areassumed to be 10D and 5D respectively, where D is fan casing diameter. We shouldsatisfy these recommended minimum standard values to let the flow become fullydeveloped before passing through the rotor and also at the end of downstream duct. 404 S. Sabzpoushan et al.
Fig. 3 The fan between forward and aft ducts in a simple test geometry
3 Grid Distribution and Boundary Conditions
Figure 4 shows rotor and stator blade sections accompanied with the generatedmesh. The boundary layer mesh and mesh concentration have been used whereverneeded. The boundary layer mesh is controlled suitably to achieve a maximumthickness of about 10 cm. Evidently, it is unavoidable facing with some limitations in mesh generation. Thesources can be due to the geometric constrains. For example, we are faced with verytiny gap at the blade tip clearance, which has considerable effects on the achievedfan performance; especially the fan noise generation [8, 9]. This tiny gap makes it sodifficult to connect the boundary layers generated on the upper and lower surfacesto the blade tip. However, there are two possible approaches to resolve this problem.One is to reduce the number of layers at the blade’s tip and casing. The other one isto ignore the layer’s growth rate and to compress them. This sometimes requires tochange the first layer height. The first approach needs a sudden detraction of layers,which is harmful for the numerical solution, especially for capturing the separation Numerical Investigation on Periodic Simulation of : : : 405
Fig. 4 Two sample sections of mesh distributions around the rotor blade and the stator vane.(a) Rotor blade leading edge. (b) Stator vane
Fig. 5 The mesh cluster between the rotor blade tip and its casing
and tip vortices. Hence, the second approach is chosen, see Fig. 5. Controlling y+ inthe range of 1–100, we can employ k-! SST turbulence model. As another concern,it is important for the surface mesh on the blade to preserve the real shape of leadingedge [5, 10]. Finally, Fig. 6 enforces this point that the surface mesh on both sidesof an interface with any pitch ratio must be as similar as possible. If we choose a slice of rotor disc (and that of the stator disc if it exists) insteadof the entire rotor and stator discs, the number of elements (and consequently therequired computer memory) will decrease considerably. This needs to apply suitableperiodic conditions. After performing some mesh independency efforts, we achievedto a grid resolution of approximately 600,000 elements. This is where the entirerotor disc solution domain needed about 1,700,000 elements to result in mesh-independent solution. Such mesh studies were conducted based on monitoring both 406 S. Sabzpoushan et al.
Fig. 6 Similar mesh generations at the stationary and rotational domains’ interface
0.510
0.500
0.490
0.480 Y (m)
0.470 Coarse (400,000) medium (600,000) 0.460 fine (1,000,000)
0.450
0.440
0.430 0.0 20.0 40.0 60.0 80.0 X-velocity (m/s)
Fig. 7 The mesh refinement study via monitoring the boundary layer velocity profile in threedifferent grid resolutions
the mass flow rate magnitude and the boundary layer velocity profile. Figure 7 showsthe results of mesh refinement study for the sliced rotor considering the boundarylayer velocity profile inside the duct wall. This figure also verifies our estimation onthe boundary layer thickness which is about 6 cm, whereas the boundary layer meshhad been generated with a total thickness of 10 cm. Here, we would like to mention some points on the chosen boundary conditions.The boundary conditions at the inlet and outlet sections of the duct are not fixed byneither mass flow rate, nor velocity magnitude and direction. So, the solution can begradually converged during the numerical procedure. We determine the turbulenceintensity of 1 % at the inlet section and the static temperature of 298 K at thisboundary. Determining the total pressure at the inlet and the average static pressure Numerical Investigation on Periodic Simulation of : : : 407
at the outlet could be other possible choices as the boundary conditions. Thesetwo alternative choices require more flow parameters and constraints to be definedat these boundaries and hence lower the problem generality [11]. The interfaceconnections are chosen in a manner to enforce equal area-weighted average ofnormal velocities at both sides of the interface. To achieve more similarity with thereal conditions, the rotor casing (shroud) is forced to have a counter-rotating RPMrelative to the rotational domain in order to be motionless in stationary referenceframe [10, 12, 13]. The ambient pressure is equal to the value reported in standardatmosphere table for Tehran altitude, which is about 87,000 Pa.
4 The Results and Discussion
According to the experimental tests performed at the atmospheric pressure of87,000 Pa under the standard conditions, the air mass flow rate for a single fan atthe maximum rotational speed is about 53~54 kg/s. As seen in Fig. 8, a high pitchratio would lead to invalid solution with about 60 % deviation from the exact value.However, it promptly drops below 10 % after this point and approaches rapidly tothe exact solution. Thus, this can not only help us to find the proper pitch ratiomagnitude, but also can be used as a type of validation for the proposed simulationcase. Figure 9 shows the tangential velocity contours just after the exit of rotor (beforeentering the stator) and right after the stator. To elaborate the swirl and the effectof stator on its decay, Fig. 10 presents the streamlines passing the fan and comparesthe flow swirl in the absence and presence of stator. The symmetry in these contours
Numerical simulation Experimental test 90 85 80 75 Air mass flow rate (kg/s)
70 65 60 55 50 45 Full rotor disc simulation
40 0 0.2 0.4 0.6 0.8 1 Pitch ratio (rotating-to-stationary)
Fig. 8 Effect of pitch ratio on the calculated mass flow rate through the axial fan 408 S. Sabzpoushan et al.
(a) (b)
Fig. 9 Tangential velocity contours, indicating the swirl strength. (a) Just at the downstream ofrotor. (b) right after the stator
and non-swirling flow at the upstream of the rotor (see streamlines in Fig. 10) alsoimply that the current periodic simulation performed reliably. To provide more qualitative validations, one can check the pressure change overthe fan. The current calculations showed that the area-weighted average of staticpressure (gauge) over two sections somewhere at middle lengths of upstream anddownstream ducts would be about 156 and 23 Pa, respectively. This value is about180 Pa at inlet and 2 Pa at outlet. They indicate that the static head of fan would beapproximately 180 Pa. Figure 11 shows the static pressure contours at two sectionsjust before and after the rotor disc. According to available fan performance curves,this is exactly what we expect for the static head of such a fan with a relatively lowincidence angle (about 17ı ) of rotor blades. Figures 12 and 13 show two important detected phenomena, which supportthe current periodic boundary condition approach in axial fan simulation. Thefirst one is small vortices, which can potentially cause flow to separate from theblade surface. The second one is the flow leakage from the blade pressure sideto its suction side through the gap at tip clearance. It is also known as downwashphenomenon.
5 Conclusion
We introduced a new periodic boundary condition implementation, which wasrelatively different from the past classical ones. This new method can be readilyused to simulate a single fan or a set of several parallel fans despite facing withasymmetric flow conditions. The major obstacle is the pitch ratio, which occurs atthe interfaces between the rotor domain and the stationary upstream and downstreamducts’ domains. After practicing various pitch ratios, we eventually found anoptimum magnitude, which would provide suitable mesh sizes with guaranteednumerical accuracy. Our study showed that the number of mesh elements required to Numerical Investigation on Periodic Simulation of : : : 409
(a)
(b)
Fig. 10 The streamlines inside the duct at a rotor speed of 900 rpm (a) without stator (b) withstator
simulate a full-geometry fan in such applications is more than 11 times the numberof elements required using the proposed periodic approach. Indeed, this reductionin the number of elements is so essential in many real industrial applications. Forexample, it is important when one wishes to simulate the set of several combinedfans, e.g., a set of four parallel fans, utilized in a huge air supply ventilation system. 410 S. Sabzpoushan et al.
Za Pressure Contour 4 –266.91 b Pressure Contour 5 363.20 Z
–275.37 326.86 –283.84 290.52 –292.30 –300.77 X Y 254.18 X Y –309.24 217.84 –317.70 181.50 –326.17 145.16 –334.63 108.82 –343.10 –351.56 72.48 –360.03 36.14 –368.49 0.20 –376.96 36.54 –385.43 72.88 –393.89 109.22 –402.36 –410.82 145.56 –419.29 181.89 [Pa] [Pa]
Fig. 11 Contours of gauge static pressure. (a) Just before passing the rotor. (b) Just after passingthe rotor
Fig. 12 Demonstration of small vortices appearing close to the rotor blade face
Velocity in Stn Frame Vector 1 [m s^–1] 7.
27
47
66
86 31
.1
.0
.9
.8 8
Rotor blade tip
Flow direction
Fig. 13 Demonstration of flow leakage from the high pressure side to the low pressure sidethrough the tip clearance gap Numerical Investigation on Periodic Simulation of : : : 411
Acknowledgements The authors would like to greatly thank the financial supports received fromthe Deputy of Research and Technology of Sharif University of Technology (SUT).
References
1. Fidalgo, J.: A Study of Fan-Distortion Interaction with the NASA Rotor-67 Transonic Stage. In: ASME Turbo Expo, June 2010, Glasgow (2010) 2. Denton, J.D.: Some Limitations of Turbomachinery CFD. In: ASME Turbo Expo, June 2010, Glasgow (2010) 3. Bhasker, C.: Simulation of three dimensional flows in industrial components using CFD techniques. In: Minin, I. (ed.) Computational Fluid Dynamics Technologies and Applications. InTech (2011). ISBN:978-953-307-169-5, doi:10.5772/19909 4. Cevik, F.: Design of an axial flow fan for a vertical wind tunnel for paratroopers. Master thesis, Mechanical Engineering Department, Middle East Technical University (2010) 5. le Roux, F.N.: The CFD simulation of an axial flow fan. Master thesis, Department of Mechanical and Mechatronic Engineering, University of Stellenbosch (2010) 6. AMCA 210: Laboratory Methods of Testing Fans for Aerodynamic Performance Rating. Air Movement and Control Association International, Inc., Arlington Heights, IL (1999) 7. Cory, W.B.: Fans and Ventilation: A Practical Guide. Elsevier-Roles Ltd, Amsterdam (2005) 8. Srinivas, G., Srinivasa Rao., P.: Numerical simulation of axial flow fan using Gambit and Fluent. IJRET 3(3), 586–590 (2014) 9. Raj, S.A., Pandian, P.P.: Effect of tip injection on an axial flow fan under distorted inflow. IJASER 3(1), 302–309 (2014)10. Dwivedi, D., Dandotiya, D.S.: CFD analysis of axial flow fans with skewed blades. IJETAE 3(10), 741–752 (2013)11. Augustyn, O.P.H.: Experimental and numerical analysis of axial flow fans. Master thesis, Stellenbosch University (2013)12. Raj, A.S., Pandian, P.P.: Numerical simulation of static inflow distortion on an axial flow fan. IJMERR 3(2), 20–25 (2014)13. Sarmiento, A.L.E., Gamboa, Y.F.Q., Oliveira, W., Camacho, R.G.R.: Performance analysis through computational fluid dynamics of axial rotor with symmetric blades used in tunnel ventilation. HIDRO and HYDRO, PCH NOTICIAS and SHP NEWS 60(1), 22–25 (2014) Numerical Analysis of Turbulent ConvectiveHeat Transfer in a Rotor-Stator Configuration
D.-D. Dang and X.-T. Pham
Abstract This paper presents the numerical analysis of convective heat transferof a rotor-stator configuration, which is typically found in hydro-generators. TheReynolds Averaged Navier Stokes (RANS) turbulence models based on the eddy-viscosity approximation were employed. Different steady and unsteady multipleframes of reference models were used to deal with the flow interaction in therotor-stator system. The fluid flow and heat transfer analysis were performed usingconjugate heat transfer methodology, in which the governing equations for the fluiddynamics, heat conduction with additional constraints on the fluid-solid interfacewere simultaneously solved. The computed convective heat transfer coefficient wascompared against available experimental data to assess the suitability of turbulencemodels.
1 Introduction
During the operation of hydro-generators, the heat generated by the electromagneticand electrical loses causes the internal heating in the solid components. The solidtemperatures that are too high might reduce the lifetime or at worst result in thebreakdown of the machine. In order to maintain the temperature of the solid parts within a safe margin,the heat generated is removed by a circulation of cooling air. The most importantparameter governing the thermal performance of the hydro-generators is thereforethe convective heat transfer coefficient (CHTC) on the boundary between thecooling fluid and the solid surface. Traditionally, CHTCs were approximated bythe convective empirical correlations which are available in literature and widely
D.-D. Dang () • X.-T. PhamEcole de technologie superieure, Montreal, QC H3C 1K3, Canadae-mail: [emailprotected],[emailprotected]
© Springer International Publishing Switzerland 2016 413J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_38 414 D.-D. Dang and X.-T. Pham
used for thermal analysis. However, those correlations were originally establishedfor simple configurations, and might not be applicable or insufficiently accurate forthe case of complex geometries. Prediction of convective heat transfer coefficient in hydro-generators is cruciallyimportant but not a trivial task due to the complexity of flow dynamics in themachine. The flow field in the rotor-stator system exhibits a number of phenomenawhich are challenging for numerical modelling such as rotation, turbulence andunsteady nature. With respect to methodology, Lumped-Parameter Thermal Network (LPTN),Finite Element Analysis(FEA) and Computational Fluid Dynamics(CFD) are themost common approaches for thermal analysis of electrical machines [2]. Althoughthe analytical LPTN and FEA have advantage of being very fast and efficient, theseapproaches require the knowledge of CHTCs in prior to the computation, which arechallenging to obtain and therefore not always available. These methods are beyondthe scope of the current research that focuses mainly on CFD application. Pickering et al. [11] are considered as the first authors who applied CFD topredict the temperature in the solid of electrical machines. The numerical analysisof an air-cooled, 4-pole generator case concluded that CFD showed to be a potentialtool for thermal analysis of electrical machines. However, because of limitedcomputing resources at the time, only simulations with a coarse mesh near thepole surface could be performed. Depraz et al. [3] studied the cooling system of alarge hydro-generator using the three-dimensional flow simulation. The comparisonbetween the numerical results by CFD and traditional two-dimensional networkbased flow calculation showed a good agreement. Li [7] employed the conjugateheat transfer (CHT) methodology to predict the temperature distribution in the solidpole of the large hydro-generators. The CHT calculations used a coarse mesh onthe wall surface for wall-functions utilization. However, the comparison with theexperimental data was not carried out in his work. Recently Toriano et al. [12]proposed a hybrid method combining the numerical and experimental techniques toevaluate the heat transfer coefficient on the pole-face of a scaled rotating prototype. Although numerous factors might affect the heat transfer prediction by conjugateheat transfer approach, the aim of the current study is limited to investigate thesensitivity of implemented numerical parameters on the prediction of fluid flowand heat transfer in the rotor-stator system. In particular the influence of turbulencemodels and steady-state Multiple Frames of Reference models are analyzed.
2 Computational Domain and Meshes
Because of the limited access on the real generator for experimental measurement,a scaled prototype was built. Since it is not realizable from the computationalperspective to perform simulations of the entire scaled rotating model with an Numerical Analysis of Turbulent Convective Heat Transfer in a Rotor-Stator. . . 415
Fig. 1 Computational domain of scaled rotating model
adequately fine grid; calculations were carried out in two steps. First, the simulationwith the full scaled model (Fig. 1) has been computed with a coarse 86 M hexahedralcells grid using the standard k turbulence model. This calculation was then usedto generate the conditions at the inlet boundary for the following second step ofthe calculation. The second configuration was compromised to only one section of10 degree single-pole in (r; ) plane which allowed to perform with a finer meshup to 1.6 M hexahedra cells (Fig. 2). The extensive parametric analysis will beperformed on this simplifed 2-D configuration using various turbulence and multipleframe of reference models. 416 D.-D. Dang and X.-T. Pham
Fig. 2 Computational domain of two-dimensional simplified configuration
3 Mathematical Model and Numerical Method
3.1 Governing Equations
The governing equations for incompressible steady state flow [4] are expressed inthe Cartesian tensor notation for a rotating coordinate system as follows. Continuity equations:
@ .Ui / D 0I (1) @xi
Momentum transport equation: @ @P @ @Ui @Uj .Ui Uj / D C . C / ui uj @xj @xi @xj @xj @xi 2"ijp ˝p Uj ˝j Xj ˝i ˝j Xi ˝j (2) „ ƒ‚ … „ ƒ‚ … centrifugalforce Coriolisforce Numerical Analysis of Turbulent Convective Heat Transfer in a Rotor-Stator. . . 417
The two last terms in momentum equations were added to account for the Cori-olis and centrifugal forces in the rotating frame of reference. These contributionshowever are eliminated in the stationary frame of reference. Energy equation: @ @ @T @ Uj htot D f uj h C Ui ij ui uj C SE ; (3) @xj @xj @xj @xj
where the specific total enthalpy for ideal gas is given by:
P U2 1 htotal D Cp T C C i C u2i (4) 2 2
The heat conduction equation in the solid has the same form as the energyequation for fluid Eq. 3, except that the velocity components are set to be zero, andthe solid thermal conductivity s was substituted. On the fluid-solid interface, additional constraints need to be added to ensure theequilibrium of heat flux and the temperature continuity through the two mediums.
Tsolid D Tfluid @T @T s jsolid D f jfluid (5) @n @n
3.2 Turbulence Models and Boundary Layer Modelling
The presence of Reynolds stresses ui uj in Eqs. 2–3 means that the derivedequations are not closed, which necessitates additional equations for closure. Theturbulent stresses and turbulent heat flux are obtained using the effective viscosityapproximation: 2 @Ui @Uj ui uj D kıij t C (6) 3 @xj @xi
t @h uj h D (7) Prt @xj
The turbulent eddy-viscosity t in Eqs. (6) and (7) is calculated by the turbulencemodels. Several turbulence model bases on the eddy viscosity approximation areimplemented to calculate the flow dynamics and heat transfer. Standard k (SKE) [6] model was developed for high-Re flow and has been widely used inindustrial simulation because it compromises between the robustness and accuracy.However, SKE was consistently found inappropriate for the low-Re flows or flowswith separation and reattachment of boundary layer. To improve the prediction, the 418 D.-D. Dang and X.-T. Pham
Shear Stress Transport (SST) model [8] was also implemented. Menter [9] presentedseveral test cases to show that the SST k ! predicts flows with separation in thepresence of adverse pressure gradients more accurately than the SKE model.Boundary layer modelling Modelling of the near-wall region is of major impor-tance for CFD simulations, especially for the cases in which the prediction of theflow quantities on the wall (skin friction, convective heat transfer) is important.In principle, low-Reynolds number (LRN) and wall-functions are two commonapproaches for the near-wall modelling. The LRN model refers to an approach thatresolves the entire boundary layer using a very fine mesh in the near wall regions.The grids used for the low-Re number model require typically a dimensionless walldistance of the wall-adjacent cell about unity, i.e. yC < 1. The grid for the low-Renumber is shown in Fig. 3 (right), in which the wall-space of the first cell was setas 6.103 mm. Because of high computational cost associated with LRN model, thewall-functions are often used instead. The idea behind the wall-functions approachis to place the first computational node into the logarithmic layer and employ asemi-empirical dimensionless profile to obtain the wall shear stress. The advantageof this approach is that the boundary layer can be resolved with an adequately smallnumber of grid points. The wall functions approach requires the yC value of the firstnode between 30 and 300, i.e., 30 < yC < 300. In the present CFD code, the LRNand wall-functions models are employed for the !base and based turbulencemodels, respectively.Rotor-stator interaction The fluid flows in a rotor-stator system having non-axisymmetric components are always unsteady, which is a challenge for CFDsimulations [10]. In an effort to make the rotor-stator analysis practical with limitedcomputing resource, several steady-state Multiple Frames of Reference (MFR)models have been developed. In these models, an interface is defined betweenrotating and stationary components in a manner so that the steady-state calculationsin each frame are supported while maintaining as much interaction between thecomponents as possible [5]. There are two steady-state interface techniques are
Fig. 3 Grids for differentboundary layer models:wall-functions (left) andlow-Re number model (right) Numerical Analysis of Turbulent Convective Heat Transfer in a Rotor-Stator. . . 419
available in the present CFD code. The first model is called Mixing Plane, where theupstream flow velocity profile is first averages circumferentially before transferringto the downstream region. In this context, any non-uniformity in the circumferentialdirection will not be preserved in the next region. The second type of steady-stateinterface is called Frozen Rotor, where the flow profile variation is now preservedacross the interface, however, the relative position between the rotor and stator isfixed in time and space. Also, the transient sliding interface is also available to modelthe unsteady flow due to the relative motion between the rotating and stationarycomponents. In this model, the flow field variation is fully taken into account.
3.3 Boundary Conditions and Convergence Criteria
The boundary conditions for the configuration illustrated in Fig. 2 are definedas follows. The fluid velocity at the inlet boundary was set to be uniform, andnormal to the inlet surface. The average value of the inlet profile was derived fromthe 3D simulation on the coarse grid with SKE turbulence model. For the casewith rotational speed 50 rpm, Uinlet D 2:10 m/s. The inlet temperature was set to298 K (25 ı C). For all wall boundaries, no-slip conditions with zero velocity wereassumed. The circumferential periodicity feature of the geometry was specified inthe numerical model by defining rotational periodic interfaces. On the rotor-statorinterface, the general grid interface (GGI) was defined, which allows performingcalculations of non-conforming meshes in a conservative manner. The governingequations presented in Sect. 2 were solved by CFD code ANSYS CFX-15.0.Considering the recommendation of ASME V&V numerical accuracy guideline [1],all equations were solved using second-order accuracy schemes. Convergence was judged by examining the residual levels as well as by moni-toring the relevant variables at critical locations. The used criterion requires that themaximum residual of all equations dropped at least by the order of 104 and thevariables of interest at considered locations keeps constant at the steady-state value.
4 Results and Discussions
The flow and turbulence structure in the inter-pole duct calculated using mixing-plane model with the SST k ! is illustrated in Fig. 4. The flow structure observedare typical for the flow in the duct with forward facing-step, in which flow withuniform velocity at the inlet develops in upstream region until it impinges on thestep; a circulation bubble is formed in the step corner due to the adverse pressure 420 D.-D. Dang and X.-T. Pham
Fig. 4 Velocity contour (left) and turbulent kinetic energy contour (right) predicted by mixing-plane
Fig. 5 Normalized radial velocity in the inter-pole duct predicted by different MFR models
gradient caused by step blockage. The boundary layer separates over the step andreattaches to the pole surface thereafter. Due to the system rotation the flow structureis asymmetric in the inter-pole duct; the Coriolis force have the effect of shrinkingthe recirculation size on the leading edge and enlarging the bubble size on thetrailing edge. The steady state models frozen rotor and mixing-plane was employed for aninherently unsteady flow in rotor-stator interaction, which is necessary to verify. Thecalculated result using these models are compared with the time-averaged transientsimulation as reference result. Figure 5 shows the normalized radial velocity in the Numerical Analysis of Turbulent Convective Heat Transfer in a Rotor-Stator. . . 421
Fig. 6 Average heat transfer coefficient on the pole-face computed by different turbulence models
duct at r D 1.12 m for different MFR models. It is observed that the mixing-planemodel shows a better agreement with the time-average transient model meanwhilethe result computed by FR model significantly varies depending the relative positionof rotor and stator. This results implies that the circumferential flow variation thateach passages rotates during a full revolution is small at the rotor-stator interface.Since the simulation using mixing-plane model requires less computational effortthan transient simulations, this model will be employed for the further parametricanalysis. Figure 6 reports the comparision of average CHTC on the pole face predictedby different turbulence models closure, including the standard k (SKE), the Re-Normalisation Group k (RNG), the standard k ! (SKO) and SST k ! (SST)models. The averaged value is calculated by Eq. 8. R qw dA hD R A (8) A Tw Tref dA
where the Tref is the reference temperature, which is fluid temeprature at 5 mm awayfrom the pole face for both numerical and experimental approach. The experimentaldata is extracted from Toriano et al. [12]. It can be seen that all turbulence modelspredict higher CHTC in comparison with experimental data. The results calculatedby SST k ! shows the best agreement with the experiments, the relative error iscalculated as 11 %. In Fig. 7, the dimensionless temperature T profiles are compared for differentpositions at the pole-face along lines normal to the surface, as functions of y , 422 D.-D. Dang and X.-T. Pham
Fig. 7 Profiles of dimensionless temperature T as functions of y along lines normal to the poleface
defined by:
1=4 C k1=2 .Tw T/ Cp T D (9) qw
1=4 C k1=2 y y D (10)
According to Eq. 9, the higher T values results in a lower wall heat flux incase of the same level of predicted turbulence kinetic energy(TKE). However, theTKE predicted by wall-functions approach are found much higher than the low-Re number model. The example of the TKE profiles at two positions 2ı and 3ıare illustrated in Fig. 8. Overall, this results in the significant over-estimation ofthe CHTC by the wall function, as showed in Fig. 6. The maximum discrepancybetween the CHTC predicted by wall-functions and experimental data is up to 50 %. Numerical Analysis of Turbulent Convective Heat Transfer in a Rotor-Stator. . . 423
Fig. 8 Normalized turbulence kinetic energy in the air-gap predicted by different turbulencemodels
5 Conclusion
High resolution steady-state RANS CFD simulations of conjugate heat transfer in arotor-stator configuration was performed. The focus was the sensitivity analysis ofnumerical parameters on the fluid flow and convective heat transfer prediction onthe pole-face. The results showed that the mixing-plane model demonstrated a goodagreement with the transient model than the frozen rotor, whose results significantlyvaries depending on the relative position of the rotor and stator. The heat transfercoefficient prediction by different turbulence models shows that the SST with low-Re model for the near-wall region is more appropriate than the wall-functions, whichover-estimates up to 50 % this coefficient on the pole face on average.
Acknowledgements The support of the Natural Sciences and Engineering Research Councilof Canada(NSERC) and Fonds de recherche du Quebec Nature et technologies(FRQNT) aregratefully acknowledged. We are also grateful for the funding and computing infrastructureprovided by Institut de recherche d’Hydro-Quebec(IREQ). 424 D.-D. Dang and X.-T. Pham
References
1. ASME Standard for verification and validation in computational fluid dynamics and heat transfer.: ASME V&V 20-2009, American Society of Mechanical Engineers (2009) 2. Boglietti, A., Cavagnino, A., et al.: Evolution and modern approaches for thermal analysis of electrical machines. IEEE Trans. Ind. Electron. 56(3), 872–882 (2009) 3. Depraz, R., Zickermann, R., Schwery, A., Avellan, F.: CFD validation and air cooling design methodology for large hydro generator. In: Proceedings of 17th International Conference on Electrical Machines ICEM, Crete Island, Greece (2006) 4. Ferziger, J.H., Peric, M.: Computational Methods for Fluid Dynamics. Springer, Berlin/Hei- delberg (2001) 5. Galpin, P., Broberg, R., Hutchinson, B.: Three-dimensional Navier-Stokes predictions of steady state rotor-stator interaction with pitch change. In: Proceedings of 3rd Annual Conference of the CFD Society of Canada, Banff, Alberta (1995) 6. Launder, B.E., Spalding, D.B.: The numerical computation of turbulent flows. Comput. Methods Appl. Mech. Eng. 3(2), 269–289 (1974) 7. Li W, Guan C, Chen Y (2014) Influence of rotation on rotor fluid and temperature distribution in a large air-cooled hydrogenerator. IEEE Trans. Energy Convers. 28, 117–24 8. Menter, F.R.: Improved Two-Equation k-omega Turbulence Models for Aerodynamic Flows. NASA TM-103975 (1992) 9. Menter, F.R., Kuntz, M., Langtry, R.: Ten years of industrial experience with the SST turbulence model. Turbul. Heat Mass Transf. 4, 625–32 (2003)10. Moradnia, P., Chernoray, V., Nilsson, H.: Experimental assessment of a fully predictive CFD approach, for flow of cooling air in an electric generator. Appl. Energy 124, 223–30 (2014)11. Pickering, S.J., Lampard, D., Shanel, M.: Modelling ventilation and cooling of the rotors of salient pole machines. In: IEEE International Electric Machines and Drives Conference, Cambridge, MA, pp. 806–808 (2001)12. Torriano, F., Lancial, N. et al.: Heat transfer coefficient distribution on the pole face of a hydrogenerator scale model. Appl. Therm. Eng. 70(1), 153–162 (2014) Determining Sparse Jacobian Matrices UsingTwo-Sided Compression: An Algorithmand Lower Bounds
Daya R. Gaur, Shahadat Hossain, and Anik Saha
Abstract We study the determination of large and sparse derivative matrices usingrow and column compression. This sparse matrix determination problem has richcombinatorial structure which must be exploited to effectively solve any reasonablysized problem. We present a new algorithm for computing a two-sided compressionof a sparse matrix. We give new lower bounds on the number of matrix-vectorproducts needed to determine the matrix. The effectiveness of our algorithm isdemonstrated by numerical testing on a set of practical test instances drawn fromthe literature.
1 Introduction
The determination of the Jacobian matrix F 0 .x/ of a mapping F W <n ! <m is akey computation in Newton’s method for solving nonlinear optimization problems.Using forward difference (FD), the product of the Jacobian matrix with a vector smay be approximated as ˇ @F.x C ts/ ˇˇ 1 ˇ D F 0 .x/s As ŒF.x C s/ F.x/ b; (1) @t tD0
with one extra evaluation of F at .x C s/, where > 0 is a small increment,assuming F.x/ has already been evaluated. Algorithmic (or Automatic) Differenti-ation (AD) [7] gives b D F 0 .x/s and c D w> F 0 .x/ in forward mode and reversemode, respectively. The numerical values are accurate up to the machine round-off and the cost of the matrix-vector product is a small multiple of the cost ofone function evaluation. The key observation is that the product of the Jacobianmatrix of function F, evaluated at a given point x, with a given vector s can beobtained as As or A> s at a cost that is a small multiple of the cost of evaluatingthe function F at x [7]. Therefore, in what follows, we express the computational
D.R. Gaur • S. Hossain () • A. SahaDepartment of Mathematics and Computer Science, University of Lethbridge, Lethbridge,AB, Canadae-mail: [emailprotected]; [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 425J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_39 426 D.R. Gaur et al.
Fig. 1 Algorithm for FD 1 for j 1 to nestimation of the Jacobian 2 do 1matrix of F at x 3 B(:, j) [F(x + I(:, j)) − F(x)]
Fig. 2 A row compression a11 0 0 0 0 a16 1 0 0 0matrix S a21 0 a23 a24 a25 0 1 0 0 0 a31 0 a33 0 0 0 0 1 0 0 B= 0 a42 0 a44 0 0 0 0 1 0 a51 0 0 0 a55 0 0 0 0 1 0 a62 0 0 0 a66 0 1 0 0
A S
a11 a16 0 0 a21 a23 a24 a25 a31 a33 0 0 = a42 0 a44 0 a51 0 0 a55 a62 a66 0 0
cost of determining a Jacobian matrix in terms of the number of matrix-vectorproducts (MVPs) of the kind As or A> s. The complete Jacobian can be determined,for example, by differencing along n coordinate directions as shown in Fig. 1 whereI denotes the identity matrix. We use colon notation of [5] to specify submatricesof a matrix. In AD reverse mode, the Jacobian matrix can be determined row-by-row as vector-matrix products s> A. Many large-scale practical optimizationproblems often contain useful structure. For example, the component functions Finonlinearly depend only on a small subset of the n independent variables givingrise to partially separable structure [6]. It is usually the case that zero-nonzeropattern of the Jacobian is a priori known or can be computed easily. Consider thematrix A in Fig. 2. Define matrix S where S.W; 1/ D e1 C e2 ; S.W; 2/ D e3 C e6 ;S.W; 3/ D e4 ; S.W; 4/ D e5 and compute the product B D AS using, for example,the algorithm in Fig. 1. It is clear from Fig. 2 that the FD estimation of the unknownelements of matrix A can simply be read-off, i.e., without incurring any floating-point operations, from the compressed matrix B at a cost of only 4 (instead of6) additional evaluations of function F. Alternatively, we may define a columncompression matrix W where W.W; 1/ D e2 C e6 ; W.W; 2/ D e4 C e5 ; W.W; 3/ De1 ; W.W; 4/ D e3 and compute the compressed matrix C> D W > A in reverse modeAD. In either case, 4 matrix-vector products are required to completely determinethe matrix. We say that matrix A is directly determined if all the nonzero unknownscan be read-off from the compressed matrices B or C as defined above. The Jacobianmatrix determination problem (JMDP) based on matrix-vector product calculationcan be stated as below (See [10]). Determining Sparse Jacobian: Algorithm and Lower Bound 427
Given the specification of the sparsity pattern for matrix A, obtain matrices S 2 f0; 1gnp and W 2 f0; 1gmq such that matrix A is directly determined and p C q is minimum.
A group of columns satisfying the property that no two of them contain nonzeroentries in the same row position is called structurally orthogonal. The observationthat a group of structurally orthogonal columns can be approximated with onlyone extra function evaluation was reported first in a seminal paper by Curtis,Powell, and Reid [3]. Their row compression algorithm1 was further analyzed byColeman and Moré [1] who gave the first graph coloring interpretation of thecompression problem. The classical arrow-head matrix example demonstrates thatone-sided compression (i.e., either column or row) may not yield full exploitation ofsparsity [9]. Hossain and Steihaug [9], and Coleman and Verma [2] independentlyproposed techniques for two-sided compression (i.e., combined row and columncompression) for the sparse Jacobian determination problem. Unfortunately, sparseJacobian determination using compressions (one-sided or two-sided) is NP-hard [2]implying that exact methods are impractical except for small problem instances.Greedy heuristics complete direct cover (CDC) by Hossain and Steihaug [9] andminimum nonzero count ordering (MNCO) by Coleman and Verma [2] producematrices S and W by considering the matrix rows and columns in specified orders.Recently, Juedes and Jones [11] proposed an approximation algorithm for minimum 2star bi-coloring (ASBC) with an approximation guarantee of O.n 3 / of the optimal.An easy to compute “good” lower bound on the number of groups allows one tomeasure the effectiveness of such algorithms. For row compression (of A or A> ) asatisfactory lower bound on the number of matrix-vector products (or AD passes)is given by max .A/ D maxi i where i denotes the number of nonzero entries in nnz.A/row i. For two-sided compressions, [8] proposed the expression d maxfm;ng e wherennz.A/ denotes the number of nonzero entries in matrix A. The bound is derivedusing a consistency argument in solving the system of linear equations defined bythe products B D JS and C D W > J. Juedes and Jones [11] derive the same boundusing a graph theoretic argument. In this paper we propose a new and more effectivelower bound which generalizes the above lower bound. We propose a new heuristicalgorithm for two-sided compression of sparse Jacobian matrices. We demonstratethe effectiveness of our lower bound and algorithm by extensive computationalexperiments on a standard set of test instances. The remainder of the paper isorganized as follows. In Sect. 2, we derive the new lower bound on the numberof matrix-vector products which is followed by the description of the new two-sidedcompression algorithm. Section 3 contains results of numerical experiments fromthe lower bound computation and the two-sided compression algorithm on a setof test matrices. Test results indicate that the new lower bound is superior to theexisting one. On many of the test matrices, our algorithm produced better results.The paper is concluded in Sect. 4 with comments on future research.
1 A row compression is a one-sided compression where only seed matrix S is defined. 428 D.R. Gaur et al.
2 Two-Sided Compression
As discussed in Sect. 1 the example in Fig. 2 can be determined using 4 MVPs usingeither row or column compression with one-sided compression. This is also theminimum possible since the maximum number of nonzero entries in any row of Aor A> , max .A/ D max .A> /, is 4. However, a closer examination of Fig. 2 revealsthat the last two columns of the compressed matrix B are quite sparse – only twoof the entries are nonzero. In other words, there are available sparsity which are notexploited by one-sided compressions. On the other hand, with the following two-sided compression 0 1 0 1 1 0 1 B1 0C B1C B C B C B C B C B0 1C B0C SDB C; WDB C B0 1C B0C B C B C @0 1A @0A 0 1 0
we can compute the MVPs 0 1 a11 a16 Ba a C a C a C B 21 23 24 25 C B C B a31 a33 C AS D B C; W > A D a11 C a21 0 a23 a24 a25 a16 B a42 a44 C B C @ a51 a55 A a62 a66
such that all the nonzero entries are determined uniquely. A lower bound on thenumber of MVPs for one-sided compressions is given by the easily evaluatedexpression minfmax .A/; max .A> /g which has been found to be a good approxima-tion to the chromatic number of graphs associated with the matrices on an extensiveset of test instances [1]. On the other hand, only a handful of published literatureconsider two-sided compressions. The Ph.D. dissertation [9] derived the bound nnz.A/LB2S D d maxfm;ng e on the number of MVPs in a two-sided compression whichis also independently derived and analyzed experimentally in [11]. In this paper wegeneralize this lower bound and show that the new lower bound is always at least asgood and in many test instances it yields better result. Determining Sparse Jacobian: Algorithm and Lower Bound 429
2.1 A New Lower Bound
Following [9] the bound LB2S can be derived intuitively in the following way.Consider a two-sided compression S 2 f0; 1gnp ; W 2 f0; 1gmq . With an MVPof type As; s 2 f0; 1gn , at most m nonzero entries can be uniquely determined.Similarly, with an MVP of type w> A; w 2 f0; 1gm, at most n nonzero entries canbe uniquely determined. If there are nnz.A/ nonzero entries in matrix A, then itsfollows that nnz.A/ nnz.A/ pm C qn maxfm; ng. p C q/ implying . p C q/: maxfm; ng
In one-sided compression a row with the maximum number of nonzero determines alower bound for the number of MVPs. In graph-theoretic terminology this representsa subgraph where each pair of vertices are connected by an edge. Our proposal is toextend this key observation to the two-sided scenario. In other words, we claim thata dense submatrix of a given matrix A yields a lower bound on the number of MVPsto uniquely determine matrix A. Clearly, a bound of the type LB2S on a submatrix 0 0of A is also a bound on the whole matrix A. Now consider a submatrix A0 2 <m nsufficiently dense such that neither the rows nor the columns of the submatrix arestructurally orthogonal. A trivial example is a completely dense submatrix. Then, weneed at least minfm0 ; n0 g MVPs to uniquely determine matrix A. In graph-theoreticterminology, this completely dense submatrix corresponds to a complete bipartitesubgraph Km0 ;n0 of the bipartite graph associated with matrix A. We want to find thelargest dense submatrix of the kind described above. Unfortunately, the complexityof deciding whether or not a bipartite graph contains a complete bipartite subgraphof size k is NP-complete [4]. Instead of trying to find the largest dense submatrixof matrix A we therefore settle for a reasonably dense submatrix which is easyto compute. The above discussion can be succinctly stated as: find permutationmatrices P and Q such that the largest dense submatrix of A is included in someleading submatrix of the permuted matrix PAQ. The k-th leading submatrix ofmatrix A is denoted by A.1 W k; 1 W k/. To move dense rows and columns towardthe top left corner of matrix A we use the following simple procedure. Order the rows in non increasing order of the number of nonzero entries followed by the same ordering procedure applied to the columns. Matrices P and Q represent the row and column ordering, respectively.
We then apply the formula LB2S on each leading submatrix of permuted matrixPAQ. The largest LB2S value is our lower bound LB2SX on the number of MVPs todetermine matrix A. Thus,
LB2SX D maxfLB2Si g i 430 D.R. Gaur et al.
and nnzi .PAQ/ LB2Si D ; i
where nnzi .:/; i D 1; : : : ; minfm; ng denotes the number of nonzero entries in thei-th leading submatrix of the argument. The above discussion can be summarized in the following lemma.Lemma 1 A lower bound on the number of matrix-vector products to directlydetermine A 2 <mn is given by
LB2SX D maxfLB2Si g i
with nnzi .PAQ/ LB2Si D ; i
where nnzi .:/; i D 1; : : : ; minfm; ng denotes the number of nonzero entries in thei-th leading submatrix of the argument and P and Q are permutation matrices. Ifm D n, then we have nnz.A/ LB2SX LB2S D : maxfm; ng
2.2 A Heuristic for Two-Sided Compression
The heuristic algorithm that we outline here is inspired by the Recursive LargestFirst (RLF) coloring heuristic due to [12]. The main idea of our compressionalgorithm is to group columns and rows such that for each nonzero aij ,1. there is a column group k containing column j such that there is no column j0 ¤ j also in group k for which aij0 ¤ 0, or2. there is a row group l containing row i such that there is no row i0 ¤ i also in group l for which ai0 j ¤ 0This condition is termed as direct cover condition in [8] and the nonzero aij meetingthis is said to be directly covered or simply covered, for brevity. The completedirect cover (CDC) algorithm of [9] groups the columns and rows while respectingthe cover condition until all the nonzero entries are covered. The resulting columngroups and row groups define the matrices S and W such that the Jacobian matrix isdirectly determined. The columns and rows are assigned to groups according to thenumber of nonzero entries that are yet to be covered: higher the number of entriesearlier they are grouped. In our algorithm 2SIDEDCOMPRESSION, as depicted in Determining Sparse Jacobian: Algorithm and Lower Bound 431
Fig. 3, we too take into account the number of entries that are yet to be covered informing the groups. First, we introduce terminology that will enable us to describethe algorithm. Given a sparse matrix A let I D f1; : : : ; mg and J D f1; : : : ; ng denotethe set of row indices and the set of column indices, respectively. For row index iand I I define, ˚ fbdnI .i/ D i0 ji0 2 I and there is an index j 2 J for which aij ¤ 0; ai0 j ¤ 0 are not covered :
Analogously, for column index j and J J define, ˚ fbdnJ . j/ D j0 jj0 2 J and there is an index i 2 I for which aij ¤ 0; aij0 ¤ 0 are not covered : ˇ ˇAlso, let degI .i/ D jfbdnI .i/j and degJ . j/ D ˇfbdnJ . j/ˇ denote the cardinalityof the respective index sets. Thus, degI .i/ denotes the number of rows corresponding to set I that cannotbe grouped with row i. Similarly, degJ . j/ denotes the number of columnscorresponding to set J that cannot be grouped with column j. On return from 2 SIDEDCOMPRESSION each column is given a label from the setf0; 1; : : : pg indicating the column group it is included in and each row is given labelfrom the set f0; 1; : : : qg indicating the row group it is included in. A column (or row)that is labeled with group number 0 indicates that its nonzero entries are directlycovered by other group(s). Variable ncovered represents the number of nonzeroentries that have been directly covered. Initially, each column/row is labeled withgroup number 0. The while -loop in line 5 is iterated until all the nonzero entries ofmatrix A are directly covered. Variables Iu and Ju contain the indices of rows andcolumns, respectively, that are yet to be included in a group other than group 0. Thepseudocode in lines 7–16 and lines 17–25 compute the next row group and columngroup, respectively. The if -construct in lines 26–34 chooses the group which coversthe most nonzero entries (Fig. 3).
3 Numerical Experiments
In this section we provide numerical evidence to demonstrate the effectiveness of themethods developed in Sect. 2. For numerical experiments we choose the set of testinstances used in ([11] Table 2). Table 1 compares the lower bound on the numberof MVPs given by LB2SX and LB2S. In Tables 1 and 2, max .:/ represents themaximum number of nonzero entries in any row of the matrix given in the argument.Table 1 includes the test matrices of ([11] Table 2) where the two bounds differ.Out of 32 instances, LB2SX is strictly better on 14 of them; on the remaining 16instances LB2SX and LB2S yield the same value. On problem eris1176 algorithm 432 D.R. Gaur et al.
Fig. 3 An algorithm for two-sided compression of a sparse Jacobian matrix
Table 1 Lower bound on the number of MVPs in two-sided compression of sparse JacobianMatrix m n max .A/ max .A> / LB2S LB2SXabb313 313 176 6 26 5 6arc130 130 130 2 12 10 16bp0 822 822 266 20 4 7bp200 822 822 283 21 5 7bp400 822 822 295 21 5 7bp800 822 822 304 21 6 8bp1000 822 822 308 21 6 8eris1176 1176 1176 99 99 16 80lund_a 147 147 21 21 17 18lund_b 147 147 21 21 17 18str_0 363 363 34 34 7 17str_200 363 363 30 26 9 17str_400 363 363 33 34 9 18will_57 57 57 11 11 5 6 Determining Sparse Jacobian: Algorithm and Lower Bound 433
Table 2 Number of MVPs in two-sided compression heuristicsMatrix m n max .A/ max .A> / ASBC 2SIDED COMPRESSIONabb313 313 176 6 26 17 10ash219 219 85 2 9 8 4ash292 292 292 14 14 19 14ash331 331 104 2 12 10 6ash608 608 188 2 12 11 6ash958 958 292 2 13 12 6bp0 822 822 266 20 16 17bp200 822 822 283 21 17 19bp400 822 822 295 21 19 20curtis54 54 54 12 16 12 11eris1176 1176 1176 99 99 92 80fs_541_1 541 541 11 541 18 13fs_541_2 541 541 11 541 18 13ibm32 32 32 8 7 9 8lund_a 147 147 21 21 26 24lund_b 147 147 21 21 27 23shl_0 663 663 422 4 7 4shl_200 663 663 440 4 7 4shl_400 663 663 426 4 6 4str_0 363 363 34 34 25 27str_200 363 363 30 26 31 32will199 199 199 6 9 9 7
2SIDEDCOMPRESSION is optimal (see Table 2) since the number of MVPs neededto directly determine the matrix is same as the lower bound given by LB2SX. Thematrix is pattern symmetric and will require at least 99 MVPs to determine it by anyone-sided compression method. With a two-sided compression, however, at least80 MVPs will be needed to determine the matrix. The LB2S gives a lower bound of16 MVPs which is much lower than the optimum number of MVPs (D 80) requiredto directly determine the matrix. Table 2 compares the number of MVPs needed byalgorithm ASBC and our heuristic algorithm 2SIDEDCOMPRESSION . The instancesare chosen from ([11] Table 2) where the number of MVPs required by the twomethods are different. Out of 22 instances, algorithm 2SIDEDCOMPRESSION yieldsbetter result than ASBC on 17 (marked in bold face) of them. For the instanceswhere ASBC is better, the difference is at most 2.
4 Conclusion
In this paper we have considered the problem of sparse Jacobian matrix determina-tion with two-sided compressions. The new lower bound on the number of MVPs 434 D.R. Gaur et al.
generalizes the lower bound of [9, 11]. The results from numerical experiments ona standard set of test instances provide strong evidence that the proposed lowerbound is superior to that of [9, 11]. The new lower bound is easy to compute,never smaller than the bound proposed in [9, 11], and strictly larger on about50 % of the test instances. The Heuristic algorithm 2 SIDEDCOMPRESSION saves,on average, 13 % MVPs compared with the ASBC algorithm and, on average it iswithin approximately 1:5 times the optimal. There are a number of possible extensions to this research that we plan to pursuein future. In our lower bound calculation we use the simple strategy of reordering ofrows and columns according to the nonzero density. More sophisticated orderingtechniques can be considered here. For example, lining up the nonzero entriestogether in rows and columns may reveal the dense submatrices which can beidentified more easily. This will yield the added benefit of utilizing data localityto improve cache memory performance in a computer implementation. Also, thecurrent computer implementation of the compression algorithm can be improved byemploying more advanced data structures for manipulating sparse matrices.
Acknowledgements This research is supported in part by Natural Sciences and EngineeringResearch Council of Canada (NSERC) Discovery Grant (Individual).
References
1. Coleman, T.F., Moré, J.J.: Estimation of sparse Jacobian matrices and graph coloring problems. SIAM J. Numer. Anal. 20(1), 187–209 (1983) 2. Coleman, T.F., Verma, A.: The efficient computation of sparse Jacobian matrices using automatic differentiation. SIAM J. Sci. Comput. 19(4), 1210–1233 (1998) 3. Curtis, A.R., Powell, M.J.D., Reid, J.K.: On the estimation of sparse Jacobian matrices. IMA J. Appl. Math. 13(1), 117–119 (1974) 4. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP- Completeness. Freeman, San Francisco (1979) 5. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996) 6. Griewank, A., Toint, Ph.L.: On the unconstrained optimization of partially separable objective functions. In: Powell, M.J.D. (ed.), Nonlinear Optimization, pp. 301–312. Academic, London (1982) 7. Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2008) 8. Hossain, A.K.M.S.: On the computation of sparse Jacobian matrices and newton steps. Ph.D. Dissertation, Department of Informatics, University of Bergen (1998) 9. Hossain, A.K.M.S., Steihaug, T.: Computing a sparse Jacobian matrix by rows and columns. Optim. Methods Softw. 10, 33–48 (1998)10. Hossain, S., Steihaug, T.: Graph models and their efficient implementation for sparse Jacobian matrix determination. Discret. Appl. Math. 161(2), 1747–1754 (2013)11. Juedes, D., Jones, J.: Coloring Jacobians revisited: a new algorithm for star and acyclic bicoloring. Optim. Methods Softw. 27, 295–309 (2012)12. Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84, 489–505 (1979) An h-Adaptive Implementationof the Discontinuous Galerkin Methodfor Nonlinear Hyperbolic Conservation Lawson Unstructured Meshes for GraphicsProcessing Units
Andrew Giuliani and Lilia Krivodonova
Abstract For computationally difficult problems, mesh adaptivity becomes anecessity in order to efficiently use computing resources and resolve fine solutionfeatures. The discontinuous Galerkin (DG) method for hyperbolic conservation lawsis a numerical method adapted to execution on graphics processing units (GPUs).In this work, we give the framework of an efficient h-adaptive implementation ofthe modal DG method on NVIDIA GPUs, outlining implementation considerationsin the context of GPU computing. Finally, we demonstrate the effectiveness of ourimplementation with a computed example.
1 Introduction
A relatively popular approach to numerical simulation is mesh adaptivity. Notably,in general it is not possible to know a priori how to optimally distribute on adomain the numerical degrees of freedom (DOFs) e.g. the number of elements andlocal order of approximation, for finite element methods. This is especially truefor time-dependent simulations. For many difficult problems in computational fluiddynamics (CFD), computing resources will quickly be expended if DOFs are notadded satisfactorily. Fortunately, mesh adaptation strategies exist that automaticallyadd or remove DOFs solely in areas of the solution where they are needed [12].One approach is spatial refinement, or h-adaptivity, which attempts to selectivelysubdivide or merge mesh cells thereby locally increasing or decreasing solutionresolution. H-adaptivity can be implemented in varying fashions; we discuss threepossible ways: component grids, and block or cell-based refinement. Berger considered component grids in detail on structured regular meshes [1, 2].This method achieves h-adaptivity through a hierarchy of independent meshes that
A. Giuliani () • L. KrivodonovaUniversity of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canadae-mail: [emailprotected],[emailprotected]
© Springer International Publishing Switzerland 2016 435J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_40 436 A. Giuliani and L. Krivodonova
are overlay one another. An error indicator first flags elements that do not satisfysome criterion. These elements are then clustered into groups based on spatialproximity and other criteria. Then, additional rotated rectangular grids with finermesh spacing are created and overlay these groups. This is done multiple timesuntil an error tolerance is reached, creating a series of embedded grids After anappropriate number of timesteps, the mesh hierarchy is redetermined through aprocess called ‘regridding’. Since Berger [1, 2], a simplified approach has beenapplied to GPUs in [3, 10] for the shallow water equations. Other approaches includeblock and cell-based refinement strategies. In block-based refinement, cells are grouped into blocks of a predefined size. Ifany elements in a group are flagged for refinement, then all elements in that groupare refined and replaced by a number of refined blocks. This allows connectivityinformation to be stored with respect to blocks in a quadtree data structure for 2Dand octree for 3D [9, 11]. Block-based implementations have been discussed in [14]. In cell-based refinement, elements are refined individually rather than in groupswhich maintains high-resolution locality. This strategy can be implemented both onstructured and unstructured meshes, though many cell-based approaches to date dealwith Cartesian grids [7]. A trade off of refinement locality is that more connectivityinformation is required than in block-based refinement. This is because data mustbe stored on the level of elements, rather than blocks. A quadtree and octree datastructure can be used again to represent parent-child relationships for determiningmesh connectivity [5, 13]. The advantage of this strategy is that irregular boundariescan be dealt with very easily through cut-cells, higher order elements or curvedboundary conditions. GPUs are useful tools in scientific computing applications. The compute capacityof NVIDIA GPUs is quite impressive; single and double precision arithmeticthroughput can exceed 1 TFLOP/s depending on the model. When this is consideredin conjunction with the price of such devices, GPUs are an attractive scientificcomputing platform. We now focus on the main subject of this work: an efficient cell-based h-adaptive DG-GPU algorithm in NVIDIA’s CUDA C on unstructured triangularmeshes for nonlinear hyperbolic conservation laws. We begin with the derivationof a DG method for these partial differential equations (PDEs). We then outlinethe implementation details and present a computed example, examining the solver’sperformance characteristics.
2 The Discontinuous Galerkin Method
In two dimensions, hyperbolic conservation laws are PDEs of the form d u C rxy F.u/ D 0; (1) dtwith the solution u.x; t/ D .u1 ; u2 ; : : : ; uM /| defined on ˝ Œ0; T such that x D.x; y/, x 2 ˝ R2 , T is a final time and F.u/ D .F1 .u/; F2 .u// is the flux function. h-Adaptivity for the DG Method on Unstructured Meshes for GPUs 437
Additionally, the initial condition
u.x; 0/ D u0 .x/;
along with appropriate boundary conditions are applied. The DG method is a high-order method without an extensive stencil that can successfully capture shocksand discontinuities in the numerical solution of conservation laws. As opposed tothe standard finite element method, continuity at the interface between adjacentelements is not imposed. This method can be formulated by first dividing S the domain ˝ into an unstruc-tured mesh of N triangles such that ˝ D NiD1 ˝i . We note that nonconformingtessellations are permitted. In our implementation, the difference in refinement levelbetween adjacent cells must not exceed 1, see Fig. 1. We define Sp .˝i / to be the space of polynomials of degree at most p on ˝i .Additionally, fevj gjD1::Np is a set of orthonormal basis functions for Sp .˝i /, where Nprepresents the number of basis functions. The weak form of the conservation law is obtained by multiplying Eq. (1) by atest function evj 2 Sp .˝i / and integrating on an element ˝i . After integrating byparts, we obtain Z Z Z ut e vj d˝i F.u/rxy e vj d˝i C vj F.u/ ndl D 0; e (2) ˝i ˝i @˝i
where n is the outward facing normal on @˝i . Each element ˝i is mapped to the canonical triangle ˝c , having vertices at.0; 0/; .1; 0/; .0; 1/. We use the transformation 0 1 0 10 1 x xi;1 xi;2 xi;3 1rs @yA D @yi;1 yi;2 yi;3 A @ r A; (3) 1 1 1 1 s
e7 e7 e13 e3 Ω2 Ω4 Ω2 e1 e10 e8 e2 e2
Ω5 e4 Ω1 e6 e12 e11 Ω0 Ω3 e5 e0 e9 (x2,1 , y2,1 )
Fig. 1 Examples of admissible meshes: (left) two element mesh (right) ˝5 is refined yielding afive element mesh 438 A. Giuliani and L. Krivodonova
where .xi;z ; yi;z /zD1;2;3 are the three original vertices of element ˝i in physical spaceand .r; s/ are the transformed coordinates in the computational space. The Jacobianof the transformation is denoted by Ji and we require that det Ji be positive, i.e.that the vertices be ordered counter-clockwise. Vertex ordering also defines theside numbers q D 1; 2 : : : Nisides of each cell, where Nisides is the number of sidesan element has; for example si;q refers to the qth side of ˝i beginning clockwisefrom .xi;1 ; yi;1 /. In the refined mesh of Fig. 1, s2;4 is edge e11 with the indicated.x2;1 ; y2;1 /. Additionally, we map each edge of the mesh ek , i.e. si;q , to the canonical intervalIc D Œ1; 1 with the transformation 1 x x x .1 / D k;1 k;2 2 1 ; (4) y yk;1 yk;2 2 .1 C /
where is the transformed coordinate in the computational space and .xk;z ; yk;z /zD1;2are the two original vertices of ek . The determinant of the Jacobian of (4) is q 1 li;q D .xk;1 xk;2 /2 C .yk;1 yk;2 /2 : 2As for other Galerkin methods, the solution on element ˝i is approximated by PNpa linear combination of the basis functions e vj , i.e. Ui D jD1 ci;j v ej with ci;j D 1 2 M |Œci;j ; ci;j ; : : : ; ci;j ; : : : ; ci;j , the modal degrees of freedom. Finally, we separate the m
second term in (2) into line integrals along the edges of the element. It follows thatequation (2) becomes
Z d 1 ci;j D F.Ui / .Ji1 rrs vj / det Ji d˝c dt det Ji ˝c
XZ Nisides vj;q F.Ui / ni;q li;q d ; (5) qD1 I0
where vj and vj;q are values of the basis functions mapped from the physicalelement ˝i to the canonical element ˝c and from side si;q to the canonical sideI0 , respectively. Finally, ni;q is the outward facing normal of si;q . The right-handside of equation (5) is composed of two terms: a volume and surface integral.As continuity between elements is not imposed, the solution Ui is multivalued inthe surface integral. We therefore introduce a numerical flux F.Ui ; Uk / to allowinformation to be exchanged between adjacent cells ˝i and ˝k . We can evaluatethese terms exactly for a linear flux and approximately for a nonlinear flux with h-Adaptivity for the DG Method on Unstructured Meshes for GPUs 439
numerical quadrature. The equation now becomes
Z d 1 ci;j D F.Ui / .Ji1 rrs vj / det Ji d˝c dt det Ji ˝c
XZ Nisides vj;q F.Ui ; Ukq / ni;q li;q d : (6) qD1 Ic
Because equation (6) is a system of ordinary differential equations (ODEs) for thedegrees of freedom ci;j , it can be solved in time with a standard ODE solver, e.g.Runge-Kutta (RK) method. As the volume and surface integral contributions can be computed independentlyof one another, the method is predisposed to applications on highly parallelGPU architectures [6]. Developing algorithms for this computing platform is anactive area of research. An efficient h-adaptive, block-based structured mesh DG-GPU implementation has been presented in [3, 10]. Cell-based h-adaptivity onunstructured meshes in the context of GPUs can be more difficult due to theinevitable irregularity of memory accesses and operations both in time stepping andmesh adaptivity operations. We will now describe the implementation details of ourefficient h-adaptive DG-GPU algorithm in NVIDIA’s CUDA C; this implementationconsists of a time stepping and mesh adaptation module.
3 Time Stepping Details
With the time stepping module, we calculate the right-hand side of (6), without meshmodification. Every n regular timesteps, the adaptive module is executed whichrefines and coarsens select elements. The time stepping aspects of the solver areoutlined in more detail in [6]. This module consists of three compute kernels thatefficiently evaluate the right-hand side of (6) for a standard RK ODE time integrator.The first kernel, eval_volume, launches one thread per element ˝i ; each threadevaluates the volume integral terms for its allocated element. Likewise, the secondkernel, eval_surface, launches one thread per edge ek ; each thread evaluatesthe surface integral terms for its allocated edge. Finally, eval_rhs launches onethread per element and appropriately sums the volume and surface terms over themesh. An essential aspect of these final two kernels is the use of connectivity dataon the GPU.DOF storage In order fully utilize the compute capacity of the GPU, the modalDOFs must be stored in a specific fashion. See [6] for further information.Connectivity We store connectivity information in GPU DRAM memory. We storesimple pointers in linear arrays as integers, which encode the element and edge IDs˝i and ek , respectively. The bidirectional pointers are displayed in Fig. 2. Elements 440 A. Giuliani and L. Krivodonova
e0 Ω0 e1 e7 e2 e13 e3 Ω1 e3 Ω4 Ω2 e4 e1 e2 Ω2 e6 Boundary e7 Ω1 e4 e6 e12 e11 Ω3 e9 Ω0 Ω3 e11 e0 e9 Ω4 e12 e13
Fig. 2 (Left) nonconforming mesh, (right) connectivity data
point to their edges and edges point to their two elements; note that ghost cellsenforce boundary conditions. Due to the nonconforming nature of our meshes, eachelement requires Nisides pointers to its respective edges, see ˝2 in Fig. 2 (N2sides D4 ¤ 3).
4 Mesh Adaptation
The mesh adaptation module of our implementation works to optimally distributethe spatial DOFs on the computational domain. We note that work is completeddirectly in GPU video memory with minimal memory transfer between host anddevice.Tree structures A tree structure for both elements and edges is implemented,keeping track of parent-child relationships. These data structures, along with otherinformation, allow us to redetermine the mesh connectivity described in Sect. 3 afterelements and edges are refined and coarsened. In the tree, each element and edgepoint to their children; likewise, each element and edge also point to their parents. In Fig. 1, element ˝5 is refined into four new elements. We denote ˝5 as theparent of the four child elements ˝3 , ˝0 , ˝4 and ˝1 . Similar terminology is usedfor the parent edges: e5 , e8 and e10 and their children. The bidirectional pointersthat would be stored in GPU DRAM for this example are presented in Fig. 3. Wehighlight the elements and edges in the trees without children, corresponding tothose components which are active in the current mesh.Error indicator In order to determine which elements need to be refined andwhich need to be coarsened, an error indicator is used. In the literature, thereare a substantial number of indicators available. The number of times an element h-Adaptivity for the DG Method on Unstructured Meshes for GPUs 441
e0 e1 Ω0 e2 e3 Ω1 e5 e4 Ω5 e7 e9 Ω3 e8 e11 Ω2 Ω4 e10 e12 e13
Fig. 3 Tree data stored for the two sequence of meshes in Fig. 1. Elements and edges highlightedin blue are active in the current mesh
1 Ωi 0 0 -1 -1 -1 -1
Fig. 4 (Left) initial configuration around ˝i along with the values of aflag for each element,(right) mesh after refinement and smoothing; dotted line indicates the elements added
is refined and coarsened as determined from the error indicator is stored as aninteger in an array aflag. A positive integer indicates refinement, whereas anegative integer indicates coarsening. Our implementation allows a maximum levelof refinement of lmax D 5.Flag smoothing Flag smoothing is an operation which enforces that the flags inaflag do not create elements that have an interelement difference in refinementlevel greater than the maximum allowed, i.e. jli lj j 1 where li and lj arethe refinement levels of the adjacent elements ˝i and ˝j , respectively. We musttherefore modify aflag so that after the refinement and coarsening operation, anincompatible mesh is not created. For example, in Fig. 4 on the left we have the mesh configuration around anarbitrary element ˝i with value of aflag on each element. The two elements beingrefined once will induce an interelement jump in refinement level of 2 with ˝i . Thesmoothing operation will supress this jump by refining element ˝i once as shownin the mesh on the right. 442 A. Giuliani and L. Krivodonova
Refinement and coarsening After aflag is smoothed, the elements and edgesmay now be refined and coarsened. We implement refinement and coarseningkernels which modify the element and edge trees. Solution coefficients are rede-termined using an L2 projection and geometric data is adapted accordingly. Thefinal step in the mesh adaptivity algorithm is redetermining the mesh connectivitydetailed in Sect. 3. This is done in part with the element and edge trees presented inSect. 4.Memory management In refining and coarsening the mesh, we will naturally haveto add or remove data from lists, e.g. the modal degrees of freedom C. A naivesolution would be to simply replace the undesired element by a null placeholder.Unfortunately, this would lead to ‘holes’ in our arrays which is not an efficient useof memory and further would limit the effectiveness of coalesced memory accesses.We must therefore not only remove elements from arrays but also shift those thatremain so that the reduced array is dense. On GPUs, this problem does not have a simple solution. Indeed, parallelizingthis operation, called a ‘reduction’, while avoiding race conditions must be donewith great care. There are application programming interfaces (APIs) that have thistask optimized; we use CUDPP in our implementation [4]. The details of a reductionare not presented here for brevity, an involved examination of this issue can be foundin [8].
5 Computed Example
We now present an example demonstrating the effectiveness of this implementation.Consider the two dimensional advection equation
d u C rxy Œ2yu; 2xu D 0; (7) dt
on the domain ˝ D Œ1; 1 2 , subject to the initial condition .x 0:2/2 y2 u0 .x; y/ D 5 exp : (8) 2 0:152
This is called the rotating hill test problem, whereby a Gaussian pulse is advectedaround the origin; the exact solution is expressed as
u.x; y; t/ D u0 .x cos.2t/ C y sin.2t/; x sin.2t/ C y cos.2t//: (9)
This problem would benefit from h-adaptivity as much of the variation in thesolution is spatially concentrated in a small, moving region of the domain. h-Adaptivity for the DG Method on Unstructured Meshes for GPUs 443
We apply our adaptive code to this problem and advance the solution until a finaltime of T D 1, i.e. one full rotation around the origin. The DG spatial discretizationuses the upwind numerical flux and is integrated in time with an RK time integratorof appropriate order. The adaptive routines are executed every 100 regular timesteps,allowing an element to be refined a maximum of 3 times. The initial mesh wascomposed of 1080 elements; if uniformly refined three times, it would comprise69,120 elements. The numerical solution and final adaptive mesh are plotted inFig. 5 for p D 1. In Table 1, we present the computed L2 error of the final solutionfor both the adaptive and uniformly refined meshes. The advantage of h-adaptivity isapparent as only a fraction of the computational work is required to achieve an errorcomparable to that of a uniformly refined mesh. In all simulations, the refinementand coarsening subroutines comprise at most 2 % of the total runtime.
Fig. 5 Final refined mesh at T D 1 with the solution isolines for p D 1 444 A. Giuliani and L. Krivodonova
Table 1 L2 error for p D 1; 2; 3 of the h-adaptive solution, and uniformly refined solutionp Average number of elements Adaptive mesh L2 error Uniformly refined mesh L2 error1 18,195 4.684E-4 4.453E-42 24,906 4.373E-6 4.226E-63 34,785 4.514E-8 4.274E-8
6 Conclusion
In conclusion, we have presented the framework of an efficient h-adaptive imple-mentation of the DG method for hyperbolic conservation laws on GPUs. Weplan on implementing a combined hp-adaptive solver, and extending support to 3dimensional unstructured tetrahedral meshes with local time stepping.
Acknowledgements This research was supported in part by the Natural Sciences and EngineeringResearch Council of Canada (NSERC) grant 341373-07 and the NSERC CGS-M grant.
References
1. Berger, M.: Adaptive mesh refinement for hyperbolic partial differential equations. Ph.D. thesis, Stanford University (1982) 2. Berger, M.J., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53(3), 484–512 (1984) 3. Brodtkorb, A.R., Sætra, M.L., Altinakar, M.: Efficient shallow water simulations on GPUs: implementation, visualization, verification, and validation. Comput. Fluids 55, 1–12 (2012) 4. CUDA Data Parallel Primitives Library. http://cudpp.github.io/. CUDPP 2.2 5. Flaherty, J.E., Loy, R.M., Shephard, M.S., Szymanski, B.K., Teresco, J.D., Ziantz, L.H.: Adaptive local refinement with octree load balancing for the parallel solution of three- dimensional conservation laws. J. Parallel Distrib. Comput. 47(2), 139–152 (1997) 6. Fuhry, M., Giuliani, A., Krivodonova, L.: Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws. Int. J. Numer. Methods Fluids 76(12), 982–1003 (2014) 7. Gao, X.: A parallel solution-adaptive method for turbulent non-premixed combusting flows. Ph.D. thesis, University of Toronto (2008) 8. Harris, M., Sengupta, S., Owens, J.D.: Parallel prefix sum (scan) with CUDA. In: Nguyen, H., NVIDIA Corporation (eds.) GPU Gems 3. Addison-Wesley, Upper Saddle River (2008) 9. Ivan, L., Sterck, H.D., Northrup, S.A., and Groth, C.: Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids. J. Comput. Phys. 255(0), 205–227 (2013)10. Sætra, M.L., Brodtkorb, A.R., Lie, K.-A.: Efficient GPU-implementation of adaptive mesh refinement for the shallow-water equations. J. Sci. Comput. 63(1), 23–48 (2014). doi:10.1007/s10915-014-9883-4. ISSN:1573-769111. Susanto, A.: High-order finite-volume schemes for magnetohydrodynamics. Ph.D. thesis, University of Waterloo (2014) h-Adaptivity for the DG Method on Unstructured Meshes for GPUs 445
12. Wang, Z.: Adaptive High-order Methods in Computational Fluid Dynamics. Advances in Computational Fluid Dynamics. World Scientific, Singapore/Hackensack (2011)13. Zeeuw, D.L.D.: A quadtree-based adaptively-refined cartesian-grid algorithm for solution of the Euler equations. Ph.D. thesis, The University of Michigan, Ann Arbor (1993)14. Zhang, J.Z.: Parallel anisotropic block-based adaptive mesh refinement finite-volume scheme. Ph.D. thesis, University of Toronto (2011) Extending BACOLI to Solve the MonodomainModel
Elham Mirshekari and Raymond J. Spiteri
Abstract BACOLI is a numerical software package that solves systems ofparabolic partial differential equations (PDEs) in one spatial dimension. It is basedon high-order B-spline collocation and features adaptivity in space and time. Themonodomain model of cardiac electrophysiology is a multi-scale model that coupleselectrical activity in myocardial tissue at the tissue scale with that at the cellularscale. This leads to a (parabolic) reaction-diffusion PDE coupled with a set ofnonlinear (non-parabolic) PDEs that do not involve spatial derivatives. In this paper,we extend BACOLI to solve this more general class of problem, of which themonodomain model is one example. We demonstrate that the extended BACOLIsoftware package outperforms the Chaste software package, which is a powerful,widely used, and well-respected software package for heart simulation, in termsof execution time on the monodomain equation with two cell models of varyingstiffness.
1 Introduction
BACOLI is a numerical software package that solves systems of parabolic partialdifferential equations (PDEs) of the form
ut .x; t/ D f .t; x; u.x; t/; ux .x; t/; uxx .x; t//; a x b; t0 t tf ; (1a)
where u.x; t/ 2 RMu is the unknown vector function, subject to separated boundaryconditions of the form
Ba .t; u.a; t/; ux .a; t// D 0; Bb .t; u.b; t/; ux .b; t// D 0; t0 t tf ; (1b)
E. MirshekariDepartment of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, Canadae-mail: [emailprotected]. Spiteri ()Department of Computer Science, University of Saskatchewan, Saskatoon, SK, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 447J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_41 448 E. Mirshekari and R.J. Spiteri
and the initial condition
u.x; t0 / D u0 .x/; a x b: (1c)
BACOLI is one of only a few software packages to feature adaptive error controlin both space and time [18]. The software package is designed to produce anapproximate solution such that the error is estimated to be less than a user-suppliedtolerance [9]. Adaptive error control increases both the efficiency as well as thereliability of the simulation. The monodomain model is a multi-scale mathematical model for the evolutionof the electrical potential in myocardial tissue that relates the propagation of ioniccurrents at the tissue scale with their generation at the cellular scale. Because oftheir global nature, the mathematical models of the propagation at the tissue leveldepend on spatial derivatives. On the other hand, because of their local nature, themodels of a single cell do not. The monodomain model can be written as
@v Cm C Iion .s; v; t/ D r .i rv/; @t 1C (2a) @s D f .s; v; t/; @twith boundary condition
O i rv/ D 0; n:. (2b)
where v is the transmembrane potential, s is a (model-dependent) vector of cellularstates such as gating variables and ionic concentrations, Iion is the ionic currentacross the membrane per unit cell membrane area, is the area of cell membraneper unit volume, i is the conductivity, Cm is the capacitance of the cell membraneper unit area, and nO is the unit normal vector to the boundary [14]. The system (2)is supplemented by appropriate initial conditions. The cell models considered in this paper are those of Luo–Rudy [10] (consideredto be mildly stiff) and the epicardial variant of ten Tusscher and Panfilov [15](considered to be stiff). The cell models do not have any dependency on thespatial derivatives and thus cannot be solved directly using BACOLI. We extendBACOLI so that it can solve PDEs of the form (2a). The remainder of this paper unfolds as follows. A review of the BACOLIsoftware package is provided in Sect. 2. Some basic information on the of the heartis provided in Sect. 3. The performance in terms of execution time of the extendedversion of BACOLI is demonstrated on the monodomain model with two differentcell models in Sect. 4. Extending BACOLI to Solve the Monodomain Model 449
2 Review of BACOLI
This section describes some basic concepts behind the BACOLI software package,including its structure and the modifications to extend it to solve multi-scale systemslike the monodomain model.
2.1 Basic Concepts
BACOLI uses the following discretization strategy. Consider the interval Œa; b divided into Nx subintervals Œxi1 ; xi , where i D 1; 2; : : : ; Nx ; with a partition of the form
W a D x0 < x1 < : : : < xNx D b:
On subinterval i, i D 1; 2; : : : ; Nx , the unknown function is approximated by alocal polynomial. The spline s.x/ obtained from joining the local polynomials isa piecewise polynomial with certain properties; e.g., it can have several continuousderivatives [6]. The standard B-representation of a spline is as a linear combination of B-splines.A B-spline is piecewise polynomial that has minimal support with respect to itsdegree, continuity, and knot sequence. For the knot sequence X1 X2 : : : XDCMC1 , we have
X D s.x/ D Bj;M .x/aj ; jD1
where Bj;M .x/ is B-spline j of degree M, D is the number of the B-spline basisfunctions, and the coefficients aj are the unknown coefficients that are definedaccording to the choice of method, e.g., collocation . There are different types ofknots mentioned in [7]. The type that is considered in BACOLI is open uniform,defined as 8 ˆ <Xk D X1 ; ˆ if k M C 1; X Xk D constant; M C 1 k < D C 1; ˆ kC1 :̂X D X ; k D C 1; k DCMC1
where k D 1; 2; : : : ; D C M C 1; i.e., at each boundary, the multiplicity of the knotsis .M C 1/. 450 E. Mirshekari and R.J. Spiteri
With the choice of open uniform knots, the associated B-splines have twoproperties at the left boundary:
B1;M .x1 / D 1; and B01;M .x1 / D B02;M .x1 /:
Similar properties hold at the right boundary.
2.2 Structure of the BACOLI Software Package
The exact solution of (1), u.x; t/, is approximated by a linear combination of theseB-spline functions as
X D U.x; t/ D Bj;M .x/yj .t/; jD1
where yj .t/ is the vector of unknown coefficients of the B-spline j. In order todetermine yj .t/, the BACOLI software package requires U.x; t/ to satisfy (1a) ateach of the M 1 Gaussian collocation points of each subinterval [18]. The collocation equations defining fyk .t/gD kD1 are of the form
d U.l ; t/ D f .t; l ; U.l ; t/; Ux .l ; t/; U xx .l ; t//; (3) dt
where l D 1 C .i 1/.M 1/ C j; i D 1; 2; : : : ; Nx ; and j D 1; 2; : : : ; M 1.There are at most .M C1/ B-spline basis functions that do not vanish at each internal jfcollocation point [5]. Generally, on subinterval i, only the B-splines fBj;M gjDj0 do notvanish at the internal l , where j0 D .i 1/.M 1/ C 1 and jf D i.M 1/ C 2.Taking advantage of this property,
X jf U.l ; t/ D Bj;M .l /yj .t/: (4) jDj0
Inserting (4) into (3) results in the system of ODEs
X jf Bj;M .m /y0j .t/ D f .t; m ; U.m ; t/; Ux .m ; t/; U xx .m ; t//; (5) jDj0
where m D 1; 2; : : : ; M 1. Once the system of ODEs is constructed, the discretizedboundary conditions are added to this system of ODEs to form a system of initial-value differential algebraic equations (DAEs). In other words, BACOLI treats the Extending BACOLI to Solve the Monodomain Model 451
boundary conditions directly. For simplicity, on the spatial interval Œ0; 1 , at the leftboundary,
U.0; t/ D B1;M .0/y1 .t/; Ux .0; t/ D B01;M .0/y1 .t/ C B02;M .0/y2 .t/:
Similarly, at the right boundary,
U.1; t/ D BD;M .1/yD .t/; Ux .1; t/ D B0D;M .1/yD .t/ C B0D1;M .0/yD1 .t/:
Coupling the ODEs in (5) with the discretized boundary conditions gives an index-1initial-value system of DAEs of the form
Ba .t; U.0; t/; Ux .0; t// D 0;
X jf Bj;M .l /y0j .t/ D f .t; l ; U.l ; t/; Ux .l ; t/; Uxx .l ; t//; (6) jDj0
Bb .t; U.1; t/; Ux .1; t// D 0;
where l , l D 1; 2; : : : ; M 1, is collocation point l in subinterval i, i D 1; 2; : : : ; Nx .This is a system of DMu equations in DMu unknowns. The system (6) is alsosupplemented by appropriate initial conditions. This system of DAEs is integrated with respect to the independent variable t inorder to determine the unknown coefficients yj .t/. The variable-stepsize, variable-order DAE solver DASSL [13] is applied to integrate the resulting initial-valueDAEs with respect to time. After solving (6) with DASSL, an approximate solutionis available. The BACOLI software package has two options for the estimation of thespatial error. One is through the use of a super-convergent interpolant (SCI) [1], andthe other is a low-order interpolant (LOI) [2]. The SCI option generates a numericalsolution in the space of polynomials of degree M and an error estimate for thatsolution. On the other hand, the LOI option generates a numerical solution in thespace of polynomials of degree M C 1 and an error estimate for that solution.Because one can consider the SCI option to be a standard error control mode, weuse SCI in this paper. More specifically, in the SCI option, at any time step, based on the solutioncomputed through the solver DASSL, U.x; t/, which is of degree M, an interpolant, Q t/, of degree .M C 1/ is constructed using some super-convergent values. TheU.x;difference between U.x; t/ and U.x; Q t/ gives an asymptotic estimate of the errorfor the approximate solution U.x; t/. Finally, BACOLI computes a spatial errorestimate, and if necessary, performs a spatial adaptation in the form of remeshing. 452 E. Mirshekari and R.J. Spiteri
2.3 The Extended BACOLI Software Package
The BACOLI software package has been extended to solve system of equationsgiven by
ut .x; t/ D f .t; x; v.x; t/; u.x; t/; ux .x; t/; uxx .x; t//; vt .x; t/ D g.t; x; v.x; t/; u.x; t//; a x b; t t0 ;
subject to the separated boundary conditions
Ba .t; v.a; t/; u.a; t/; ux .a; t// D 0; Bb .t; v.b; t/; u.b; t/; ux .b; t// D 0;
and the initial conditions
u.x; t0 / D u0 .x/; v.x; t0 / D v0 .x/;
where u.x; t/ and v.x; t/ are the unknown scalar and vector functions of sizes 1 andMv , respectively. Full details on the modifications and a detailed list and descriptionof the modified subroutines may be found in [12].
3 Electrophysiology Background
The heart can be modelled as an electric dipole with separated positive and negativecharges that create an electrical field. The body behaves like a conductor when itis exposed to such an electrical field. In each heart cycle, an electrical current runsthrough the body. This electrical current causes an electrical potential at each pointof the body, giving rise to a time-varying potential difference throughout the body.The potential variations during a heart cycle represent the electrical activation ofthe heart muscle cells. The electrocardiogram (ECG) is a time-dependent readingof specific potential differences with respect to a reference potential [14]. Figure 1represents recorded variations of the transmembrane potential as a function of timefor the cell model of Bondarenko et al. [3, 4]. A control volume is a fixed regionconsidered in space in order to study the energies or fluids that pass its boundary [8].Continuum modelling of a tissue is a method of modelling based on the concept ofthe control volume, in which specific sample points are considered that relate toa quantity that is the average of its neighbouring cells. In continuum modelling,instead of considering individual cells, only those sample points are considered. The monodomain model is a continuum-based, multi-scale mathematical modelfor the electrical activity in heart tissue [14]. The standard formulation of thismodel is (2). In this paper, the constants , i , and Cm are taken to be 1400 cm1 , Extending BACOLI to Solve the Monodomain Model 453
20 Transmembrane potential (mV) 0
–20
–40
–60
–80
–100 0 10 20 30 40 50 60 70 Time (ms)
Fig. 1 Transmembrane potential as a function of time steps for the cell model of Bondarenkoet al. [3, 4]
1:75 mS/cm, and 1 F/cm2 , respectively, as per the default values from the Chastesoftware package (see below).
4 Numerical Results
This section presents the numerical results of solving the monodomain equationcoupled with the Luo–Rudy I cell model [10] and the epicardial variant of the 2006cell model of ten Tusscher and Panfilov [15]. BACOLI accepts two tolerances, arelative and an absolute tolerance. The boundary conditions for both cases are insulating boundary conditions on adomain chosen as Œ0; 1
vx .0; t/ D 0; vx .1; t/ D 0:
In order to measure the errors of the numerical solution obtained with theextended BACOLI software package, a reference solution is generated withChaste, a powerful, widely used, and well-respected software package for heartsimulation. This software applies a semi-implicit method [19] to the monodomainmodel (2a), and the cell models (which reduce to ordinary differential equations ateach discrete point in space) are solved with Heun’s method. The reference solutionis generated by comparing increasingly accurate solutions for which the time step ishalved and the number of spatial mesh points is doubled. The process of generatingthese increasingly accurate solutions continues until a desired number of matchingdigits are obtained. In our numerical experiments, the comparison is made at 21equally spaced time steps in the temporal domain and 101 equally spaced spatialpoints. 454 E. Mirshekari and R.J. Spiteri
In order to measure the accuracy and efficiency of any numerical method, onecan compute an average of the error at Nt points in the temporal domain t 2 Œt0 ; tf and at Nx points in the spatial domain, i.e., the average is over N D Nt Nx space-timepoints. After generating the reference solution for all N points, the mixed root meansquare (MRMS) [11] error is computed as v u N u1 X vO i vi 2 eMRMS Dt ; N iD1 1 C jvO i j
where vO i and vi , respectively, denote the reference solution and the numericalsolution for the voltage at space-time point i.
4.1 Monodomain Equation with Luo–Rudy I Cell Model
The Luo–Rudy I cell model consists of 8 variables and simulates the action potentialof a guinea pig ventricular cell [10]. The initial values are taken from [16]. Table 1shows some accuracy results. The first column contains the spatial points for theoutput. The second column represents the reference solution generated by Chaste.This reference solution has 4 matching digits and MRMS error 4:86 104 %between two consecutive phases. The third column represents the solution generatedwith the extended BACOLI at tolerance 1 108 and SCI spatial error estimate. Theresults are reported at the time tf D 5 ms. The matching digits are 7 at x D 0:0 and6 elsewhere. For the extended BACOLI software package, the absolute and relative tolerancesare 1:25 102 , and the number of collocation points is 3. These tolerances areroughly the largest that gives a numerical solution with MRMS error of about5 %. Because it is possible to achieve MRMS error of about 5 % with relativelylarge tolerances, it is most efficient for the extended BACOLI to use a relativelysmall number of collocation points. The initial mesh is set to have 4 uniformsubintervals in order to minimize any bias in the extended BACOLI softwarepackage determining a suitable mesh. The resulting numerical solution has MRMSerror 4:15 %, and that for Chaste is almost the same. The results show that with
Table 1 Solution of the x Reference solution Extended BACOLI solutionLuo–Rudy I for the voltage(mV) after 5 ms, obtained 0.0 19.72929206746164 19.7292917671303982with Chaste and the 0.2 18.86929313940373 18.8692734805216737extended BACOLI software 0.4 18.85464593756643 18.8546280990272841package (SCI scheme and 0.6 19.29529722142660 19.2952427151648926tolerance 1 108 ) 0.8 16.61051308307339 16.6104957169274741 1.0 20.52934860573765 20.5293347833165818 Extending BACOLI to Solve the Monodomain Model 455
the extended BACOLI software package it takes 0:18 s and with Chaste it takes0:50 s. Therefore, in this case there is a speed-up of a factor of around three inthe extended BACOLI software package. Such a saving becomes significant whensuch simulations are run many times sequentially as may be required for parameteror other types of optimization; for example, running the Luo–Rudy I model amillion times consecutively with the extended BACOLI software package wouldtheoretically save about 320; 000 s or 3:7 days on a six-day run.
4.2 Monodomain Equation with Ten Tusscher Cell Model
The epicardial variant of the model of ten Tusscher and Panfilov [15] consists of19 variables and simulates the human ventricle [15]. The initial values are takenfrom [17]. Table 2 shows some accuracy results. The first column contains thespatial points for the output. The second column represents the reference solutiongenerated by Chaste. This reference solution has 3 matching digits and MRMSerror 3:92 103 % between two consecutive phases. The third column representsthe solution generated with the extended BACOLI at tolerance 1 108 and SCIspatial error estimate. The results are reported at the time tf D 5 ms. The matchingdigits vary between 3 and 6. For the extended BACOLI software package, the absolute and relative tolerancesare 1 102 , and the number of collocation points is 3. These tolerances are roughlythe largest that gives a numerical solution with MRMS error of about 5 %. Forconsistency with the previous experiment, the initial mesh size is taken to be uniformwith 4 subintervals. The resulting numerical solution has MRMS error 4:53 % andthat for Chaste is 5:60 %. The results show that with the extended BACOLIsoftware package it takes 1:16 s and with Chaste it takes 2:58 s. Therefore, in thiscase there is a speed-up of a factor of around two in the extended BACOLI softwarepackage.
Table 2 Solution of the x Reference solution Extended BACOLI solutionepicardial variant of themodel of ten Tusscher and 0.0 15.73671511358962 15.7366969196549213Panfilov [15] for the voltage 0.2 15.88153751011188 15.8815217931186137(mV) after 5 ms, obtained 0.4 16.07890988375955 16.0788979751286440with Chaste and the 0.6 16.40115130209244 16.4010866279204457extended BACOLI software 0.8 17.36927102074231 17.3668857822675768package (SCI scheme andtolerance 1 108 ) 1.0 20.73937330189026 20.7486443492494281 456 E. Mirshekari and R.J. Spiteri
5 Conclusions
Modifications to the BACOLI software package were made to extend it to solvea more general class of problems. The extended BACOLI software package cansolve problems in one dimension that consist of a scalar parabolic PDE coupled withany number of PDEs that have no dependency on spatial derivatives. The practicalapplication of such problems described in this paper is the mathematical model ofthe electrical activity of the heart known as the monodomain model. The examplesconsidered in this paper are the monodomain equation coupled with the Luo–Rudy Icell model and the monodomain equation coupled with the epicardial variant of thecell model of ten Tusscher and Panfilov [15]. For these examples, we demonstrateda speed-up of factors of 2–3 in execution time of the extended BACOLI softwarepackage in comparison with the Chaste software package for accuracies typicallyrequired in practice.
References
1. Arsenault, T., Muir, P.H., Smith, T.: Superconvergent interpolants for efficient spatial error estimation in 1D PDE collocation solvers. Can. Appl. Math. Q. 17(3), 409–431 (2009) 2. Arsenault, T., Smith, T., Muir, P.H., Pew, J.: Asymptotically correct interpolation-based spatial error estimation for 1D PDE solvers. Can. Appl. Math. Q. 20(3), 307–328 (2012) 3. Auckland Bioengineering Institute.: The CellML project. http://www.cellml.org/ 4. Bondarenko, V.E., Szigeti, G.P., Bett, G.C.L., Kim, S.J., Rasmusson, R.L.: Computer model of action potential of mouse ventricular myocytes. Am. J. Physiol.-Heart C. 287(3), H1378– H1403 (2004) 5. de Boor, C.: A Practical Guide to Splines. Springer, New York (1978) 6. de Boor, C., The MathWorks Inc.: Spline Toolbox User’s Guide. The MathWorks, Inc., Natick (1999) 7. Dodgson, N.: B-splines. http://www.cl.cam.ac.uk/teaching/2000/AGraphHCI/SMEG/node4. html (Aug 2013) 8. Engineers Edge LLC.: Control volume – fluid flow. http://www.engineersedge.com/fluid_flow/ control_volume.htm (Feb 2014) 9. Holder, D., Huo, L., Martin, C.F.: The Control of Error in Numerical Methods. Springer, New York (2007)10. Luo, C., Rudy, Y.: A model of ventricular cardiac action potential. Circ. Res. 68(6), 1501–1526 (1991)11. Marsh, M.E., Torabi Ziaratgahi, S., Spiteri, R.J.: The secrets to the success of the Rush – Larsen method and its generalizations. IEEE Trans. Bio-Med. Eng. 59(9), 2506–2515 (2012)12. Mirshekari, E.: Extending BACOLI to solve multi-scale problems. Master’s thesis, Department of Mathematics and Statistics, University of Saskatchewan (2014)13. Petzold, L.R.: A description of DASSL: a differential/algebraic system solver. Technical report, Sandia National Labs., Livermore (1982)14. Sundnes, J., Lines, G.T., Cai, X., Nielsen, B.F., Mardal, K.A., Tveito, A.: Computing the Electrical Activity in the Heart. Springer, Berlin (2006)15. ten Tusscher, K.H.W.J., Panfilov, A.V.: Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291(3), 1088–1100 (2006) Extending BACOLI to Solve the Monodomain Model 457
16. The CellML Project.: A model of the ventricular cardiac action potential. http://models.cellml. org/exposure/2d2ce7737b42a4f72d6bf8b67f6eb5a2/luo_rudy_1991.cellml/@@cellml_ codegen/F77 (Feb 2014)17. The CellML Project: Alternans and spiral breakup in a human ventricular tissue model (epicardial model). http://models.cellml.org/exposure/a7179d94365ff0c9c0e6eb7c6a787d3d/ ten_tusscher_model_2006_IK1Ko_epi_units.cellml/@@cellml_codegen/F77 (Apr 2014)18. Wang, R., Keast, P., Muir, P.H.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50(2), 239–260 (2004)19. Whiteley, J.P.: An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans. Bio-Med. Eng. 53(11), 2139–2147 (2006) An Analysis of the Reliability of Error ControlB-Spline Gaussian Collocation PDE Software
Paul Muir and Jack Pew
Abstract B-spline Gaussian collocation software for the numerical solution ofPDEs has been widely used for several decades. BACOL and BACOLI are recentlydeveloped packages of this class that provide control of estimates of the temporaland spatial errors of the numerical solution through the use of adaptive time-stepping/adaptive time integration method order selection and adaptive spatial meshrefinement. Previous studies have investigated the performance of the BACOLand BACOLI packages, primarily with respect to work-accuracy measures. Inthis paper, we investigate the reliability of the BACOL and BACOLI packages,focusing on the relationship between the requested tolerance and the accuracyachieved. In particular, we consider the effect, on the reliability of the software,of (i) the degree of the piecewise polynomials employed in the representation of thespatial dependence of the approximate solution, (ii) the type of spatial error controlemployed, and (iii) the type of spatial error estimate computed.
1 Introduction/Background
B-spline Gaussian collocation software for the numerical solution of PDEs has beenwidely used for several decades. Although there has been some development ofsoftware of this type for 2D PDEs, see, e.g., [9] and references within, softwarefor the 1D case is at a more advanced state, and it is this case that we focus on inthis paper. (However, since the approach for the 2D case considered in [9] buildson the numerical algorithms used in the 1D case, the questions we consider in thispaper will be relevant for the 2D case.) A characterizing feature of such softwareis the representation of the numerical solution as a linear combination of knownB-spline [4] basis functions (piecewise polynomials of a given degree p) in spacewithin unknown time-dependent coefficients. The B-spline basis is chosen to have
P. Muir () • J. PewSaint Mary’s University, Halifax, NS, B3H 3C3, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 459J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_42 460 P. Muir and J. Pew
C1 -continuity and depends on a spatial mesh, fxi g; i D 0; : : : ; NINT, that partitionsthe spatial domain. The approximate solution has the form
X NC U.x; t/ D yi .t/Bi .x/; (1) iD1
where yi .t/ is the time-dependent coefficient of the i-th B-spline basis function,Bi .x/, and NC D NINT.p 1/ C 2. The B-spline coefficients are determined bysolving collocation conditions, obtained by requiring the approximate solution toexactly satisfy the PDEs at the images of the Gauss points of a given order oneach subinterval of the spatial mesh, and additional equations that depend on theboundary conditions. Assuming a system of PDEs of the form ut .x; t/ D f t; x; u.x; t/; ux .x; t/; uxx .x; t/ ; a x b; t t0 ; (2)
with initial conditions
u.x; t0 / D u0 .x/; a x b; (3)
and boundary conditions bL t; u.a; t/; ux .a; t/ D 0; bR t; u.b; t/; ux .b; t/ D 0; t t0 ; (4)
the collocation conditions are a system of ODEs the form U t .l ; t/ D f t; l ; U.l ; t/; U x .l ; t/; U xx .l ; t/ ; (5)
for l D 2; : : : ; NC 1. The collocation points are l D xi1 C hi j ; l D 1 C p1.i 1/.p 1/ C j; i D 1; : : : ; NINT; j D 1; : : : ; p 1; fi giD1 are the images ofthe Gauss points on Œ0; 1 , and hi D xi xi1 . (Thus, kcol, the number of collocationpoints per subinterval, is p 1.) About 35 years ago, the B-spline Gaussian collocation package, PDECOL [10]was developed. The B-spline basis was implemented using the de Boor B-splinepackage [4] and the ODE solver, GEARIB [6], was used to solve the collocationconditions (5) together with the time differentiated boundary conditions. Temporalerror control was provided by the ODE solver. PDECOL used a fixed spatial meshwhich meant that no control of the spatial error was provided. About a decade later, a modification of PDECOL, called EPDCOL [8] wasreleased; EPDCOL replaced the band matrix solver employed by PDECOL withan almost block diagonal (ABD) solver, COLROW [5], more appropriate for thestructure of the linear systems arising from the B-spline collocation process. Savingsin execution time of about 50 % were obtained. The next update to this family of PDE solvers, BACOL [18], released a littleover a decade later, was a new implementation of the B-spline Gaussian collocation Reliability of Error Control B-Spline Gaussian Collocation Software 461
algorithm, with several additional features. BACOL employed a Differential-Algebraic Equation (DAE) solver, DASSL [3], so that the boundary conditions (4)could be treated directly together with the ODEs (5) arising from the collocationprocess. Temporal error control was provided by DASSL. The most significant newfeature was the implementation of spatial error estimation and control. The spatialerror estimates were obtained by computing a second (higher order) collocation N t/, based on B-splines of degree p C 1, which was then used togethersolution, U.x;with U.x; t/ to compute a set of spatial error estimates of the form s Z 2 b Uj .x; t/ U N j .x; t/ Ej .t/ D dx; j D 1; : : : ; NPDE; (6) a ATOLj C RTOLj jUj .x; t/j
where Uj .x; t/ is the jth component of U.x; t/, U N j .x; t/ is the jth component of NU.x; t/, ATOLj and RTOLj are absolute and relative tolerances corresponding to thejth component of the solution, and the number of PDEs is NPDE. At the end of eachtime step, t, the collocation solution, U.x; t/ was accepted if
max Ej .t/ 1: (7) 1 jNPDE
Otherwise, the time step was rejected and BACOL computed a second set of spatialerror estimates of the form, v uNPDE Z 2 uX xi Uj .x; t/ U N j .x; t/ OEi .t/ D t dx; i D 1; : : : ; NINT; jD1 xi1 ATOLj C RTOLj jUj .x; t/j (8)which were then used within an adaptive mesh refinement (AMR) process forwhich the goal was to determine a spatial mesh for the current time such thatthe estimated spatial errors were (i) approximately equidistributed over the meshsubintervals and (ii) less than the user tolerance [20]. BACOL was shown to havesuperior performance compared to a number of comparable packages, especially forproblems with sharp moving layers and higher accuracy requirements [19]. In BACOL, the spatial error estimate is used to control the error in U.x; t/, theapproximate solution returned by the code. We refer to this as Standard (ST) errorcontrol. A minor modification of BACOL would allow it to return U.x; N t/ and inthis case, the computation of the returned solution would be controlled based on anerror estimate for a collocation solution that is of one lower order. We refer to this asLocal Extrapolation (LE) error control; see, e.g., [7], in the context of Runge-Kuttaformula pairs for the numerical integration of ODEs. The most recent update to this family of solvers is a modification of BACOLknown as BACOLI [12]. The key feature of BACOLI is the avoidance of the com- N t/, which represented a substantial cost within BACOL. BACOLIputation of U.x;instead computes only one collocation solution and one low cost interpolant that isused in the computation of the spatial error estimate. There are two options available 462 P. Muir and J. Pew
for the interpolant. One involves a superconvergent interpolant (SCI) [1] which isbased on the presence of certain points within the spatial domain where U.x; t/ isof at least one order of accuracy higher. The SCI is of the same order of accuracy N t/ and leads to a spatial error estimate for U.x; t/, similar to the situation inas U.x;BACOL. This option therefore provides ST error control. The other option involvesa lower order interpolant (LOI) [2], of one order of accuracy lower than U.x; t/. TheLOI is based on interpolation of U.x; t/ at certain points such that the interpolationerror of the LOI agrees asymptotically with the error of a collocation solution of oneorder lower than U.x; t/. This option therefore provides LE error control. BACOLIhas been shown to be about twice as fast as BACOL [12]. When considering the effectiveness of a numerical software package, the mostcommon analysis involves work-precision (i.e., work-accuracy) diagrams that showthe relationship between the CPU time and the accuracy. A number of such studiesinvolving BACOL and BACOLI have been performed; see, e.g., [12, 19]. However,an equally important measure of the quality of a software package is its reliability ,and a key component of this measure involves assessing the relationship between thetolerance requested and the accuracy achieved. This depends on the quality of theerror estimates and the algorithms that adapt the computation to attempt to controlthe error estimates with respect to the user-specified tolerances. The purpose of this paper is to present an experimental analysis of the reliabilityof the error control PDE solvers BACOL and BACOLI on a standard test problemchosen from the literature. (Additional results of this type for several other testproblems are reported in [11] and some preliminary results of this type for BACOLwere reported in [17].) In particular, we are interested in how the choice of thedegree, p, of the B-spline basis and the choice of error control mode (ST or LE)impact on the reliability of the solvers. We also examine the effectiveness of theinterpolation-based spatial error estimates provided by the SCI and LOI schemes.
2 Numerical Results
In this section we present numerical results for four code combinations identified inthe previous section:• BACOL with ST error control : BAC/ST• BACOL with LE error control : BAC/LE• BACOLI with the SCI scheme/ST error control : SCI/ST• BACOLI with the LOI scheme/LE error control : LOI/LE We consider the One Layer Burgers Equation (OLBE) with D 104 :
ut D uxx uux ; (9) Reliability of Error Control B-Spline Gaussian Collocation Software 463
with boundary conditions at x D 0 and x D 1 (t > 0) and an initial condition att D 0 .0 x 1/ such that the exact solution is ! 1 1 1 x 2t 4 u.x; t/ D tanh : 2 2 4
The solution has a sharp layer region around x D 0:25 when t D 0. As t goes from0 to 1, the layer moves to the right and is located around x D 0:75 when t D 1. Aplot of the solution for D 104 is given in [12]. We apply each of the four code combinations identified above to the OLBEwith D 104 , for a range of kcol values (recall that the degree of the piecewisepolynomials in space is p D kcol C 1) and for 91 tolerance values (with ATOL1 DRTOL1 ) uniformly distributed from 101 to 1010 . For each experiment, we reportthe L2 -norm of the error at the final time, Tout D 1. Additional results, includingresults for other test problems, are given in [11].
2.1 Reliability vs. kcol
In this subsection, we consider, for the LOI/LE code, the relationship between theerror and the tolerance, for kcol D 3; 5; 7; 9. We provide plots of the tolerance vs.the error and plots of the tolerance vs. the error tolerance ratio (ETR), i.e., the errordivided by the tolerance. See Figs. 1, 2, 3, and 4. In Fig. 1a, we see that there is goodcorrelation between the requested tolerance and the achieved accuracy; the pointsare clustered around the reference line corresponding to equal values of the toleranceand error. In Fig. 1b, considering the horizontal reference line corresponding to anETR value of 1, we see that the ETR values vary over the range of 91 tolerancevalues, with some errors greater than the tolerance and some less than the tolerance.The other horizontal line represents the average of the ETR values. This figure also
Fig. 1 (a) Tolerance vs. Error, LOI/LE, kcol D 3, (b) Tolerance vs. ETR, LOI/LE, kcol D 3 464 P. Muir and J. Pew
Fig. 2 (a) Tolerance vs. Error, LOI/LE, kcol D 5, (b) Tolerance vs. ETR, LOI/LE, kcol D 5
Fig. 3 (a) Tolerance vs. Error, LOI/LE, kcol D 7, (b) Tolerance vs. ETR, LOI/LE, kcol D 7
Fig. 4 (a) Tolerance vs. Error, LOI/LE, kcol D 9, (b) Tolerance vs. ETR, LOI/LE, kcol D 9 Reliability of Error Control B-Spline Gaussian Collocation Software 465
shows that maximum ETR is 5.99, the minimum is 0.08, and that on average theerror is 1.86 times the tolerance. Figures 2, 3, and 4, give similar results for thekcol D 5; 7; 9 cases, and show maximum, minimum, and average ETR values of{3.35, 0.15, 1.13}, {3.78, 0.01, 1.03}, and {5.72, 0.01, 0.90}, respectively. Similarresults are reported in [11] for the other code combinations and test problems.
2.2 Reliability vs. Error Control Mode
In this subsection we consider the BAC/ST and BAC/LE codes for the kcol D 5case. From the results for these codes we can assess the effect that the error controlmode has on reliability since this is the only difference between these codes. FromFigs. 5 and 6 we can see that the two codes appear to have quite similar performance.For the BAC/ST code, the maximum, minimum, and average ETR values are 4.49,0.28, and 1.79, while for the BAC/LE code, the corresponding values are 4.49,
Fig. 5 (a) Tolerance vs. Error, BAC/ST, kcol D 5, (b) Tolerance vs. ETR, BAC/ST, kcol D 5
Fig. 6 (a) Tolerance vs. Error, BAC/LE, kcol D 5, (b) Tolerance vs. ETR, BAC/LE, kcol D 5 466 P. Muir and J. Pew
Fig. 7 (a) Tolerance vs. Error, SCI/ST, kcol D 5, (b) Tolerance vs. ETR, SCI/ST, kcol D 5
0.18, and 2.01. Similar results are reported in [11] for the other kcol values andtest problems.
2.3 Reliability vs. Error Estimation Scheme
In this section we compare BAC/ST with SCI/ST and BAC/LE with LOI/LE, forkcol D 5. This will allow us to assess the effectiveness of the interpolation-based error estimation schemes. The plots for the SCI/ST code are given in Fig. 7.A comparison of Figs. 5 and 7 show that the BAC/ST and SCI/ST codes havecomparable reliability, while a comparison of Figs. 6 and 2 show that the BAC/LEand LOI/LE codes also have comparable reliability. From Fig. 7b, we see that forthe SCI/ST code the maximum ETR is 6.85, the minimum ETR is 0.10, and theaverage is 1.63. Generally similar results for the other kcol values and test problemsare reported in [11]. (The only exception, as documented in [12] and [11], is for theSCI/ST, kcol D 3 case, where issues associated with the quality of its error estimateshave been observed.)
3 Discussion/Conclusions/Future Work
Several conclusions can be drawn from the results presented in the previoussection:• The codes considered in this paper appear to be reliable; the error is generally well correlated with the requested tolerance. We see that in almost all cases the error is within the correct order of magnitude of the requested tolerance. Furthermore the error is, on average, a small multiple of the requested tolerance. This suggests that the error estimates computed by the codes are generally of Reliability of Error Control B-Spline Gaussian Collocation Software 467
good quality and that the new SCI and LOI error estimation schemes are of comparable quality to the original error estimation scheme employed by BACOL.• The relationship between the error and tolerance for each of the two versions of BACOL that employ different error control modes (ST vs. LE) is quite similar. This suggests that the choice of error control mode does not have a strong impact on the reliability of the solver.• The degree p of the B-spline basis as specified by the choice of kcol (p D kcolC1) does appear to have an impact on the reliability. A comparison of the results for the kcol D 3; 5; 7, and 9 cases shows that as kcol is increased, the reliability of the codes tends to increase; that is, the error is, on average, almost equal to the tolerance for higher kcol values. For the kcol D 9 case, the error is often significantly less than the tolerance. (Of course, while it is desirable for the error to be about the same size as or even slightly less than the tolerance, if the error is too much less than the tolerance this can lead to inefficiency in the computation.) While the entire family of software packages discussed in this paper leave kcolas a parameter that must be chosen by the user, it is in fact not clear how tomake an appropriate choice of this parameter for a given problem. The correlationbetween higher kcol values and improved reliability that can be seen from theresults presented in this paper, as well as earlier results investigating the relationshipbetween the choice of kcol, the tolerance, and the efficiency of the solvers [12], willprovide a basis for further analysis that will allow us to develop a new release ofBACOLI in which the code itself will choose kcol appropriately to improve theefficiency and reliability of the computation. Since BACOL and BACOLI employ DASSL to compute the time-dependent B-spline coefficients, the overall reliability of the computations performed by thesecodes is dependent on the reliability of DASSL. An example of the relationshipbetween the tolerance and the error for DASSL can be seen in Figure 1 of [16];it shows generally good correlation between the error and the tolerance, but withvariations in the ETR of approximately one order of magnitude in either direction,similar to those reported here for BACOL and BACOLI. Furthermore, for DASSL and the codes considered in this paper, the relationshipbetween the tolerance and the error is not particularly smooth. That is, while thecorrelation between a given tolerance and the corresponding error is generally good(i.e., the error is, on average, within a small multiple of the tolerance), a smallchange in the tolerance does not necessarily lead to a similar small change in theerror. This is related to what is referred to as the “conditioning” or “stability” ofthe software [16]. The papers [14, 15], and [13] describe how control theoretictechniques can be employed within ODE or DAE software to provide an improvedcondition number for the software. We plan to introduce these type of techniquesinto the BACOL and BACOLI packages. Finally, as mentioned earlier, since the approach for the 2D case considered in[9] builds on the 1D case, we expect the results of this paper to be relevant for thenumerical treatment of 2D PDEs. 468 P. Muir and J. Pew
References
1. Arsenault, T., Smith, T., Muir, P.H.: Superconvergent interpolants for efficient spatial error estimation in 1D PDE collocation solvers. Can. Appl. Math. Q. 17, 409–431 (2009) 2. Arsenault, T., Smith, T., Muir, P.H., Pew, J.: Asymptotically correct interpolation-based spatial error estimation for 1D PDE solvers. Can. Appl. Math. Q. 20, 307–328 (2012) 3. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1989) 4. de Boor, C.: A Practical Guide to Splines. Volume 27 of Applied Mathematical Sciences. Springer, New York (1978) 5. Díaz, J.C., Fairweather, G., Keast, P.: Algorithm 603. COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination. ACM Trans. Math. Softw. 9(3), 376–380 (1983) 6. Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice- Hall, Englewood Cliffs (1971) 7. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Volume 8 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993) 8. Keast, P., Muir, P.H.: Algorithm 688: EPDCOL: a more efficient PDECOL code. ACM Trans. Math. Softw. 17(2), 153–166 (1991) 9. Li, Z., Muir, P.H.: B-spline Gaussian collocation software for two-dimensional parabolic PDEs. Adv. Appl. Math. Mech. 5, 528–547 (2013)10. Madsen, N.K., Sincovec, R.F.: Algorithm 540: PDECOL, general collocation software for partial differential equations. ACM Trans. Math. Softw. 5(3), 326–351 (1979)11. Muir, P.H., Pew, J.: Tolerance vs. error results for a class of error control B-spline Gaussian collocation PDE solvers. Saint Mary’s University, Department of Mathematics and Computing Science Technical Report Series. http://cs.smu.ca/tech_reports (2015)12. Pew, J., Li, Z., Muir, P.H.: A computational study of the efficiency of collocation software for 1D parabolic PDEs with interpolation-based spatial error estimation. Saint Mary’s University, Dept. of Mathematics and Computing Science Technical Report Series. http://cs.smu.ca/tech_ reports (2013)13. Pulverer, G., Söderlind, G., Weinmüller, E.: Automatic grid control in adaptive BVP solvers. Numer. Algorithms 56(1), 61–92 (2011)14. Söderlind, G.: Digital filters in adaptive time-stepping. ACM Trans. Math. Softw. 29(1), 1–26 (2003)15. Söderlind, G., Wang, L.: Adaptive time-stepping and computational stability. J. Comput. Appl. Math. 185(2), 225–243 (2006)16. Söderlind, G., Wang, L.: Evaluating numerical ODE/DAE methods, algorithms and software. J. Comput. Appl. Math. 185(2), 244–260 (2006)17. Wang, R.: High order adaptive collocation software for 1-D parabolic PDEs. Ph.D. thesis, Dalhousie University (2002)18. Wang, R., Keast, P., Muir, P.H.: BACOL: B-spline Adaptive COLlocation software for 1D parabolic PDEs. ACM Trans. Math. Softw. 30(4), 454–470 (2004)19. Wang, R., Keast, P., Muir, P.H.: A comparison of adaptive software for 1D parabolic PDEs. J. Comput. Appl. Math. 169(1), 127–150 (2004)20. Wang, R., Keast, P., Muir, P.H.: A high-order global spatially adaptive collocation method for 1-D parabolic PDEs. Appl. Numer. Math. 50(2), 239–260 (2004) On the Simulation of Porous Media Flow Usinga New Meshless Lattice Boltzmann Method
S. Hossein Musavi and Mahmud Ashrafizaadeh
Abstract We propose a new meshless lattice Boltzmann method (MLLBM) forthe simulation of nearly incompressible flows in porous media. As for the standardlattice Boltzmann method, the collision and the streaming operators are split. Whilethe collision equation remains unaltered, the streaming equation is discretized usingthe Lax-Wendroff scheme in time, and the meshless local Petrov-Galerkin schemein space. The Poiseuille flow is solved to validate the numerical scheme. The methodis then used to simulate a flow in a porous medium. Although in general, meshlessmethods suffer from high computational costs, the present method shows promisingaccuracy and performance at a lower memory and computational costs for complexgeometries, when compared with standard lattice Boltzmann methods.
1 Introduction
Since the beginning of the development of the lattice Boltzmann method (LBM),a great number of studies have been conducted to extend the capabilities of thestandard lattice Boltzmann method to handle nonuniform and unstructured grids[1–6]. A major trend in these studies is to apply some standard numerical tech-niques, such as the finite difference (FD), the finite volume (FV), and the finiteelement (FE) methods, for the discretization of the Boltzmann equation. However,for geometrically complex problems, such as flows in porous media, where the timeand computational requirements of creating good quality meshes are significant, theidea of developing a meshless lattice Boltzmann solver naturally arises. In this study, we implement our newly proposed meshless lattice Boltzmannmethod [7] for the simulation of nearly incompressible flows in porous media.In our formulation, we split the collision and the advection equations followingthe standard lattice Boltzmann method [8]. The collision equation is the same asfor the standard LBM, however, the advection equation is discretized using the
S.H. Musavi • M. Ashrafizaadeh ()Department of Mechanical Engineering, Isfahan University of Technology,Isfahan 8415683111, Irane-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 469J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_43 470 S.H. Musavi and M. Ashrafizaadeh
Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin (MLPG)method in space. The meshless feature of the proposed method makes it a more flexible latticeBoltzmann solver, especially for cases where using the usual mesh generationtechniques introduces significant numerical errors into the solution, or whereimproving the mesh quality is a complex and time consuming process. Thesefeatures of the present method are well illustrated in a number of test cases describedin the next sections.
2 Formulation
The discrete Boltzmann equation (DBE) with the BGK collision approximation is
@fi @fi 1 eq C ci;˛ D . fi fi /; i D 1; : : : ; nQ ; (1) @t @x˛
where fi is the particle distribution function, is the relaxation time towardsequilibrium, nQ is the number of discrete microscopic velocities and ci is thediscrete microscopic velocity. For the D2Q9 lattice we have
c0 D 0 ; ci D cos.i 1/ ex C sin.i 1/ ey ; i D 1; 3; 5; 7 ; 4 4 p ci D 2Œcos.i 1/ ex C sin.i 1/ ey ; i D 2; 4; 6; 8 ; 4 4 (2) eqand fi is the equilibrium distribution, that is ! eq ci;˛ u˛ .ci;˛ u˛ /2 u2˛ fi D ti 1C C : (3) c2s 2c4s 2c2s
As usual, we break Eq. (1) into two steps, namely the collision step:
1 eq fQi D fi . fi fi /; (4) and the advection step:
@fi @fi C ci;˛ D 0: (5) @t @x˛ On the Simulation of Porous Media Flow Using a New Meshless Lattice. . . 471
We discreatize Eq. (5) in time using the Lax-Wendroff scheme, which reads
@fin ıt2 @2 fin finC1 D fin ıtci;˛ C ci;˛ ci;ˇ : (6) @x˛ 2 @x˛ @xˇ
where ıt is the time step size and the superscript n is the time step number. In order to apply the meshless local Petrov-Galerkin scheme to discretize Eq. (6)in space, first, the local weak form of Eq. (6) on the control volume ˝I of point I isderived by taking its inner product with a local test function WI over ˝I , and usingintegration by parts, so that we obtain Z Z Z @fin ıt2 @WI @fin WI finC1 d˝ D WI fin d˝ ıtWI ci;˛ C ci;˛ ci;ˇ d˝ ˝I ˝I ˝I @x˛ 2 @xˇ @x˛ 2 Z ıt @fin C WI ci;˛ ci;ˇ nˇ d; (7) 2 I @x˛
where I is the boundary of the control volume ˝I and nˇ is the unit outward normalvector of I . Equation (7) is the local weak form of Eq. (6). In the next step, the field variable fi is to be expressed in terms of nodal valuesfi;J by a local interpolation scheme, that is
X Ns fi .x; t/ D J .x/fi;J .t/ D ˚ T .x/fs .t/; (8) JD1
where Ns is the number of nodal points in a local interpolation domain of point xcalled the support domain, ˚ T .x/ D f1 .x/ 2 .x/ : : : Ns .x/g is the transpose ofthe vector of shape functions, and fs .t/ D ffi;1 .t/ fi;2 .t/ : : : fi;Ns .t/gT is the vector ofthe nodal values of fi in the support domain. In this study, we make use of the localradial point interpolation method (LRPIM) [9] which uses the local radial basisfunctions (RBF) augmented with polynomials as the basis function; thus ˚ T .x/in the interpolation equation (8) is the transpose of the vector of LRPIM shapefunctions given as [9]
Q̊ T .x/ D fRT .x/ pT .x/gG1 ; (9)
where RT .x/ D fR1 .x/ R2 .x/ : : : RNs .x/g is the transpose of the vector of radialbasis functions (RBF), pT .x/ D f1 x y : : : pm .x/g is the transpose of the vector of Tmonomial basis functions, m is the number of monomial basis functions, Q̊ .x/ Df˚ .x/ Ns C1 .x/ : : : Ns Cm .x/g is the transpose of the extended vector of the shape T
functions, and G is a symmetric matrix defined in Ref. [9]. Substituting Eq. (8) in 472 S.H. Musavi and M. Ashrafizaadeh
Eq. (7) we write,
NI h Z X i NI h Z X WI J d˝ fi;J nC1 D WI J d˝ JD1 ˝I JD1 ˝I Z ıt2 @WI @J ıtWI C ci;ˇ ci;˛ d˝ ˝I 2 @xˇ @x˛ Z i ıt2 @J C WI ci;˛ ci;ˇ nˇ d fi;Jn ; (10) 2 I @x˛
where NI is the number of nodal points involved in the interpolation of the fieldvariable on the inner and the boundary points of the control volume ˝I . Byintroducing the mass matrix as Z MIJ D WI J d˝; (11) ˝I
and the stiffness matrix as Z ıt2 @WI @J Ki;IJ D ıtWI C ci;ˇ ci;˛ d˝ ˝I 2 @xˇ @x˛ Z ıt2 @J C WI ci;˛ ci;ˇ nˇ d; (12) 2 I @x˛
we rewrite Eq. (10) as follows,
X NI X NI nC1 MIJ fi;J D ŒMIJ C Ki;IJ fi;J n : (13) JD1 JD1
To complete the discretization process, the integrals of equations (11) and (12) areto be evaluated numerically. The Gauss quadrature scheme is employed for thispurpose. We have
X NG MIJ D k WI .xk /J .xk /jJ˝I j; (14) kD1 On the Simulation of Porous Media Flow Using a New Meshless Lattice. . . 473
and
X ıt2 @WI ˇˇ @J ˇˇ ˝I NG Ki;IJ D k ıtWI .xk / C ci;ˇ ˇ ci;˛ ˇ jJ j kD1 2 @xˇ xk @x˛ xk
@ ˇ b ıt2 X NG Jˇ C k WI .xk / ci;˛ ˇ .ci;ˇ nˇ /jJI j; (15) 2 kD1 @x˛ xk
where k is the Gauss weighting factor for the Gauss quadrature point xk , J˝I andJI are the mapping Jacobian matrices for the domain and the boundary integrations,respectively, and NG and NGb are the number of Gauss points used for the domainand the boundary integrations, respectively. Now, Eq. (13) becomes the fully discretized equation for the nodal point I.Writing this equation for all of the nodal points in the computational domain(I D 1; : : : ; N), and assembling the resulting equations in a global system ofequations, we can write
MfnC1 i D ŒM C Ki fni ; i D 1; : : : ; nQ ; (16)
where M, K, and fi are the global mass matrix, stiffness matrix, and particledistribution vector, respectively. Equation (16) is a system of N equations with Nunknowns which should be solved separately for each direction i after imposing theboundary conditions. The advection equation of the particle distributions is a hyperbolic equation,which requires boundary conditions for the incoming particles at the boundary(ci;ˇ nˇ < 0). In this study, we impose boundary conditions using the bounce-backscheme of non-equilibrium distributions, i.e. eq eq fi fi D fi fi ; (17)
where fi is the outgoing particle distribution along the opposite direction of theincoming distribution fi . Substituting the equilibrium distribution of Eq. (3) in theabove equation, we obtain
fi D fi C 2b ti .ci;˛ ub;˛ /=c2s ; (18)
where b and ub;˛ are the macroscopic density and velocity at the boundary. If fiin Eq. (18) is considered to be the post-collision (pre-streaming) distribution, thenEq. (18) becomes an explicit essential boundary condition for the discretized systemof Eq. (16). The coefficient matrix in Eq. (16) is a sparse matrix which can be efficientlysolved using sparse iterative solvers such as BiCGStab. However, the explicit natureof the standard lattice Boltzmann method, and the diagonally dominant characterof the mass matrix, motivated us to find rational ways of diagonalizing the mass 474 S.H. Musavi and M. Ashrafizaadeh
matrix, and thus save much of the computational time. In this study, we use row-sumlumping, in which the sum of the elements of each row of the mass matrix is used asthe diagonal element. As a result, our meshless lattice Boltzmann method becomesan explicit solver for the fluid flow problems. The maximum value for the timestep leading to a stable solution is determined using the Courant number, CFL Dmax fjci jg ıt=ıxmin , where ıxmin is the minimum point spacing in the domain. For astable solution the CFL number should be smaller than 1.0.
3 Results
3.1 Poiseuille Flow
The first test case considered in this study is the pressure driven fully developed flowbetween two parallel plates. In solving this test case using the proposed method,we consider a square computational domain in the xy plane and discretize it using9 9, 17 17, 23 23, and 65 65 uniform point distributions. The constantpressure boundary condition is imposed in the inlet and outlet and the no-slip andthe impermeability boundary conditions are imposed at the walls. For each pointdistribution, time iterations continue until a steady state is reached. The result of ourmethod for the velocity distribution is depicted and compared with the analyticalsolution in Fig. 1.
0.8
0.6 U
0.4 Presentstudy Analyticalsolution
0.2
0 0 0.2 0.4 0.6 0.8 1 YFig. 1 The velocity distributions for the Poiseuille flow; line: analytical solution, symbols: presentstudy On the Simulation of Porous Media Flow Using a New Meshless Lattice. . . 475
0 10
10-1
E 10-2
10-3
-4 10 10 -2 10 -1 10 0 hFig. 2 Numerical convergence of the meshless Lattice Boltzmann method in L2 error norm withrespect to the point spacing for the Poiseuille flow
In order to determine the order of accuracy of the numerical scheme, we employthe following relative L2 error norm,
PNe !1=2 ID1 .UaI UnI /2 ED PNe 2 ; (19) ID1 UaI
where UaI and UnI are the analytical and numerical solutions of the velocity atpoint I, respectively, and Ne is the fixed number of points used for the erroranalysis. For this test case, a 10 10 uniform point distribution is used for the erroranalysis. The variation of the above error norm with respect to the point spacingh is sketched in the logarithmic diagram of Fig. 2. The rate of the convergence ofour numerical method, computed by the linear regression of the data in Fig. 2, isR D 2:15.
3.2 Flow in a Porous Medium
One of the most common applications of the standard lattice Boltzmann method isthe simulation of the flow in porous media. In order to illustrate the ability of thepresent meshless lattice Boltzmann method to deal with complex geometries, we 476 S.H. Musavi and M. Ashrafizaadeh
simulate a pressure-driven fluid flow in a typical two-dimensional porous mediumshown in Fig. 3 and compute the permeability of the medium. One of the commondefinitions of the Reynolds number for the porous media flow is
vDp Re D ; (20)
where and are the density and viscosity of the fluid, v is the superficial flowvelocity, and Dp is a representative grain size for the porous media, which is usuallychosen to be the average grain diameter. Experimental studies have illustrated thatthe flow regimes with low Reynolds numbers (typically less than 10) are Darcian,meaning that the superficial velocity-pressure gradient relation obeys the Darcy’slaw k v D rp; (21)
where k is the permeability of the medium measured in m2 or Darcy, where1 Darcy D 9:869233 1013 m2 . In our simulation, the no-slip boundary condition is imposed on the top andthe bottom walls and on the grain surfaces using the bounce-back scheme and theconstant pressure boundary condition is imposed on the left and right boundaries.The pressure differences between inlet and outlet boundaries in our simulations are
Fig. 3 Domain and a typical point distribution for the flow in a porous medium On the Simulation of Porous Media Flow Using a New Meshless Lattice. . . 477
set so that the Reynolds number of the flow is low and the flow remains in theDarcian regime. The domain is discretized using arbitrary sets of nodal points as shown in Fig. 3.First, to investigate the convergence of the numerical method with respect to thedomain discretization, we discretize the domain with 2601, 7364, 25284, and 90043arbitrary distributed nodal points and compute the superficial velocity for each case.The results are depicted in Fig. 4. As shown in this figure, the difference betweenthe cases with 25284 and 90043 points is less than 1 % and therefore we use the casewith 25284 points in all the following simulations. However, as observed from thefigure, the standard LBM requires 640000 grid points for the accurate simulation ofthe flow. It means that the required number of points in our method is about 1/25thof that of the standard LBM. Next, to obtain the permeability of the medium, we compute the superficialvelocity for different pressure gradients and perform the linear interpolation of theresults as is depicted in Fig. 5. From this figure, the permeability is obtained ask D 1:53 105 m2 D 1:55 107 Darcy. The pressure and velocity magnitude distributions of the porous medium floware depicted in Figs. 6 and 7, respectively, for the dimensionless pressure gradientof 0.1.
0.0014
Standard LBM Meshless LBM 0.0012 Superficial velocity
0.001 Standard LBM 640000 grid points
0.0008
Meshless LBM 25284 nodal points 0.0006
0.0004 3 4 5 6 10 10 10 10 Number of pointsFig. 4 The superficial velocity versus the number of nodal points used in the meshless LBM 478 S.H. Musavi and M. Ashrafizaadeh
0.002
0.0016
Superficial velocity 0.0012
0.0008
0.0004
0 0 0.02 0.04 0.06 0.08 0.1 0.12 Pressure gradientFig. 5 The superficial velocity with respect to the pressure gradient for the flow in the porousmedium
0.8 0.09 0.08 0.07 0.06 0.05 0.04 0.6 0.03 0.02 y
0.01
0.4
0.2
0 0 0.2 0.4 0.6 0.8 1 x
Fig. 6 Pressure distribution for the flow in the porous medium On the Simulation of Porous Media Flow Using a New Meshless Lattice. . . 479
vel
0.8 0.02 0.018 0.016 0.014 0.012 0.01 0.6 0.008 0.006 y
0.004 0.002 0 0.4
0.2
0 0 0.2 0.4 0.6 0.8 1 x
Fig. 7 Velocity magnitude distribution for the flow in the porous medium
4 Conclusions
A new meshless lattice Boltzmann method (MLLBM) has been developed for thesimulation of the nearly incompressible fluid flows. The main advantage of ourmethod with respect to the previous extensions of the lattice Boltzmann methodis to eliminate the need for any meshes. A feature that shows its superiority over thestandard lattice Boltzmann method in complex geometries, such as porous media. Two test cases have been considered in this study. First, the Poiseuille flow hasbeen solved and compared with the analytical solution, showing the second order ofaccuracy of our method. Next, the flow in a porous medium has been simulated toillustrate the capability of the method in dealing with domains with a very complexgeometry. Although the results presented in this study are for two-dimensionalcases, the extension for three-dimensional problems is straightforward and remainsfor our future works. However, our preliminary 3D developments show that the timeand memory saving for the meshless LBM method should be even higher than thatof the 2D model. 480 S.H. Musavi and M. Ashrafizaadeh
References
1. Bardow, A., Karlin, I.V., Gusev, A.A.: General characteristic-based algorithm for off-lattice Boltzmann simulations. EPL (Europhys. Lett.) 75(3), 434 (2006)2. Patil, D.V., Lakshmisha, K.N.: Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh. J. Comput. Phys. 228(14), 5262–5279 (2009)3. Li, Y., LeBoeuf, E.J., Basu, P.K.: Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh. Phys. Rev. E 72(4), 046711 (2005)4. Shi, X., Lin, J., Yu, Z.: Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int. J. Numer. Methods Fluids 42(11), 1249–1261 (2003)5. Düster, A., Demkowicz, L., Rank, E.: High-order finite elements applied to the discrete Boltzmann equation. Int. J. Numer. Methods Eng. 67(8), 1094–1121 (2006)6. Min, M., Lee, T.: A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows. J. Comput. Phys. 230(1), 245–259 (2011)7. Musavi, S.H., Ashrafizaadeh, M.: Meshless lattice Boltzmann method for the simulation of fluid flows. Phys. Rev. E 91(2), 023310 (2015)8. Wolf-Gladrow, D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, vol. 1725. Springer, Berlin (2000)9. Liu, G.R., Gu, Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer Science and Business Media, Dordrecht/New York (2005) A Comparison Between Twoand Three-Dimensional Simulationsof Finite Amplitude Sound Waves in a Trumpet
Janelle Resch, Lilia Krivodonova, and John Vanderkooy
Abstract Simplifying a three-dimensional problem of simulating sound propaga-tion in musical instruments is frequently done by exploiting axial symmetry andreducing the problem to one or two dimensions. We examine if such dimensionreduction is valid. We numerically solve the equations of motion of compress-ible gases using the discontinuous Galerkin method to model nonlinear soundpropagation inside a trumpet. The numerical results in two and three dimensionsare then compared with experimental data. Experiments were carried out on atrumpet in which the sound pressure waves of the Bb3 and Bb4 notes played at fortewere recorded. We found that it is crucial to consider the problem with all threespatial dimensions to ensure reflections in the bell region are properly modelled.Additionally, the shape of flare must be carefully approximated to compute thepropagating waves accurately.
1 Introduction
In order to accurately model sound propagation in brass musical instruments, higheramplitude propagating waves must be considered. For musical instruments suchas the trombone or trumpet, pressure variations inside the instrument can be asignificant fraction of atmospheric pressure if loud, high frequency notes are played.Assuming the pressure disturbance entering the narrow tubing of the bore is largeenough, nonlinear behaviour can cause waveform distortion. In particular, the crestof the wave will travel faster than the trough causing the wave to steepen. Wavesteepening will excite higher harmonic components of the sound pressure waveswhich influences the timbre of the sound by giving it a more ‘brassy’ effect [3]. The
J. Resch () • L. KrivodonovaDepartment of Applied Mathematics, University of Waterloo, 200 University Avenue W.,Waterloo, ON N2L 3G1, Canadae-mail: [emailprotected]; [emailprotected]. VanderkooyDepartment of Physics and Astronomy, University of Waterloo, 200 University Avenue W.,Waterloo, ON N2L 3G1, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 481J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_44 482 J. Resch et al.
linear acoustic equations used to model sound propagation outside an instrumentare not suitable to model the sound propagation inside. Instead, wave propagationinside of the instrument should take compressibility into account which can bemodelled using the Euler equations. We solve these equations numerically using thediscontinuous Galerkin method (DGM). This method is particularly useful becauseit can handle unstructured meshes and has excellent dissipative and dispersiveproperties. In certain regimes (not considered here) shock waves can be producedinside an instrument and this method can handle such nonlinear effects [2]. Previously, in [5], we modelled nonlinear wave propagation in a 2D trumpet. Inaddition, we considered the consequences of neglecting the third spatial dimensionsince the spreading of the waves in 2D and 3D differs. We calculated that the2D – 3D dimensionality difference (which we call the dimension factor) in ourresults would be approximately 14 dB. We arrived at this value by assuming thatthe axial pressure in both 2D and 3D is a good measure of the total energy leavingthe bell. A full derivation of the dimension factor value can be found in [5]. Aftertaking this amplitude difference into account, we obtained a good match betweenthe experimental and numerical data, for the frequency components that propagateout of the trumpet bell. Here we present some of our preliminary results for extending the model to 3D.In particular, we will be able to verify and further investigate the influence thatspatial dimensions have on such models. This is an important aspect to considersince many mathematical descriptions of sound wave propagation in the literatureare reduced to 2D or even 1D.
2 Experimental Data and Numerical Setup
The experimental data that we collected from a Bb Barcelona BTR-200LQ trumpetshown in Fig. 1 is discussed in detail in [5]. We took one period of the soundpressure measurements obtained from a quarter-inch microphone carefully mounted
Fig. 1 Placement of microphones on the Barcelona BTR-200LQ trumpet A Comparison Between 2D and 3D Simulations of Finite Amplitude Waves in. . . 483
on the shank of the trumpet mouthpiece, and applied Fourier analysis to the datato reconstruct the waveform as a sum of 30 cosine waves. This experimental dataobtained at the mouthpiece was then used as a boundary condition on the pressurefor our simulations. The musical notes recorded and then simulated were the Bb3 andBb4 played at forte. A half-inch microphone was also placed outside the trumpet onthe central axis about 17 cm away from the bell. We compared our simulation resultswith the pressure measurements obtained from this microphone outside the bell. For all simulations, we assumed that the initial flow was at rest and relatedpressure and velocity through the 1D expression derived from linear acoustic theory.Since there is little return from the reflections at the bell, at least for the higherfrequencies, the plane wave velocity expression is a reasonable approximation. Forthe boundary conditions, we prescribed reflecting boundary conditions on the innerand outer walls of the trumpet mesh (excluding the mouthpiece). Since we modelleda trumpet in a box, we used pass-through boundary conditions on the computationaldomain.
3 The Influence of the Geometry and Dimension of the Bell
Several numerical simulations have been carried out to determine if the modelconsidered is adequate. Some such computational experiments are discussed in[5]. We have discovered that considering the proper geometry of brass musicalinstruments is critical to simulate wave propagation with any accuracy. We havealso investigated the importance of the bell geometry. One of the most significantproperties of the bell is how it influences the reflections of the sound pressure waves.The location at which the harmonic waves reflect in the bell is dependent on theirfrequency. For higher frequency waves, a larger portion of the energy will be lost orbe completely transmitted from the bell [1]. In the literature, it is common for the trumpet flare to be approximated by severaldifferent functions or combination of functions, e.g., exponential functions, hyper-bolic functions, etc. However, we were uncertain if such geometric descriptionswould be sufficient to accurately model wave propagation through the trumpet bell.If the propagating sound pressure waves are particularly sensitive to the bell’scurvature, simulations may produce exaggerated discrepancies if the bell shapeis poorly approximated. Simulations performed in 2D should be relevant for thefrequency components that are transmitted from the bell. Therefore, to reduce runtime, we first examined our problem by neglecting the third spatial dimension andfocused on the flare shape. The first mesh, which will be called mesh 1, represented a 2D trumpet wherethe bell was approximated by an elliptic function. To improve this approximation,bell measurements were taken at several locations along the flare, which was theninterpolated by the obtained set of points and cubic splines. The measurements were 484 J. Resch et al.
done at the university’s machine shop by sampling the diameter of the bore nearexpansion, 1.02 m from the mouthpiece. This flare shape was used to construct a2D trumpet mesh called mesh 2. To obtain more precise points to approximate thetrumpet shape, a photo of the trumpet flare was taken. The grabit software fromMath Works Inc. was used to accurately trace out the shape of the trumpet bell. Thisthird bell shape was used to create a 2D mesh which we will call mesh 3. Since thisrepresentation of the bell was most accurate, it was further used to extend mesh 3into a full 3D mesh. This 3D mesh will be referred to as mesh 4. The total numberof cells is 8190, 8467, 8038 and 14,362 for mesh 1, mesh 2, mesh 3 and mesh 4,respectively. The meshes consist of triangular or tetrahedral elements with adaptivesizes to accurately resolve the geometric features of the trumpet. More details ofthe mesh construction can be found in [5]. The computational domain and a samplemesh are shown in Fig. 2. Once these flare shapes were obtained, simulations were carried out on themeshes of a trumpet that is 1.48 m in total length. The diameter of the bore isconstant except near the end of the instrument where it slowly increases and thenflares rapidly to give the bell shape. This expansion for all meshes begins 1.02 mfrom the mouthpiece end. In [5], we justified that the bends of the instrument do not
Fig. 2 Mesh 3: Two-dimensional computational domain and trumpet mesh constructed by tracingout the shape of the bell in Matlab A Comparison Between 2D and 3D Simulations of Finite Amplitude Waves in. . . 485
Fig. 3 Initial and reflected wave of simulated pressure pulse through mesh 1, mesh 2, mesh 3 andmesh 4
greatly influence the wave propagation in the instrument, especially in comparisonto the bell. Therefore, for all the simulations discussed, the bends will not bemodelled, i.e., the bends and coils of the trumpet will be unwrapped and the tubingwill be straightened out. Before simulating the musical notes however, we considered a simpler problem.We sent a pressure pulse down the trumpet meshes from the mouthpiece to examinetheir acoustic behaviour. As expected, the variations in the bell geometry greatlyinfluenced wave propagation. The pulse generated at the mouthpiece was given by 2 1:0 C .0:01 0:01 cos.1500t//; if t < 1500 pD 1:0; else
which corresponds to a unipolar pressure pulse with an amplitude of approx-imately 2000 Pa. In Fig. 3, we show the pressure simulated at a point located atthe mid-length of the cylindrical bore. The first peak located at approximatelyt D 0:0025 corresponds to the initial pulse moving from the mouthpiece towardsthe bell. The second, inverted peak seen at about t D 0:007 corresponds to thesignal travelling back to the mouthpiece after it has been reflected by the bell.In Fig. 4, we also plotted the reflected transfer data which is calculated from thefrequency content of the reflected pulse divided by that of the incident pulse. Thiscurve represents the power reflected by the bell meshes. 486 J. Resch et al.
Fig. 4 Corresponding reflected transfer function of pressure pulses depicted in Fig. 3
Table 1 The number of cells, and the frequency components that are mostly confined to theinstrument or transmitted from the bell are summarized for each meshMesh name Cell count Reflected freq. (Hz) Transmitted freq. (Hz)Mesh 1 8190 600 800Mesh 2 8467 600 1000Mesh 3 8038 400 1000Mesh 4 14;362 900 1200
We observe in Figs. 3 and 4 a rather significant difference in the shape of thereturning pressure pulse for each mesh considered, especially when comparingthe 2D and 3D results. We found that different frequency components wereconfined to the instrument and transmitted from the bell for each mesh. This wasaccomplished by examining the relationship between the reflected transfer functionand its corresponding compliment, the transmission transfer function. Frequencycomponents less than approximately 550 Hz for mesh 1, 600 Hz for mesh 2, 400 Hzfor mesh 3, and 900 Hz for mesh 4, are mostly reflected before or within the bell [5].For frequency components greater than 800 Hz for mesh 1, 1000 Hz for mesh 2 andmesh 3, and 1200 Hz for mesh 4, the waves are being mostly transmitted from thebell. The properties and number of cells for each mesh is summarized in Table 1. A Comparison Between 2D and 3D Simulations of Finite Amplitude Waves in. . . 487
Fig. 5 Two-dimensional numerical and experimental results for the Bb3
Further simulations were then carried out using the measured Bb3 and Bb4 pressurewaveforms. In particular, we solved the full 2D and 3D sets of compressible Eulerequations with the initial and boundary conditions mentioned above on the meshes.Each note was initialized with 30 harmonics. The frequency spectra of the 2D simulation results for the Bb3 and Bb4 notesoutside of the trumpet bell can be seen in Figs. 5 and 6, respectively. Figures 5and 6 also depict the frequency spectra of the experimental data. A comparisonof the numerical and experimental data shows that the numerical amplitude isapproximately 19C dB off, where 14 dB are expected since the third spatialdimension is neglected (see analysis in [5]). However, we are not certain why weobtained an additional decibel difference of roughly 4C dB. As we will discuss inthe next section, a possible explanation is that the subtleties of the bore geometrynear the mouthpiece were not considered. Energy losses were also neglected andsome conjecture that this could make a difference of several decibels (up to 6 dB)[4]. Nonetheless, to easily compare the harmonic distribution of the experimentaland 2D numerical data presented in Figs. 5 and 6, we shifted the experimental databy the decibel difference stated in the plots. 488 J. Resch et al.
Fig. 6 Two-dimensional numerical and experimental results for the Bb4
We can see in Figs. 5 and 6 that the performed simulations on mesh 1, i.e., the2D trumpet mesh where the bell is approximated by an elliptic function, gives thepoorest results in shape and amplitude. The 2D simulation results carried out onmesh 2 and mesh 3 however show improvement in both aspects, i.e., in amplitudeand harmonic distribution. In particular, the spectral data aligns relatively wellwith the shifted experimental curve, with the exception of the lowest frequencycomponents. The first three harmonics for the Bb3 and the fundamental frequencyfor the Bb4 deviate most from the experimental data. However, we postulate that thelowest frequencies are mostly confined within the instrument and consequentially,are greatly influenced by the reflections that take place near the bell. We alsopredicted that we would observe some amplitude differences and harmonic distri-bution discrepancies from examining the pulse results. We further hypothesized thatthe experimental and numerical data will better match (specifically for the lowestharmonic components of the notes) if the equivalent simulations were carried out in3D, i.e., on mesh 4. A Comparison Between 2D and 3D Simulations of Finite Amplitude Waves in. . . 489
Fig. 7 Three-dimensional numerical and experimental results for the Bb3
The frequency spectra of the measured Bb3 and Bb4 pressure waveforms outsidethe trumpet bell are plotted in Figs. 7 and 8, respectively. The result of solvingthe full 3D system on mesh 4 is also plotted for the Bb3 and Bb4 notes in Figs. 7and 8, respectively. Overall, we see that the 3D simulations show greater similarityto experimental data. There are however still some discrepancies, particularly inthe amplitude of the 3D solution curves. The Bb3 and Bb4 notes are overestimated byapproximately 10 dB. For comparative purposes, we again shifted the experimentalcurves by the amplitude difference. When this shift is considered, the shape of theexperimental data and numerical solutions are in better agreement compared to theresults in Figs. 5 and 6. More specifically, the resulting lower frequency componentsfor both simulated notes demonstrate that the bell reflections are more accuratelymodelled when the third spatial dimension is considered. In the next section, weconjecture on possible sources for the 10 dB difference obtained in our 3D numericalsimulations. 490 J. Resch et al.
Fig. 8 Three-dimensional numerical and experimental results for the Bb4
4 Discussion Regarding the Geometry of the Bore Near the Mouthpiece
Neglecting subtleties of the geometry near the mouthpiece of the trumpet may bean explanation for the observed amplitude discrepancies in our simulation results.In reality, the tube of the trumpet near the mouthpiece does not maintain constantradius. The radius of the trumpet bore from the mouthpiece slowly increases forapproximately 22 cm. The tubing then remains roughly at a constant radius until thebore begins to widen at 102 cm from the mouthpiece. Furthermore, the geometryof the trumpet mouthpiece itself is also quite complex and varies in shape anddynamics for each unique mouthpiece model. We tried to avoid some of thesepotential effects by measuring the sound pressure waveforms at the shank of themouthpiece. However, the mouthpiece throat’s radius is approximately 0.4 theshank’s radius; and the radius of the trumpet bore at 20 cm from the shank isapproximately 1.42 the shank’s radius. It is thus possible that neglecting thesegeometric attributes of the trumpet may cause an overestimation in the simulatedamplitude. We are currently investigating how the geometry of the bore between themouthpiece and the first bend of the trumpet influences the wave propagation beingmodelled. We speculate that in addition to the geometric shape of the tube near A Comparison Between 2D and 3D Simulations of Finite Amplitude Waves in. . . 491
the mouthpiece, the rate at which the radius increases may be important to consider.Just as the rapid flare of the trumpet influences the wave propagation, specifically theharmonic reflections, we hypothesize that the increase in radius at the mouthpiecemay have a similar effect. We are also currently studying the influence of suchreflections near the mouthpiece.
5 Conclusion
We have presented our 2D and 3D simulation results of modelling sound productionin a trumpet. For comparative purposes, a plot of the experimental data and allsimulation results presented here for the Bb3 and Bb4 notes can be found in Fig. 9.Although neglecting the third spatial dimension is frequently done, it is evidentthat such an assumption will not accurately model the spreading of the waves inall spatial directions. Furthermore, in Figs. 3, 4, 5, and 6, we can see that wavepropagation is very sensitive to the geometry of the flare, specifically where thereflections at the bell are sensitive even to minor changes to its shape. We haveshown that it is not sufficient to merely approximate the bell geometry by a smoothcurve, it must be constructed with maximal agreement to physical geometry toobtain good results. Our 3D simulation results presented in Figs. 7 and 8 are significantly moreaccurate than our 2D simulations. In addition, these findings have initiated aninvestigation to determine the importance of the tube profile near the mouthpieceof the trumpet on accuracy of the model. While the tube appears to be straight fromthe outside, it has a complex geometry inside that cannot be inferred. We would needto cut open an instrument to measure the true geometry and are currently lookingfor a less invasive solution. Preliminary results indicate that better match of theexperimental and numerical data can be achieved.
Fig. 9 Numerical and experimental results for the Bb3 and Bb4 492 J. Resch et al.
References
1. Benade, A.H.: Fundamentals of Musical Acoustics. Dover Publications, New York (1900)2. Flaherty, J.E., Krivodonova, L., Remacle, J.F., Shephard, M.S.: Some aspects of discontinuous Galerkin methods for hyperbolic conservation laws. J. Finite Elem. Anal. Des. 38, 889–908 (2002)3. Hirschberg, A.J., Gilbert, J., Msallam, R., Wijnands, A.P.J.: Shock waves in trombones. J. Acoust. Soc. Am. 99, 1754–1758 (1995)4. Kausel, W., Moore, T.: Influence of wall vibrations on the sound of brass wind instruments. J. Acoust. Soc. Am. 128, 3161–3174 (2010)5. Resch, J., Krivodonova, L., Vanderkooy, J.: A two-dimensional study of finite amplitude sound waves in a trumpet using the discontinuous Galerkin method. J. Comput. Acous. (2012). doi:10.1142/S0218396X14500076 A Dual-Rotor Horizontal Axis Wind TurbineIn-House Code (DR_HAWT)
K. Lee Slew, M. Miller, A. Fereidooni, P. Tawagi, G. El-Hage, M. Hou,and E. Matida
Abstract This paper describes the DR_HAWT (Dual-Rotor Horizontal Axis WindTurbine) in-house code developed by the present authors, the National Renew-able Energy Laboratory (NREL) Phase VI validation test case, and a dual-rotorconfiguration simulation. DR_HAWT uses a combination of the Blade ElementMomentum theory (BEM) and the vortex filament method to accurately predictthe aerodynamic performance of horizontal axis wind turbines for a single or dual-rotor configuration. Using two-dimensional (2D) airfoil coefficients obtained fromXFOIL, the aerodynamic power prediction produced by DR_HAWT resulted in anaverage error of less than 20 % when compared to the NREL Phase VI experiment.Upon correcting the 2D wind tunnel aerodynamic coefficient data for 3D floweffects, good agreement was obtained with the exception of the deep-stall region.In addition, a dual-rotor test case consisting of a combination of a half-scaled andfull-sized NREL Phase VI rotor, is presented. The dual-rotor configuration resultedin a 16 % average increase in power generation as compared to the full-sized rotoralone.
1 Introduction
With recent advancements in wind energy technology, and the increase in oil prices,wind energy is now considered to be a favourable green-energy alternative [1]. In anattempt to maximize energy extraction while minimizing costs, a Contra-RotatingDual-Rotor Horizontal Axis Wind Turbine configuration has been proposed. It ishypothesized that by having both an upwind rotor and a downwind rotor, morepower can be produced on a single wind turbine tower.
K. Lee Slew () • M. Miller • E. MatidaCarleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canadae-mail: [emailprotected]; [emailprotected]. FereidooniNational Research Council Canada, 2320 Lester Road, Ottawa, ON K1V 1S2, CanadaP. Tawagi • G. El-Hage • M. HouZEC Wind Power, 1800 Woodward Drive, Ottawa, ON K2C 0P7, Canada
© Springer International Publishing Switzerland 2016 493J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_45 494 K. Lee Slew et al.
Due to significant improvements in computational technology and knowledge ofHorizontal Axis Wind Turbine (HAWT) aerodynamics, computational simulationtools have become a preferred method in the aerodynamic performance prediction ofwind turbines. Commercially available CFD packages offer accurate aerodynamicpredictions whilst capturing the complex flow phenomena that occur with HAWTs,however, these programs are typically computationally expensive. For this reason,Blade Element Momentum theory (BEM) is often used by the wind energy industryas a preliminary aerodynamic performance prediction tool. BEM is computationally inexpensive and yields accurate results provided accept-able airfoil aerodynamic data is available, however, BEM cannot capture all the flowphenomena which occurs in the wake of a HAWT. To reduce runtimes comparedto CFD, while providing a more accurate representation of the wake, an in-housecode, utilizing a grid-less technique, for Dual-Rotor Horizontal Axis Wind Turbines(DR_HAWT) was created by the present authors based on a combination of BEM[1] and the vortex filament method [2, 3]. DR_HAWT possesses the ability to predictthe aerodynamic performance of a single or dual-rotor configuration as well asparallel processing capabilities. To verify and validate DR_HAWT, the National Renewable Energy Laboratory(NREL) Phase VI wind turbine was simulated. The experiment was performed inthe NASA-Ames 24:4 m 36:6 m wind tunnel using a two-bladed, 10-m-diameterHAWT. The tapered and twisted blades utilize the S809 airfoil; further geometricdetails can be found in Ref. [4]. All comparisons in this paper, unless statedotherwise, are to the aerodynamic power output of the Sequence S experiment [4]. This paper will briefly describe the BEM and vortex filament methods imple-mented in DR_HAWT including the 3D correction factors used to account for theeffect of rotation. The NREL validation test case will be presented as well as adual-rotor HAWT configuration.
2 Methodology
2.1 Blade Element Momentum Theory
The well documented BEM theory [1] is a combination of Blade Element theory andthe Momentum theory. The wind turbine blade is discretized into elements along theblade span, where geometric data are known. At a specified Reynolds number (Re),velocity and angle of attack (AOA), the coefficient of lift and drag (CL and CD ,respectively) can be obtained from look-up tables for each specific airfoil. Usingthe aerodynamic coefficients, the tangential and normal forces acting on the bladeelements can be determined. These forces can then be integrated along the span ofthe blade to determine the rotor torque, thrust and power. The wake of the wind turbine is commonly accounted for in the BEM methodby induction factors. Due to the extraction of energy from the wind flow, these Dual-Rotor Horizontal Axis Wind Turbine Code (DR_HAWT) 495
induction factors decrease the velocity of the flow in the axial and tangentialdirection by means of a semi-empirical model. The BEM method, combinedwith various other correction factors such as Prandtl’s tip loss factor, providesan adequate aerodynamic prediction of single rotor configuration of HAWT [1].However, for a dual-rotor configuration, where the influence of the upwind anddownwind rotor wake is an intrinsic aspect of the aerodynamic performanceprediction, a better means of representing the wake is necessary.
2.2 Vortex Filament Method
In order to determine the influence of the wake on the aerodynamic performanceof a HAWT, particularly in a dual-rotor configuration, a vortex filament method hasbeen adapted from Strickland et al. [2, 3]. In this three-dimensional (3D), grid-lessfree vortex model, vortex filaments are used in place of the induction factors in theBEM method. These vortex filaments are free to convect in the fluid domain therebyproviding a suitable spatial and temporal representation of the wake. Similar to the BEM method, the blade is discretized into elements where thegeometric characteristics are known. Each element is defined by unit vectors in thespanwise (s), chordwise (c), and normal (n) directions, located at the aerodynamiccenter of each element. In addition, the position of the aerodynamic center of eachelement and the shed vortices are defined in a global coordinate system (i, j, andk axes) located at the center of the upwind hub. Figures 1 and 2 illustrate thecoordinate systems used in DR_HAWT. With respect to these coordinate systems, the relative velocity, Vrel , seen by ablade element as shown in Fig. 2, can be evaluated by vector sum of the bladerotational velocity, VT , free stream velocity, Vo , and the induced velocity caused bythe vortex filaments, Vind . The local flow conditions (Re, AOA) are calculated and,
Fig. 1 Global coordinate system used in DR_HAWT 496 K. Lee Slew et al.
Fig. 2 Velocity components acting on a blade element
the lift and drag coefficients are linearly interpolated from look-up tables consistingof experimental or numerical airfoil aerodynamic data. Based on the Lifting Line Theory, each blade element is associated with a boundvortex filament of strength, B . This strength can be expressed using the Kutta-Joukowski theorem as follows: 1 B D CL cjVrel j (1) 2 The bound vortex strength is a function of the CL , the chord length (c) and themagnitude of the relative velocity. Since the relative velocity takes into account theinduced velocity, a predictor-corrector method is employed for which the boundvortex strength is predicted based on the previous time step induced velocities.The predicted strength is then used to correct the induced velocities which in turncorrects the bound vortex strength. This process is repeated until the differencesbetween the predicted and corrected values are below 1 103 . Once the boundvorticity and the induced velocities have been corrected, the strength of the vorticesthat constitute the wake can be computed. Figure 3 illustrates the representation of blade elements with the wake which ismade up of vortex filaments. For a time step NT, the bound vortex strength of anelement i is written as NT i . As the blade rotates, vortices are shed in the form ofspanwise and trailing vortex filaments parallel to the span and chord of the bladerespectively. The strength of these vortex filaments are calculated based on Kelvin’sTheorem for which any change in the bound vortex strength over time must beaccompanied by an equal and opposite vortex strength in the wake. This forms thespanwise vortices (bold lines) whereas the trailing vortices (thin lines) result fromthe spanwise variation in bound vortex strength. The tip vortex is assumed to havethe same magnitude and opposite direction as the bound vortex closest to the tip. Each vortex filament in the wake convects with a local velocity. As the vortexfilament first leaves the blade, its initial velocity is assumed to be equal to the sumof the induced velocities at the blade element end and the oncoming freestream Dual-Rotor Horizontal Axis Wind Turbine Code (DR_HAWT) 497
Fig. 3 Spanwise and trailing vortices being shed from a blade
velocity. In order to calculate the distance travelled by any vortex filament in thewake for a given time step, a second order Adams-Bashforth explicit integrationformula is used. Knowing the strength and position of the shed vortices, the inducedvelocity at any point can be calculated using the Biot-Savart Law given by: Z r dl Vind D (2) 4 l jrj3
where r is the position vector from the point of interest in the fluid domain to theincremental length (dl) along the vortex filament. The total perturbation velocityinduced by all the vortex filaments represents the effect of the wake on the blades.In order to calculate the torque, thrust and power generated by the blades, the normaland tangential forces are integrated along the span of the blade by assuming linearvariation along each element [1]. Finally, the aerodynamic power produced by thewind turbine is the product of the rotational velocity and the total torque of the rotor.
2.3 Aerodynamic Coefficients
This study uses the 2D (non-rotating) airfoil coefficient data (CL and CD ) obtainedeither by using experimental wind tunnel data, acquired from Ref. [4] or XFOILwhich numerically calculates the lift and drag coefficients for a given airfoil profilebased on pressure distributions obtained using a panel-method [5]. For comparisonpurposes, the same numeric values of Reynolds numbers are used in both casesfor the aerodynamic coefficient input files. In addition, the experimental data wascorrected to obtain a better understanding of 3D flow effects due to rotation. In an attempt to model these 3D effects, several numerical correction factorshave been formulated by the likes of Du and Selig [6], Snel et al. [7], Bak et al. [8],Lindenburg [9], and more. These various models all simply correct the 2D CL and 498 K. Lee Slew et al.
CD wind tunnel data using the following equations.
CL;3D D CL;2D C f .c=r; : : :/ CL (3) CD;3D D CD;2D C f .c=r; : : :/ CD (4)
where f is the correction factor which in all cases is a function of at least the chordto radial (c=r) location ratio. CL and CD is the difference between the 2D CL andCD wind tunnel data and the 2D CL and CD if hypothetically the flow remainedattached at all AOA. Some 3D correction formulations, such as that of Du andSelig, have empirical correction factors which are built into f . The fine tuning ofthe 3D corrections is only viable when experimental 3D data, like the data obtainedfrom the NREL experiment, is available. Obtaining a suitable universal empiricalcorrection factor for the 3D corrections should be performed by simulating severaldifferent experiments. The 3D correction factors of Snel et al. and Du and Selig were implemented inDR_HAWT for AOA up to 30ı , at which point the corrected values linearly decayto the 2D value at 55ı as per Ref. [10]. Figure 4 illustrates an example of the Snelet al. 3D corrections applied to the CL of the S809 airfoil at a Reynolds numberof 6:5 105 . It should be noted that the Snel et al. correction only corrects the CLwhereas Du and Selig corrects both the CL and CD .
Fig. 4 CL curves obtained by XFOIL, 2D wind tunnel data and 3D corrections (Snel et al.) for agiven Reynolds number Dual-Rotor Horizontal Axis Wind Turbine Code (DR_HAWT) 499
2.4 Convergence Analyses
Prior to simulating the Sequence S of the NREL Phase VI experiment, a sensitivityanalysis was performed to determine the effect of changing the number of elements,time increment, and number of time steps in DR_HAWT for the performanceprediction of the NREL Phase VI wind turbine. Default values of 26 discretizedblade elements used to define the geometry of each blade in Ref. [4], a timeincrement of 0.03145 s and 200 time steps were both halved and doubled. Theoutcome of this study revealed that the default values were adequate for thissimulation. Typically, converged values of power, torque and thrust for a given oncomingwind speed are obtained after the wake has travelled 3–4 diameters downstream.Correspondingly, a single-rotor configuration for 200 time steps with a timeincrement of 0.03145 s, has a simulation runtime of approximately 60 min usingan i7 3.60 GHz processor with 16 GB of RAM; even faster runtimes can be obtainedif the code is run with parallel processing. This highlights DR_HAWT’s ability torun parametric studies of a single or dual-rotor configuration in a timely mannercompared to existing commercially available CFD packages.
3 Results and Discussion
Figure 5 illustrates the output power predicted by DR_HAWT using aerodynamiccoefficients from XFOIL, wind tunnel (2D) experiments, wind tunnel data correctedfor 3D flow effects using Snel et al. and Du and Selig, as well as the UAEexperimental data. Based on XFOIL data, DR_HAWT resulted in an average
Fig. 5 Aerodynamic power prediction produced by DR_HAWT 500 K. Lee Slew et al.
error below 20 % when compared with the experimental data, which is withinthe measurement error bars [11]. With the 2D wind tunnel data, however, largedifferences in excess of 100 % were obtained in the stall region (10–25 m/s). The large variation in aerodynamic power curves indicate that the input aero-dynamic coefficient is an influencing factor in the output results produced byDR_HAWT. As seen in Fig. 5, this mainly affects the outcome in the stall region(10–25 m/s) where separation and 3D flow effects are present on the blade. Due to the rotational nature of the wind turbine, the flow over the wind turbineblade is not 2D but it is in fact 3D which has been found to delay stall. The pressuregradient due to rotation of the blade invokes flow in the radial direction which altersthe boundary layer, hence resulting in higher lift and lower drag values. For thisreason, wind turbines generally will obtain higher power values than predicted whenusing 2D wind tunnel airfoil data alone [10]. Although closer to the experimentallyobtained aerodynamic power, particularly in the pre-stall (below 7 m/s) and stallregimes (7–15 m/s), the 3D corrections still do not produce favourable results whendeep-stall is prominent (above 15 m/s). According to Vermeer et al., the use of 3Dcorrections at high wind velocity is questionable [12]. No matter if XFOIL, 2D wind tunnel or 3D corrected aerodynamic coefficientdata is used for the simulations, an over prediction in power up to 7 m/s windspeeds can be observed. The consistency in power values obtained by the varioussimulations is due to the fact that the majority of the blade is in the linear, pre-stall, region of the lift and drag coefficient curves. As illustrated in Fig. 4, thislinear region does not vary substantially between the different methods of obtainingthe aerodynamic airfoil data since the flow is attached. DR_HAWT was found toconsistently over predict the tangential force compared to experimental values [11].Under the assumption that the CL and CD are not contributing factors in the pre-stall region, this discrepancy must result from an inaccuracy in the method usedto determine the AOA. The challenge of predicting the AOA was also identifiedby Lindenburg [10], Sant et al. [13] and Jonkman [14] suggesting that furtherinvestigation into the methods used by DR_HAWT is required.
4 Dual-Rotor Configuration
In order to examine the effects of adding a second rotor to the HAWT, a half-sized, geometrically scaled NREL Phase VI rotor and an unmodified NREL UAEPhase VI rotor were simulated in the upwind and downwind positions, respectively.The upwind rotor had a rotational velocity twice that of the unmodified rotor inorder to maintain similar tip speeds. The rotors were separated coaxially by half thediameter of the unmodified rotor. The contra-rotating dual-rotor configuration wassimulated using XFOIL 2D data for wind speeds ranging from 5 to 25 m/s. As canbe seen in Fig. 6, the addition of the upwind rotor resulted in an average increasein power of 16 % over the single rotor configuration. The power generated by thedual-rotor configuration is lower than that of the combined individual single rotors Dual-Rotor Horizontal Axis Wind Turbine Code (DR_HAWT) 501
by approximately 6 %. This is as expected since the presence of the wake from onerotor onto another is typically detrimental to their performance. A visualization ofthe dual-rotor configuration post-processed in ParaView [15] can be seen in Fig. 7.
Fig. 6 Aerodynamic power comparison of a single and dual-rotor HAWT configuration
Fig. 7 Wake visualization of a dual-rotor configuration where the dots represent the end points ofa vortex filament 502 K. Lee Slew et al.
5 Conclusion
An in-house code, named DR_HAWT was created by the present authors for theaerodynamic performance prediction of a single or dual-rotor HAWT configuration.Converged values of aerodynamic loads are obtained within approximately 60 min.DR_HAWT was validated against the NREL UAE Phase VI experiment withrelatively good agreement. Power predicted by DR_HAWT using numericallyobtained data from XFOIL, fell within the experimental error bars and had an overallerror of less than 20 %. This adequate aerodynamic performance prediction and fastcomputational run times facilitate parametric studies of a single or dual-rotor HAWTconfiguration. Further analyses revealed that the input lift and drag coefficient data have a largeinfluence on the predicted aerodynamic forces. 3D flow effects were found to havea profound significance particularly in the stall region. 3D corrections formulatedby Snel et al. and Du and Selig were applied to 2D coefficient data and found toshow improvement with the exception of the deep stall region (above 15 m/s). Acontra-rotating dual-rotor HAWT configuration based on the NREL Phase VI rotor,was found to produce 16 % more power than the single NREL Phase VI rotor alone,demonstrating the potential benefits of a dual-rotor wind turbine configuration.
Acknowledgements The authors would like to thank NSERC (National Sciences and EngineeringResearch Council of Canada) and OCE (Ontario Centres of Excellence) for the financial supportas well as Scott Schreck of the National Renewable Energy Laboratory for providing experimentaldata for validation purposes.
References
1. Hansen, M.: Aerodynamics of Wind Turbines. EarthScan, London (2008) 2. Strickland, J., Webster, B., Nguyen, T.: A Vortex Model of the Darrieus Turbine: An Analytical and Experimental Study. Sandia National Laboratories, Lubbock (1980) 3. Fereidooni, A.: Numerical Study of Aeroelastic Behaviour of a Troposkien Shape Vertical Axis Wind Turbine – MASc Thesis. Carleton University, Ottawa (2013) 4. Hand, M., Simms, D., Fingersh, L., Jager, D., Cotrell, J., Schreck, S., Larwood, S.: Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaigns. National Renewable Energy Laboratory, Golden (2001) 5. Drela, M.: XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils. Dept. of Aeronautics and Astronautics, MIT, Cambridge (1989) 6. Du, Z., Selig, M.S.: A 3-D Stall-Delay Model for Horizontal Axis Wind Turbine Performance Prediction. Am. Inst. Aeronaut. Astronaut. 21, 9–19 (1998) 7. Snel, H., Houwink, R., Bosschers, J.: Sectional Predicition of Lift Coefficients on Rotating Wind Turbine Blades in Stall. Energy Research Centre of the Netherlands, Petten (1994) 8. Bak, C., Johansen, J., Anderson, P.B.: Three-Dimensional Corrections of Airfoil Character- istics Based on Pressure Distributions. In: European Wind Energy Conference & Exhibition, Athens (2006) 9. Lindenburg, C.: Modelling of Rotational Augmentation Based on Engineering Considerations and Measurements. In: European Wind Energy Conference, London (2004) Dual-Rotor Horizontal Axis Wind Turbine Code (DR_HAWT) 503
10. Lindenburg, C.: Investigation into Rotor Blade Aerodynamics: Analysis of the Stationary Measurements on the UAE Phase-VI Rotor in the NASA-Ames Wind Tunnel. Energy research Centre of the Netherlands, Petten (2003)11. Schreck, S.: Sequence S Data. National Renewable Energy Laboratory via Private Communi- cations, Golden (2015)12. Vermeer, L., Sørensen, J., Crespo, A.: Wind Turbine Wake Aerodynamics. Prog. Aerosp. Sci. 39, 467–510 (2003)13. Sant, T., van Kuik, G., van Bussel, G.: Estimating the Unsteady Angle of Attack from Blade Pressure Measurements on the NREL Phase VI Rotor in Yaw using a Free Wake Vortex Model. In: 44th AIAA Aerospace Sciences Meeting and Exhibition, Reno (2006)14. Jonkman, J.: Modeling of the UAE Wind Turbine for Refinement of FAST_AD. National Renewable Energy Laboratory, Golden (2003)15. Ayachit, U.: The ParaView Guide: A Parallel Visualization Application. Kitware, Inc., Clifton Park, NY (2015) Numerical Study of the Installed ControlledDiffusion Airfoil at Transitional ReynoldsNumber
Hao Wu, Paul Laffay, Alexandre Idier, Prateek Jaiswal, Marlène Sanjosé,and Stéphane Moreau
Abstract Reynolds-Averaged Navier-Stokes simulations have been carried out forthe self-noise study of a Controlled Diffusion airfoil at chord Reynolds numberof 1:5 105 . Two numerical setups of the anechoic open-jet facilities whereexperimental data on aerodynamics have been collected using hot-wire and particleimage velocimetry measurements are investigated to capture installation effects andto compare different turbulent models. The installation effects are observed betweendifferent wind tunnel nozzle dimensions. Different turbulent models in simulationshave been compared for such flow case. The k!SST model shows better agreementwith the experimental data at a reasonable computational cost, providing a reliableinitialisation field and boundary conditions for future direct numerical simulationstudies.
1 Introduction
In the design process of a rotating machine such as an automotive engine coolingfan, a wind turbine or an air-conditioning unit, one major evaluation index is thenoise level for a given loading. Where other noise sources can be reduced oravoided by a careful design, the trailing-edge (TE) noise, or airfoil self-noise, is theonly remaining noise source when an airfoil encounters a hom*ogeneous stationaryflow. Such a noise is generated by the interaction between the TE with pressurefluctuations convecting in a boundary layer. If the boundary layer is laminar, tonalnoise will be introduced; if the boundary layer is turbulent, pressure fluctuations arepresent over a wide range of frequencies, which leads to broadband noise radiation.The study of TE noise received much attention mainly in the 1970s and early1980s. Most of such studies during that time involved experimental measurementsof wall-pressure fluctuations and far field acoustic on various airfoils in open-jetanechoic wind tunnels [1, 2]. In recent years, Computational Fluid Dynamics (CFD)
H. Wu () • P. Laffay • A. Idier • P. Jaiswal • M. Sanjosé • S. MoreauDepartment of Mechanical Engineering, Université de Sherbrooke, Sherbrooke, QC J1K2R1,Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 505J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_46 506 H. Wu et al.
methods have become an important complementary part aside from the experimentalmethods. In practice however, simplified flow configurations are often employedin simulations whereas most trailing-edge aeroacoustics experiments have beenconducted in open-jet wind-tunnel facilities, where the airfoil is immersed in a jetdownstream of the nozzle exit. It has been seen that the proximity of the airfoilto the jet nozzle exit and the limited jet width relative to the airfoil thicknessand chord length can cause the airfoil loading and flow characteristics to deviatesignificantly from those measured in free air and hence, alter the radiated noisefield [3]. Installation effects thus take place for different jet configurations andrequire simulations to model it. As aerodynamics has direct influence on aeroacoustics, the present work aims atpresenting and quantifying the installation effects as well as studying the influenceof different turbulent models on such flow case simulation. Experimental andnumerical setups are firstly introduced with associated technical methods. Resultson a systematic CFD study, based on Reynolds-Averaged Navier-Stokes (RANS)are then presented and are compared with flows data over the Controlled Diffusion(CD) airfoil (a cambered airfoil originally developed at Valeo Motors and Actuators)installed in the anechoic wind tunnels of Ecole Centrale Lyon (ECL) and ofUniversité de Sherbrooke (UdS). The evaluation of different turbulence models andinstallation effects observed in simulations will be discussed. The results give asynthesis of the previous RANS study and provide guidance for the appropriateinitial field needed in future direct numerical simulation (DNS) studies [4, 5].
2 Experimental Setup
All the measurements were done in an open-jet anechoic wind tunnel shown inFig. 1a. The airfoil mock-up (Fig. 1b) is held between two side plates in a 30 30 cm2 test section, to keep the flow two-dimensional. This setup is very similar tothat used in [3, 6] at ECL, of which, the jet nozzle dimension is 50 25 cm2 . Theself-noise is the noise radiated by the turbulent eddies coming from the turbulentboundary layer. The setup should be free of any additional noise source that wouldmake the self-noise. In particular, very low turbulence intensity is achieved by avery high convergent ratio of about 1:25. The measured turbulence intensity is 0.3–0.4 %. The wind tunnel is also acoustically treated to achieve low background noise.According to the above mentioned criteria, it becomes clear that open-jet anechoicwind tunnel is best suited for the study of airfoil self-noise [7]. The hot wire andPIV measurements have been performed in the wake and pressure sensor probes onthe airfoil to detect airfoil loading, as described below. Numerical Study of CD Airfoil at Transitional Re Number 507
Fig. 1 (a) Front view of the CD airfoil in the anechoic wind tunnel and (b) Close side-view
2.1 Wall Pressure Measurements
The mean wall pressure on the CD airfoil was measured using a Baratron capaci-tance manometer which is connected to a pin hole on the surface of the airfoil usinga capillary tube. There are in total 21 probes, 18 of them are placed in streamwisedirection and rest 3 in spanwise direction. All the streamwise probes are placed atthe mockup mid-span as shown in Fig. 1b.
2.2 Velocity Measurements
The streamwise velocity in the wake was obtained using a TSI 1210-T1.5 singlehot-wire probe The length of the probe is 1.27 mm and a diameter of 3:8 m. Thedisplacement of the hot wire in the wake was realized using a Superior ElectricM062-FD03 Slo-Syn Stepping Motor. The minimum displacement of the hot wire(the highest spatial resolution) in the near wake was 0.05 mm. The hot-wire probeis connected to a Constant Temperature Anemometer System IFA 300. The dataacquisition was achieved using a National Instrument BNC 2090 system controlledwith Labview at the sampling frequency of 20 kHz. The relative measurement errorwas calculated taking into account parameters listed in [8]. It was found to equal to2.215 %. Planar PIV (Particle Image Velocimetry) measurements have been performedusing a single LaVision sCMOS 5.5 megapixel camera with a pixel pitch of 6.5 mand a Evergreen 70 mj ND:YAG laser [9]. The laser-sheet thickness was measuredto be about 2 mm. 700 images in double frame were recorded using sCMOScamera fitted with Nikon 50 mm lens at an acquisition rate of 15 Hz. The image 508 H. Wu et al.
magnification was about 0.07. The particle image diameter and depth of field wereadjusted using lens aperture [9], chosen to be 11 for the current experiments.This is done so that particle image diameter is greater than 2 pixels (calculatedby neglecting any lens aberrations). At this image diameter under sampling ofparticle image which can cause peak locking can be avoided [10]. The depth of fieldcalculated was about 60 mm, which is much larger than the measured laser sheetthickness so all the particles can be assumed in focus. Time between frames wasselected based on many considerations like minimization of loss-of-pair due out ofplane motion, truncation error due to constant velocity assumption of the particlebetween the two frames and minimization of relative error on the displacementestimate[11]. Finally it must be remembered that the dynamic velocity range ofPIV increases with time separation between the two frames [12]. Taking all theseconsiderations into effect, time between two frames was chosen to be about 45 s.The free stream displacement particle between the two frames was measured tobe roughly about 7.5 pixels in the object plane. Taking the smallest resolvabledisplacement fluctuation to be 0.1 pixel, the dynamic velocity range was calculatedto be about 75 [9]. The seeding density during the experiments was kept at about0.055 ppp to ensure a more that 10 particles were present for the final interrogationdomain for a better signal-to-noise ratio in cross correlation [13]. The parametersused for PIV measurements are summarized below in Table 1. All the results wereprocessed using DAVIS 8 software. To improve the accuracy in peak fitting andsub-pixel accuracy, normalized cross correlation option in DAVIS 8 was selectedfor the calculation of cross correlation. The final interrogation window size waskept at 16 16 pixel with a 50 % overlap. An adaptive window shape was selectedfor image cross correlation to take into account the effect of shear in the nearwake. The error analysis in PIV is a topic of active research and depends uponmany factors. In absence of detailed error analysis the value of error was chosento be equal to 0.1 pixel which corresponds to a typical error in measurement ofdisplacement in PIV [9]. On the other hand typical error on pulse separation hasan order of magnitude in nanoseconds while time between frames is in the order
Table 1 Parameters used in Parameters Valuethe PIV measurement Depth of focus 59 mm Seeding density > 0:055 ppp Mode Double frame Frequency 50 Hz Number of images 700 Window of interrogation 16 16 Focal length 57.41 mm FOV 240 mm Particle image diameter 2.33 pixels Magnification 0.07 Numerical Study of CD Airfoil at Transitional Re Number 509
of microseconds hence relative error is very small, compared to relative error indisplacement.
3 Numerical Setup
3.1 Flow Conditions and Physical Models
The flow conditions for CD airfoil are a freestream velocity U0 of 16 m/s measuredat the wind tunnel nozzle exit away from the airfoil. The Mach number is 0.05 andthe Reynolds number is 1:5 105 based on the airfoil chord length c D 0:1356 m.The flow is therefore turbulent or transitional. To model the turbulence in RANSsimulations, 4 turbulence models are tested: k (standard and with low Reynoldsnumber correction), k! SST, and tr k kl !. As the RANS results willserve as the initialization field for future DNS study, the transitional model tr k kl w is chosen for its capacity to capture transition process in the simulation.Air is supposed to be an incompressible perfect gas. The geometric angle of attackwith respect to the wind tunnel axis is 8ı . To evaluate the airfoil loading and theinstallation effects, the pressure coefficient Cp and the friction coefficient Cf areintroduced as shown respectively in Eq. (1),
Ps P0 n Cp D 1 2 and Cf D 1 2 (1) 2 0 U0 2 0 U0
where Ps stands for the static pressure, n for the local wall-shear stress and allthe parameters with the sub-index 0 correspond to the reference values in this flowcase. The values of the results shown afterwards in the section Results are non-dimensionalized based on c and U0 .
3.2 Numerical Parameters
To introduce the installation effects, the simulation domain includes the completenozzle geometry to assess the effect of the interaction between the jet shear layerand the airfoil (Fig. 2). A 2D domain which represents the mid-section of theExperimental Setup is generated by Centaur. Hybrid grids with a total 69,000 cellsare employed to get a balance between simulation accuracy and computational costaccording to what was reported in previous simulations [3] Quadrilaterals are usedto refine the grid close to the walls in order to capture the boundary layer aroundthe airfoil and the shear layer effect of the wind tunnel exit as shown in Fig. 2b,c. The dimensionless wall-normal grid spacing in wall units yC is smaller than1 over most of the chord length on both pressure and suction sides. The mesh for 510 H. Wu et al.
Fig. 2 Hybrid mesh (a); Zoom view of airfoil trainling-edge and nozzle exit (b) (c)
ECL installation is generated using the same methodology as in Fig. 2. The RANSsimulations are performed using ANSYS FLUENT-15. The pressure-based solverprovides two types of pressure-velocity coupling algorithms: either in a segregatedmanner (SIMPLE) or in a coupled manner (Coupled). The results demonstrateno difference in the wake velocity profiles or loading of the airfoil. The Coupledscheme takes 2 3 times longer than SIMPLE. In the present paper, calculationspresented are performed using the SIMPLE scheme only.
4 Results
4.1 Effects of Turbulence Models4.1.1 Velocity and Turbulence Kinetic Energy Field
All models are able to establish a converged wake zone except for the tr k kl !model as shown in Fig. 3. Even from the established field obtained with the k!SSTmodel that can be seen as a reference from previous studies[3], the scaled residualsof the tr k kl ! model oscillate around much higher levels, two orders ofmagnitude higher than other models. With regards to turbulence, the k over-predicts the turbulence kinetic energy (TKE) (Fig. 3). By using damping functions,the k model with low Reynolds-number correction has been implemented inANSYS Fluent and shows better behaviour than the standard k model. Yet it Numerical Study of CD Airfoil at Transitional Re Number 511
Fig. 3 Velocity (left) and turbulence kinetic energy (right) iso-contours
still over-predicts the turbulence over the whole profile. The k! SST model onlysees the transition process to turbulence close to the trailing edge. The tr k kl ! has unphysical boundary-layer development at the TE which is caused byhigh numerical instabilities. To make a fair comparison, the simulations for tr k kl ! have then been conducted in a URANS (Unsteady Reynolds AveragedNavier-Stokes) mode and then averaged to get a steady solution.
4.1.2 Airfoil Loading
The tr k kl ! model averaged data is closer to the mean wall-pressuremeasurement in the TE region and has similar prediction capacity as the k! SSTmodel at other positions along the airfoil (Fig. 4). Compared with the k! SSTmodel or other k! based models, the tr k kl ! model includes three transportequations to solve the turbulent kinetic energy (kt ), the laminar kinetic energy(kl ) and the scale-determining variable ! where ! D =kt , the ratio of isotropicdissipation and the turbulence kinetic energy respectively. In the ! equation, atransitional production term is introduced to produce a reduction in turbulencelength scale during the transition breakdown process, which essentially helpspredict the magnitude of low-frequency velocity fluctuations in the pre-transitional 512 H. Wu et al.
-Cp 0
k-ε k-ε Low-Re kω-SST tr-k-kl-ω (RANS) tr-k-kl-ω (URANS) exp -1 -1 -0.8 -0.6 -0.4 -0.2 0 x/cFig. 4 Comparison of mean wall-pressure coefficient
boundary layer that have been identified as the precursors to transition [14]. As thenumerical transitional process at UdS appears around TE, this model has shown itsadvantage on capturing such fluctuations compared with other tested models. Theunsteadiness appearing in the tr k kl ! RANS simulation at the TE can bepotentially verified in the future by measurements of wall-pressure fluctuations.
4.1.3 Velocity Profile in the Wake Region
The wake velocity profile is a consequence of the transition process on the airfoil.Here the velocity magnitude from different models are compared with both hot-wireand PIV measurements in Fig. 5. Simulation results tend to over-predict the velocitydeficit at all measuring positions but are consistent with their respective productionof turbulence. Some of the discrepancies can be attributed to the isotropic hypothesison calculating the turbulence kinetic energy made in the RANS simulation. The kwith low Reynolds number correction model improves the result from the standardk model by nearly 10 %. Yet, the k!SST model shows better overall comparisonwith the experimental data. In summary, for RANS simulations of such a flow case, the tr k kl !model must be run in an unsteady mode and its mean solution is similar to thatof the k! SST model with an airfoil loading even slightly better comparedwith experiment. The k! SST model provides the best overall prediction amongthe tested models considering its reasonable computational time. It is thus usedas a reference model for future comparisons. By using damping functions, the Numerical Study of CD Airfoil at Transitional Re Number 513
(a) k-ε (b) (c) k-ε Low-Re kω-SST 0.1 tr-k-kl-ω (RANS) 0 tr-k-kl-ω (URANS) HW PIV 0
y/c y/c y/c 0 -0.1
-0.1
-0.1 -0.2 0.5 1 0.5 1 0.5 1
Ut/U0 Ut/U0 Ut/U0
0.1 0
y/c y/c y/c 0 -0.1
-0.1
-0.1 -0.2 0 0.02 0.04 0.06 0.08 0 0.01 0.02 0.03 0.04 0.05 0 0.005 0.01 0.015 0.02 0.025 0.03 2 2 2 k/U0 k/U0 k/U0
Fig. 5 Wake velocity and TKE profiles. (a) x=c D 0:07. (b) x=c D 0:22. (c) x=c D 0:44
two-equation k model with low Reynolds number correction shows betterbehaviour in wake velocity profiles than the the standard k model.
4.2 Installation Effects
As firstly mentioned by Moreau et al. [3], installation effects are significant betweenan isolated airfoil case and one that is installed in an open-jet wind tunnel. In thispaper, the effects of nozzle jet width is specifically investigated by comparing resultsin two different setups at ECL and UdS respectively. The loading of the airfoil is changed as can be seen from the wall pressurecoefficient in Fig. 7. This is mostly caused by a different transition process. AtECL, a separation bubble turns the laminar boundary layer into a turbulent oneat the leading edge, which is hardly the case in the UdS set-up for which, thetransition process is less severe. Indeed Fig. 6 shows that the friction coefficientCf is barely negative in the UdeS case and the laminar recirculation bubble is hardlyformed. Because of the more limited nozzle jet width at UdS, the boundary layer 514 H. Wu et al.
0.2 RANS-UdS RANS-ECL
Leading-edge area 0.15 0.03
0.02
0.1 Cf 0.01
0 0.05 -0.01
-1 -0.95 -0.9
-1 -0.8 -0.6 -0.4 -0.2 0 x/c
Fig. 6 Mean friction coefficient: global and zoom in the leading-edge area
2 RANS-UdS RANS-ECL Exp-UdS Exp-ECL 1
-Cp
-1 -1 -0.8 -0.6 -0.4 -0.2 0 x/cFig. 7 Mean wall pressure coefficient
at the leading edge develops in a different way as a consequence of the larger flowconfinement. Besides, there is a difference of 15 % on Cp at the TE in Fig. 7 betweenthe experimental data and the simulation results at UdS as no flow separation iscaptured in this simulation. Consequently, the RANS simulations have a hard timepredicting the transition process correctly, and the flow separation either at theleading edge (laminar recirculation bubble) or at the TE (driven by the adversepressure gradient). Numerical Study of CD Airfoil at Transitional Re Number 515
5 Conclusion
RANS simulations have yielded the transitional flow around a CD airfoil installedin anechoic wind tunnels for future TE noise study. The results have been system-atically compared with experimental data. Different turbulence models have beentested to compare their capacity to capture transitional process. The k! SSTmodel has a better global behaviour considering the reasonable computational costand is thus considered as the reference model for future RANS simulations. Thetransitional model trkkl! slightly improves the airfoil mean loading accordingto the averaged URANS data. Installation effects for the two setups with different jetnozzle size have been studied for flow with same inlet condition. In the larger nozzlecase, a laminar recirculation bubble that triggers the boundary layer developmenton the suction side is observed while in the other case not. RANS simulations showlimitations in predicting the laminar transition process and flow separations underadverse pressure gradient.
References
1. Brooks, T.F., Hodgson, T.H.: Trailing edge noise prediction from measured surface pressures. J. Sound Vib. 78(1), 69–117 (1981) 2. Fink, M.R.: Experimental evaluation of theories for trailing edge and incidence fluctuation noise. AIAA J. 13(11), 1472–1477 (1975) 3. Moreau, S., Henner, M., Iaccarino, G., Wang, M., Roger, M.: Analysis of flow conditions in freejet experiments for studying airfoil self-noise. AIAA J. 41(10), 1895–1905 (2003) 4. Wang, M., Moreau, S., Iaccarino, G., Roger, M.: LES prediction of wall-pressure fluctuations and noise of a low-speed airfoil. Int. J. Aeroacou. 8(3), 177–197 (2009) 5. Winkler, J., Sandberg, R.D., Moreau, S.: Direct numerical simulation of the self-noise radiated by an airfoil in a narrow stream. In: 18th CEAS/AIAA Aeroacoustics Conference, Colorado Springs, pp. 2012–2059 (2012) 6. Neal, D.R.: The effects of rotation on the flow field over a controlled-diffusion airfoil. Ph.D. thesis, Michigan State University (2010) 7. Vathylakis, A., Kim, J.H., Chong, T.P.: Design of a low-noise aeroacoustic wind tunnel facility at Brunel University. In: 20th AIAA/CEAS Aeroacoustic Conference and Exhibit, Atlanta (2014) 8. Jorgenson, F.: How to Measure Turbulence with Hot Wire Anemometers. Dantec Dynamics, Skovlunde (2004) 9. Raffel, M., Willert, C.E., Kompenhans, J.: Particle Image Velocimetry: A Practical Guide. Springer, Heidelberg/New York (2013)10. Westerweel, J.: Fundamentals of digital particle image velocimetry. Meas. Sci. Technol. 8(12), 1379 (1997)11. Adrian, R.J., Westerweel, J.: Particle Image Velocimetry, Number 30. Cambridge University Press, Cambridge/New York (2011)12. Adrian, R.J.: Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Meas. Sci. Technol. 8(12), 1393 (1997)13. Keane, R.D., Adrian, R.J.: Optimization of particle image velocimeters. I. Double pulsed systems. Meas. Sci. Technol. 1(11), 1202 (1990)14. Walters, D.K., co*kljat, D.: A three-equation eddy-viscosity model for Reynolds-averaged Navier–Stokes simulations of transitional flow. J. Fluids Eng. 130(12), 121401 (2008) Part IVMathematics and Computation in Finance, Economics, and Social Sciences Financial Markets in the Context of the GeneralTheory of Optional Processes
M.N. Abdelghani and A.V. Melnikov
Abstract A probability space is considered “unusual” if the information flow isnot right or left continuous or is not complete. On these probability spaces livescertain type of stochastic processes known as optional processes including optionalsemimartingales. Optional semimartingales have right and left limits but are notnecessarily right or left continuous. Here, we present a short summary of thecalculus of optional processes and define stochastic logarithms and present someof its properties. Moreover, we develop a financial market model based on optionalsemimartingales and methods for finding local martingale deflators for this market.Finally, we present some financial examples.
1 Introduction The assumption that the stochastic basis ˝; F ; F D .F t /t0 ; P satisfy the usualconditions, where F is complete and right-continuous, is a foundation concept inthe theory of stochastic processes. Stochastic processes that are adapted to this basisform a large class of processes known as semimartingales whose paths are right-continuous with left limits (RCLL). This theory has been instrumental in generatingmany important results in mathematical finance and in the theory of stochasticprocesses. It is difficult to conceive of a theory of stochastic processes without theusual conditions and RCLL processes. However, it turns out that it is not difficult to give some examples of stochasticbasis without the usual conditions. In 1975 Dellacherie [4] started to study the theoryof stochastic processes without the assumption of the usual conditions, termed“unusual conditions”. Further developments of this theory were carried on by Lepingle [12], Horowitz[8], Lenglart [11], and mostly by Galtchouk [6, 7]. We believe that the theory optional processes will offer a natural foundationand a versatile set of tools for modeling financial markets. We can mention herefew research problems that were not treated with the methods of the calculus of
M.N. Abdelghani () • A.V. MelnikovUniversity of Alberta, Edmonton, AB, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 519J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_47 520 M.N. Abdelghani and A.V. Melnikov
optional processes but possibly should be. The first one, is a recent developmentin mathematical finance specially in pricing of derivative contracts and hedgingunder transaction costs (see [2] for details) that hints to the needed application of thecalculus of optional processes to price derivatives and hedge under transaction costs.Furthermore, in models [1] with stochastic dividends paid at random times, there isan opportunity to treat these problems in the context of optional semimartingaletheory in a natural way. Duffie [5] presented a new approach to modeling termstructures of bonds and contingent claims that are subject to default risk. PerhapsDuffie’s method could be studied with the methods of optional calculus. The aim ofthis paper is to present the theory of optional processes on unusual stochastic basis,develop new results and bring its methods to mathematical finance. The paper isorganized as follows. Section 2 presents foundation material on optional processes.Section 3 introduces stochastic exponentials and logarithms. Section 4 describesoptional semimartingale model of a financial market and two methods for findinglocal martingale deflators for these markets. Section 5 presents examples of optionalsemimartingale markets. Finally, we give some concluding remarks.
2 Foundation Suppose we are given ˝; F ; F D .F t /t0 ; P , t 2 Œ0; 1/, where F t 2 F; F s F t ; s t, a complete probability space. It is complete because F containsall P null sets. But this space is unusual because the family F is not assumedto be complete, right or left continuous. On this space, we introduce O.F/ andP.F/ the -algebras of optional and predictable processes, respectively (see [7]).A random process X D .Xt /, t 2 Œ0; 1/, is said to be optional if it is O.F/-measurable. In general, optional processes have right and left limits but are notnecessarily continuous. For an optional process we can define the following: (a)X D .Xt /t0 , a left continuous version of the process X and XC D .XtC /t0 ,the right continuous version of X; (b) The jump processes 4X D .4Xt /t0 and4Xt D Xt Xt and (c) 4C X D .4C Xt /t0 , 4C Xt D XtC Xt . A randomprocess .Xt /; t 2 Œ0; 1/, is predictable if X 2 P.F/ and strongly predictable ifX 2 P.F/ and XC 2 O.F/. We denote by P s .F/ the set of strongly predictableprocesses. An optional semimartingale X D .Xt /t0 is an optional process thatcan be decomposed to an optional local martingale M 2 M loc and an optionalfinite variation processes A 2 V , i.e. X D X0 C M C A, [7]. A semimartingaleX is called special if the above decomposition exists but with A being a stronglypredictable process (A 2 A loc the set of locally integrable finite variation processes[7]). Let S denote the set of optional semimartingales and S p the set of specialoptional semimartingales. If X 2 S p then the semimartingale decompositionis unique. Using the decomposition optional martingales and of finite variationprocesses we can decompose a semimartingale further to X D X0 C X r C X g DX0 C .Ar C M r / C .Ag C M g / where Ar and Ag are right and left continuous finite Financial Markets in the context of The General Theory of Optional Processes 521
variation processes, respectively. M r 2 M rloc is a right-continuous local martingale gand M g 2 M loc is a left-continuous local martingale. The stochastic integral with respect to optional semimartingale X is defined as abilinear form .'; / ı Xt where Z t Z t g g Yt D .'; / ı Xt D Y0 C ' Xtr C ˇ Xt D 's dXsr C s dXsC : 0C 0
where Yt is again an optional semimartingale ' 2 P.F/; and 2 O.F/, such that 2 1=2 1=2 Rt ' ŒX r ; X r 2 A loc and 2 ˇ ŒX g ; X g 2 A loc . The integral 0C 's dXsris our usual R t stochastic integral with respect to RCLL semimartingale however the gintegral 0 s dXsC is Galchuk stochastic integral [7] with respect to left-continuoussemimartingale.
3 Stochastic Exponentials and Logarithms
Stochastic exponentials and logarithms are indispensable tools of financial mathe-matics (see, Melnikov et al. [13]). They describe relative returns, link hedging withthe calculation of minimal entropy and therefore utility indifference. Moreover, theydetermine the structure of the Girsanov transformation. For optional semimartingales the stochastic exponential was defined by Galchuk[7]. If X 2 S then there exists a unique semimartingale Z 2 S such that Z t Z t g Zt D Z0 C Z ı Xt D Z0 C Zs dXsr C Zs dXsC D Z0 E .X/t ; (1) 0C 0
Two useful properties of stochastic exponentials are the inverse and productformulas. The inverse of stochastic exponential is E 1 .h/ D E .h /, such that,
X .4hs /2 X .4C hs /2 ht D ht hhc ; hc it : 0<st 1 C 4hs 0s<t 1 C 4C hs
And, the product of stochastic Pexponentials is E .U/E P .V/ D E .U C V C ŒU; V /where ŒU; V t D hU c ; V c it C 0<st 4Us 4Vs C 0s<t 4C Us 4C Vs : The stochastic logarithm is defined by the following theorem.Theorem 1 Let Y be a real valued optional semimartingale such that the processesY and Y do not vanish then the process Z Z 1 t 1 t 1 g Xt D ı Yt D dY r C dY ; X0 D 0; (2) Y 0C Ys s 0 Ys sC 522 M.N. Abdelghani and A.V. Melnikov
also denoted by X D L ogY is called the stochastic logarithm of Y. The processX is a unique semimartingale such that Y D Y0 E .X/. Moreover, if X ¤ 1 and C X ¤ 1 we also have ˇ ˇ X ˇˇ ˇ ˇ Yt ˇ 1 Ys ˇˇ Ys L ogYt D log ˇˇ ˇˇ C 2 ı hY c ; Y c it log ˇˇ1 C (3) Y0 2Y 0<st Ys ˇ Ys ˇ ˇ X ˇ C Ys ˇˇ C Ys log ˇˇ1 C : 0s<t Ys ˇ Ys
It is important to note that the process Y need not be positive for L og.Y/ to exist,in accordance with the fact that the stochastic exponential E .X/ may take negativevalues.Proof The assumptions that Y and Y don’t vanish implies that: Sn Dinf t W jYt j 1n " 1, hence 1=Y is locally bounded; likewise, Tn D inf t W jYt j 1n " 1, hence 1=Y is also locally bounded. Therefore, the stochasticintegral in (2) makes sense. Let YQ D Y=Y0 then YQ 0 D 1. By equation (2) we have 1 1that X D .1=Y/ ı Y. Therefore, 1 C Y ı X D 1 C Y ı YQ ı Y D 1 C Y YQ ı YQ D Y, Q Q Q Q Q Q Qi.e. YQ D E .X/. Furthermore, X D Y=Y ¤ 1 and C X D C Y=Y ¤ 1. Toobtain uniqueness let XQ be any other semimartingale satisfying Y D E .X/. Q SinceQY D Y=Y0 then YQ D E .X/. Q Therefore YQ D 1 C YQ ı XQ and Y D Y0 C YQ ı X.Q But since QX0 D 0 we have XQ D YYQ ı XQ D Y1Q ı .Y Y0 / D Y1Q ı Y D X and we obtain uniqueness. To deduce equation (3) we apply Gal’chuk-Ito’s lemma (see [7]) for the optionalsemimartingale log jYj. But the log function explodes at 0. To circumvent thisproblem consider for each n the C2 functions fn .x/ D log jxj on R such thatjxj 1=n. Consequently for all n, t < Tn and t < Sn we get log jYt j D log jY0 j C Y1 ı P P C YsYt 2Y1 2 ı hY c ; Y c it C 0<st log jYs j Y Ys s C 0s<t C log jYs j Ys .This result together with equation (2) yields equation (3) for t < Tn and t < Sn .Since, Tn " 1 and Sn " 1 we obtain equation (3) everywhere.Now, we present some of the properties of stochastic logarithms.Lemma 1 (a) If X is a semimartingale satisfying X ¤ 1 and C X ¤ 1 thenL og.E .X// D X X0 . (b) If Y is a semimartingale such that Y and Y do notvanish, then E .L og.Y// D Y=Y0 . (c) For any two optional semimartingales X andZ we get L og .XZ/ D L ogX C L ogZ C ŒL ogX; L ogZ ; the following identities:L og X1 D 1 L og .X/ X; X1 :
Proof (a) L og.E .X// D E .X/1 ı E .X/ D EE .X/.X/ ı X D X X0 . (b) Let Z DE .L og.Y// then by (a) we find that L og .Z/ D L og .E .L og.Y/// D L og.Y/ L og.Y0 / D L og.Y=Y0 /. Therefore, Z D Y=Y0 . (c) By the integral definition of 1the stochastic logarithm we find L og .XZ/ D XZ ı .X ı Z C Z ı X C ŒX; Z / DL ogX C L ogZ C ŒL ogX; L ogZ : Using integration by parts and the integral Financial Markets in the context of The General Theory of Optional Processes 523
1 1 1 definition the stochastic logarithm, L og DXı D 1 ı X X; X1 D X X X1 L og .X/ X; X1 :
4 Markets of Optional Processes Let ˝; F ; F D .F t /t0 ; P , t 2 Œ0; 1/, be the unusual stochastic basis andthat the financial market stays on this space. The market consists of two types ofsecurities x and X. A portfolio D .; /, is composed of the optional processes and . is the volume of the reference asset x while is the volume of the securityX. Suppose xt > 0 and Xt 0 for all t 0 and write the ratio process Rt D Xt =xt .Then, the value of the portfolio is
Yt D t C t Rt : (4)
We restrict the portfolio to be self-financing that is we must have,
Yt D Y0 C ı Rt . (5)
Reconciling equations (4) and (5) we obtain Ct D t C R ı t C Œ; R t D Y0 D C0where Ct is the consumption process with its initial value C0 . Since the ratio processR is optional semimartingale then , evolves in the space P.F/ O.F/ with thepredictable part determining the volume of Rr and the optional part determining thevolume of Rg . Also, belongs to the space O.F/. Furthermore, R1 for the integral inequation (5) to be well defined must be R-integrable, 0 s2 dŒR; R s 2 A loc . One can interpret the trading strategy in several different ways. First, even-though gives us the option to trade different parts of the ratio-process, Rrpredictably and Rg optionally, we can still trade both parts in the same way,predictably. Alternatively, in a portfolio of a sum of left-continuous and right-continuous assets (see example Sect. 5) we can trade each asset independently.The left-continuous asset will be traded with an optional strategy while the right-continuous asset will be traded by a predictable one. This certainly a possibleportfolio in the current structure of financial markets. Moreover, with a theoryof optimal portfolio with consumption and endowment streams we can interpretthe predictable trades of Rr as the optimal portfolio while the optional trades ofRg as an optimal consumption or endowment streams. Yet another way to thinkabout optional portfolios in financial markets is in terms of market-orders. Market-orders can either be predictable or optional. Optional in the sense that trades can beexecuted based on a condition, for example the stock price passing some boundary.Conditional trades are optional processes that act on the left-continuous part of theratio-process Rg . Finally, note that the ratio process is an optional semimartingale and must betransformed to a local optional martingale for any pricing and hedging theory tobe viable. In the next section we show how to find local martingale transforms that 524 M.N. Abdelghani and A.V. Melnikov
change the ratio process to some optional local martingales. A special subset of thesemartingale deflators can transform RLL optional semimartingales to RCLL localmartingale. This special subset of transforms can be useful in cases where marketsonly allow for predictable trading strategies. We will illustrate this procedure inexample Sect. 5. As in Melnikov et al. [13] for RCLL We suppose that the dynamics of securitiesin our market follows the stochastic exponential, Xt D X0 E t .H/ and xt D x0 E t .h/where x0 and X0 are F 0 -measurable random variables. h D .ht /t0 and H D .Ht /t0are optional semimartingales admitting the representations, ht D h0 C at C mt andHt D H0 C At C Mt with respect to (w.r.t) P. a D .at /t0 and A D .At /t0 arelocally bounded variation processes and predictable. m D .mt /t0 and M D .Mt /t0are optional local martingales. A local martingale deflator is a strictly positivesupermartingale multiplier used in mathematical finance to transform the valueprocess of a portfolio to a supermartingale (i.e. a local martingale). Here we willdevelop methods for finding local martingale deflators. We can write R as,
Xt Rt D D R0 E .H/t E 1 .h/t D R0 E .Ht /E .ht / D R0 E . .ht ; Ht //; (6) xt t D .ht ; Ht / D Ht ht ŒH; h t D Ht ht C hhc ; hc H c it C J d C J g ; X 4hs.4hs 4Hs / X 4 C h s 4 C h s 4 C Hs d J D ; J Dg : 0<st 1 C 4hs 0s<t 1 C 4C hs
if is a local optional martingale then R is a local optional martingale and we aredone. Otherwise, we have to find a strictly positive transformation Z 2 M loc thatwill render ZR 2 M loc . Z is known as the local martingale deflator. For a strictlypositive Z, we can define N 2 M loc with N D L og.Z/ D Z 1 ı Z or Z D E .N/.To find N we have the following theorem;Theorem 2 Given R D R0 E . .h; H// where .h; H/ as in equation (6) and Z DE .N/ where Z; N 2 M loc .P; F/ and Z > 0 then ZR 2 M loc is a local optionalmartingale if and only if .A a/ C hmc N c ; mc M c i C KQ d C KQ g D 0, where KQ dand KQ g are the compensators of the processes X .4hs 4Ns / .4hs 4Hs / X 4C hs 4C Ns 4C hs 4C HsKd D ; Kg D : 0<st 1 C 4hs 0s<t 1 C 4C hs
Proof Suppose Zt D E .N/t 2 M loc , Zt > 0 for all t such that ZR 2 M loc thenZR DR0 E .N/E . .h; H// D R0 E . .h; H; N//, where .h; H; N/ D Nt C Ht ht C ghhc ; hc H c it C Jtd C Jt C ŒN; H ŒN; h C ŒN; J d C ŒN; J g , hence, g .h; H; N/ D Nt C Ht ht C hhc ; hc H c it C Jtd C Jt X X C hN c ; H c it C 4Ns 4Hs C 4C Ns 4C Hs 0<st 0s<t Financial Markets in the context of The General Theory of Optional Processes 525
X X hN c ; hc it 4Ns 4hs 4C Ns 4C hs 0<st 0s<t X 4hs .4hs 4Hs / C 4Ns 0<st 1 C 4hs X 4 C h s 4 C h s 4 C Hs C C 4 Ns : 0s<t 1 C 4C hs
therefore, .h; H; N/ D Nt C Ht ht C hhc N c ; hc H c it C K d C K g . So,if .h; H; N/ 2 M loc then ZR 2 M loc . And if 4C .h; H; N/ ¤ 1 and4 .h; H; N/ ¤ 1 then .h; H; N/ 2 M loc , ZR 2 M loc . Now, let ustake into consideration the decomposition of H and h and write .h; H; N/ D.A a/ C .M m C N/ C h.m N/c ; .m M/c i C K d C K g : So, .h; H; N/ isa local optional martingale under P if .A a/ C hmc N c ; mc M c i C KQ d C KQ g D 0where KQ d and KQ g are the compensators of K d and K g , respectively.By finding all N 2 M loc such that the above equation (4) is valid and E .N/ > 0 wefind the set of all appropriate local optional martingale transforms Z such that ZR isa local optional martingale. Note that if Z is a local martingales transform such thatZR is a local martingale then it is true for all self financing strategies .Theorem 3 If Z is a local martingale transform of R, that is ZR is a local optionalmartingale, and is a self financing portfolio which is R-integrable then ZYt is alocal optional martingale.Proof Z is a local martingale transform of R therefore Z > 0. D .; / is self and R-integrable, then Yt D Y0 C ı Rt and Zt Yt can be written as,financingd Zt Yt D t ŒZt dRt C dZt Rt C d ŒZt ; Rt C dZt t D t d .Zt Rt / C dZt t : This leadsus to the following result Zt Yt D ıZt Rt CıZt : ıZt and ıZt Rt are local optionalmartingales therefore their sum Zt Yt is a well defined local optional martingale.Note, that we have implicitly used the factRthat is bounded, i.e. comes from the 1fact that is a self financing and also that, 0 t2 d ŒZR t 2 A loc .On the other hand, if we know that there exist a Z such that ZY is a local optionalmartingale then what can we say about the portfolio and the product ZR? It isreasonable to suppose that Z D E .N/ > 0, -self-financing, is R-integrable and is bounded. In this case, ı Zt Rt D Zt Yt ı Zt is a sum of two local optionalmartingales and therefore a local optional martingale it self, for any optional process, in particular for D 1; therefore ZR is a local optional martingale. An alternative approach can be developed using stochastic logarithms. What isinteresting about this approach is that we don’t have to define the process R asa stochastic exponential of an underlying process . All that is required is that theratio process R and its predictable version R don’t vanish, except on sets of measurezero. This approach is technically based on the following lemmas and results. 526 M.N. Abdelghani and A.V. Melnikov
Lemma 2 Suppose X D X0 C A C M, x D x0 C a C m, R D X=x and R ¤0 and R ¤ 0 a.s. P, then L og.R/ is a local optional martingale if and only if1 1 1 1X ı A x ı a C x2 Œm; m xX ı ŒM; m D 1.
Lemma 3 Suppose that R and R don’t vanish then L og.R/ 2 M loc , R 2M loc .If R is not a local martingale then there exists a local martingale Z > 0 such that ZRis a local martingale. The following lemma helps us with finding Z.Lemma 4 Let X D X0 C A C M, x D x0 C a C m, R D X=x and suppose thatR and R don’t vanish then L og.ZR/ is a local optional martingale if and only if1C X1 ıA 1x ıaC x12 ıŒm; m xX 1 1 ıŒM; m C ZX 1 ıŒZ; M xZ ıŒZ; m D 1 furthermoreif Z D E .N/ > 0 then X ıA x ıaC x2 ıŒm; m xX ıŒM; m C X1 ıŒN; M 1x ıŒN; m D 1 1 1 1
1:Corollary 1 Suppose that R and R don’t vanish then L og.ZR/ 2 M loc , ZR 2M loc .
5 Illustrative Examples
Consider a market composed of a bond x and an asset X evolving according to gxt D x0 E .h/t and Xt D X0 E .H/t where ht D rt C bLt ; h0 D 0, Ht D t C gWt C aLt ; H0 D 0. Lt D Lt t, Lt D LN t C t, and r, , , a, and b are d d
constants. W is diffusion term and L and LN are Poisson with constant intensity and respectively. Let F t be the natural filtration that is neither right or left continuous.Here the bond is modeled by a left continuous process for which we have assumedthat its jumps don’t necessarily avoid the jumps of the asset (see also [5] for bondsthat can experience defaults). We believe our model gives a better description of aportfolio of stocks and bonds than models that assume RCLL processes on usualprobability space. Given x and X the ratio process is Rt D Xx00 E .Ht ht ŒH; h t /. We want to findZ D E .N/ such that ZR is a local martingale. In Sect. 4 we showed that associatedwith the product ZR D Xx00 E . .h; H; N// is the process .h; H; N/. To compute a greasonable form for .h; H; N/ we suppose that Nt D &Wt C cLdt C Lt an optionallocal martingale for which Z an optional local martingale deflator, hence g g .h; H; N/ D & Wt C cLdt C Lt C t C Wt C aLdt rt C bLt ˝ g c g c g c c ˛ C rt C bLt & Wt C cLdt C Lt ; rt C bLt t C Wt C aLdt X 2 X b .b / 4C Lgs 2 C ac 4Lds C g 0<st 0s<t 1 C b4C Ls g D . r C & / t C .& C / Wt C .c C a/ Ldt C . b/ Lt C acLt Financial Markets in the context of The General Theory of Optional Processes 527
g g g C .b / Lt C b . b/ Lt ; Lt D . r C & / t C .& C / Wt C .c C a/ Ldt C acLt C b . b/ LN t :
because Wt and Ldt are martingales, .h; H; N/ is a martingale if and only if r C & C ca C b b2 D 0. The solution of this equation leads to infinitelymany solutions which means the market is incomplete. Here are some solutions. Let D 0 which leads to right continuous local martingale deflator. Another solution isa one which will eliminate jumps on drift that is by letting D 1=b the effects ofand c D 1=a, then & D r C b2 =. This is a simple but important example that we have alluded to in Sect. 4. Supposethat the market participants can’t trade the left-continuous part of the market ratioprocess R with an optional trading strategy. A way around this problem is totransform RLL semimartingles to RCLL local optional martingales. Consider amarket ratio process R D E .M/ > 0, Mt D rt C aWt C bDt C cGt where W is adiffusion process, D a right-continuous compensated Poisson with intensity and Ga left continuous compensated Poisson with intensity . Let N D ˛W C ˇD C G.We like to find Z D E .N/ > 0 such that ZR is RCLL. ZR D E .N/E .M/ DE .N C M C ŒN; M /. Therefore, ZR D E ..a C ˛/W C .b C ˇ/D C .c C /G C .r Ca˛ C bˇ C c /t/. If we choose D c we get rid of the left-continuous part ofZR. And, if we choose ˛ and ˇ such that r C a˛ C bˇ c2 D 0 we obtain theRCLL local optional martingale ZR we are searching for. However, we still need tomake sure that our choice Z D E .˛W C ˇD cG/ > 0. This is true, if and only if.1 C ˇ 4 Dt /.1 c 4C Gt / > 0 or .1 C ˇ/.1 c/ > 0 for all time t.
6 Conclusion
Optional processes including left continuous ones are a natural occurrence infinancial markets. For example, a defaultable bond is modeled by a left continuousprocess in [5], stochastic dividends in [16] and transaction costs in [2, 3]. Alsooptional processes appears for optimal consumption from investment [15], forconsumption from investment with random endowment [9, 14, 17]. Furthermore,optional processes naturally arise in the context of super-replication in incompletemarkets as a result of optional decomposition theorem [10]. Therefore, to studyfinancial markets with optional processes, left-continuous processes and a mixtureof right and left continuous processes the calculus of optional processes is goingto be an indispensable tool in the future study of mathematical finance. We haveintroduced the notions of the unusual basis and optional semimartingales, presenteda summary of the calculus of optional processes and introduced new results ofstochastic logarithms. We have also described the optional semimartingale modelof financial market and described a procedure of finding local martingale deflatorsfor this market. Finally we presented illustrative examples. The first example is of aportfolio of a left continuous bond and right continuous stock where we showed how 528 M.N. Abdelghani and A.V. Melnikov
to find local martingale deflators for this market. The second examples demostratesa method for finding a subset of local martingale deflators that transforms a ladlagsemimartingale to cadlag local optional martingale.
Acknowledgements The research is supported by the NSERC discovery grant #5901.
References
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Michael C.H. Choi and Pierre Patie
Abstract We provide a sufficient condition for the negative of the infinitesimalgenerator of a continuous-time finite skip-free Markov chain to have only realand non-negative eigenvalues. The condition includes stochastic monotonicity andcertain requirements on the transition rates of the chain. We also give a sample pathillustration of Markov chains that satisfy the conditions and its Siegmund dual. Weillustrate our result by detailing an example which also reveals that our conditionsare not necessary.
1 Introduction
A Markov chain on a denumerable state space is said to be upward skip-free if theonly upward transition is of unit size, yet it can have downward jump of any arbitrarymagnitude. In this paper, we consider a continuous time upward skip-free Markovchain X D .Xt /t0 on a finite state space E WD f0; 1; : : : ; ag with the infinitesimalgenerator G WD .G.i; j//i;j2E and transition semigroup P D .Pt /t0 , where 0 Pt D Pt G ; t 0:
Since X is upward skip-free, G.i; j/ D 0 for j > i C 1; i; j 2 E. The generator Gnecessarily satisfies 0 G.i; j/ < 1 for i ¤ j and for all j 2 E, X G. j; j/ G. j; k/ : (1) k¤j
In what follows, we assume that X is conservative (i.e. the equality in (1) holds) andG. j; j/ < 1 for j 2 E. We refer the interested readers to Asmussen [1], Norris [8]for a general reference in continuous-time Markov chains.
M.C.H. Choi () • P. PatieSchool of Operations Research and Information Engineering, Cornell University, Ithaca, NY,USAe-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 529J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_48 530 M.C.H. Choi and P. Patie
The spectrum of a skip-free Markov chain plays an important role in the study ofupward hitting time as well as the fastest strong stationary time, see e.g. Diaconisand Fill [3] and Fill [4]. It is shown in Fill [4] that if the eigenvalues of G are allreal and non-negative, then the law of the upward hitting time can be interpretedas a convolution of exponential random variables with parameters given by theaforementioned eigenvalues. However, it is not known under which condition(s)do a skip-free Markov chain admit real and non-negative spectrum. We aim to fillin this gap in the literature by providing a sufficient condition. The main idea is todevelop a similarity transformation of G with its so-called Siegmund dual GO whichis the generator of a birth-and-death process. The rest of the paper is organized as follow. We review Siegmund duality andthe concept of stochastic monotonicity in Sect. 2. They serve as important tools forour main result, which is presented in Sect. 3. Finally, we provide a sample pathillustration in Sect. 4.
2 Siegmund Duality and Stochastic Monotonicity
Siegmund duality of Markov processes was first introduced by Siegmund [9], whichwas further developed and applied by Diaconis and Fill [3], Huillet and Martinez [5],Jansen and Kurt [6].Definition 1 (Siegmund kernel) The Siegmund kernel HS D .HS .i; j//i;j2E isdefined to be
HS .i; j/ WD 1fijg ;
where 1 is the indicator function. Its inverse HS1 is
HS1 .i; j/ D 1fiDjg 1fiDj1g :
Definition 2 (Siegmund dual) We say that GO is the Siegmund dual (or HS -dual) ofG if GO is an infinitesimal generator such that
O j/ WD .HS1 GHS /T .i; j/ 0 ; G.i; i ¤ j:
That is, apart from the diagonal entries .G. O j; j//j2E , .G.i; O j//i¤j is elementwise non-negative. Next, we recall the definition of stochastic monotonicity for Markov chains.Definition 3 (Stochastic monotone) G is said to be stochastically monotone if X X G.h; k/ G.i; k/ ; 0 h i < j; (2) kj kj Real Spectrum for Skip-Free Markov Chains 531
X X G.h; k/ G.i; k/ ; 0 l < h i: (3) kl kl
It is known that the existence of Siegmund dual is equivalent to stochasticmonotonicity, which is shown in the following lemma. The proof of a more generalcase can be found in Asmussen [1, Proposition 4.1].Lemma 1 Suppose that .Xt /t0 is a continuous-time Markov chain on E withinfinitesimal generator G. The Siegmund dual GO exists if and only if G is stochasti-cally monotone. PProof For any i;P j 2 E, since HS GO T .i; j/ D GHS .i; j/ ; HS GO T .i; j/ D ki G. O j; k/and GHS .i; j/ D kj G.i; k/ , we have X X O j; i/ D G. O j; k/ G. O j; k/ G. ki k>i X D .G.i; k/ G.i C 1; k// (4) kj X D .G.i C 1; k/ G.i; k// (since G is conservative) : (5) k>j
O j/ 0 for i ¤ j if and only if G isIn view of (3) and (4), and (2) and (5), G.i;stochastically monotone. u t Finally, we recall a lemma shown in Huillet and Martinez [5] that will be used inthe proof of our main result Proposition 1.Lemma 2 Suppose that .Xt /t0 is a continuous-time Markov chain on E withinfinitesimal generator G. If G is irreducible on E (resp. on Enf0g, where 0 is anabsorbing state) and is stochastically monotone, then P1. GO is conservative except at 0 (resp. on E). That is, G.0; O 0/ > k¤0 G.0; O k/, and (1) holds (with G replaced by G)O for j ¤ 0. O2. a is the unique absorbing state in G.
3 Main Result
As X is upward skip-free, the infinitesimal generator G is non-symmetric and non-reversible (with respect to the stationary distribution of the chain), making theanalysis of the spectrum of G difficult. One way to overcome this is to developconditions such that its Siegmund dual GO corresponds to a birth-and-death process,which leads us to the following result. 532 M.C.H. Choi and P. Patie
Proposition 1 Suppose that .Xt /t0 is an upward skip-free Markov chain on E Df0; 1; : : : ; ag with infinitesimal generator G (and a 3), and assume that .Xt /t0 iseither irreducible on E or irreducible on Enf0g with 0 being an absorbing boundary.If1. G is stochastically monotone, and for every i 2 Œ0; a 2 , X X G.i C 1; k/ > G.i C 2; k/ ; and ki ki
2. for every i 2 Œ0; a 3 , X X G. j; k/ D G. j C 1; k/ 8j 2 Œi C 2; a 1 ; ki ki
then all the eigenvalues of G are real, distinct and non-negative.Proof Since G is stochastically monotone, its Siegmund dual GO exists by Lemma 1.For i 2 Œ2; a 1 , G.i; O i 1/ D G.i; i C 1/ > 0 since G is upward skip-free, O j/ D 0 8j i 2. We also haveconservative and irreducible on Enf0g, and G.i; PG.1; 0/ D G.1; 2/ > 0. For i 2 Œ0; a2 , condition 1 gives G.i; O O iC1/ D ki G.iC P1; k/ ki G.i C 2; k/ > 0. Condition 2 guarantees that G.i; O j/ D 0 for eachi 2 Œ0; a 3 and j 2 Œi C 2; a 1 . That is, G is an irreducible birth-and-death Oprocess when restricted to the state space f0; : : : ; a 1g. Moreover, by Lemma 2, a O and as a result the row corresponding to state a isis the unique absorbing state of G,zero. Denote by G the restriction of GO to f0; : : : ; a 1g. By breaking off the last O BD
row and last column of G,O we can write GO BD h GO D D .HS1 GHS /T ; (6) 0T 0
O a/ forwhere 0 is a column vector of zero, and h is a column vector storing G.i;i 2 Œ0; a 1 . Since GO is a similarity transformation of G, they share the sameeigenvalues. In addition, we observe from (6) that the eigenvalues of G are 0 andthat of GO BD . Now, for i 2 Œ0; a 1 , let
BD .0/ D c > 0 ; GO BD .0; 1/ : : : GO BD .i 1; i/ BD .i/ D BD .0/ ; GO BD .i; i 1/ : : : GO BD .1; 0/
then BD .i/GO BD .i; j/ D BD . j/GO BD . j; i/ on Enfa*g, where both sides are non-zeroonly when ji jj 1. Define on Enfa*g
A.i; j/ WD .i/1=2 . j/1=2 GO BD .i; j/ : Real Spectrum for Skip-Free Markov Chains 533
1=2 1=2If D is a diagonal matrix with D BD .i; i/ D BD .i/, then A D D BD GO BD D BD .Since GO BD is reversible, A is symmetric, and the spectral theorem for symmetricmatrices tells us that A has real eigenvalues. As a result, GO BD has real eigenvalues,and hence G has real eigenvalues. We will now show that G has distinct and non-positive eigenvalues. Since GO isthe infinitesimal generator of a finite Markov chain, GO is uniformizable. Let b WD O i/ < 1. Then PO WD I C 1 GO is a stochastic matrix of a discrete timemaxi2E G.i; bupward skip-free Markov chain with a being the unique absorbing state. By theresult of Kent and Longford [7, Sec. 4 Lemma 1], PO has distinct eigenvalues, whichimplies GO has distinct eigenvalues as well. If G has a positive eigenvalue , then1 C b > 1 is an eigenvalue for P, O which is not possible since a stochastic matrixhas eigenvalues less than or equal to 1. Therefore, all the eigenvalues of G are non-positive. t u Condition 1 and 2 are by no means necessary for G to admit a real spectrum. Thiscan be illustrated with the following example:Example 1 (Condition 1 and 2 are not necessary) 0 1 0:3 0:3 0 0 B 0:4 0:7 0:3 0 C GDB @ 0:5 C 0:3 0:85 0:05 A 0:1 0:2 0:4 0:7
has eigenvalues 0; 0:63; 0:89; 1:04, and does not satisfy condition 1 (since O 1/ D G.1; 0/ G.2; 0/ D 0:1 < 0) and 2 (since G.0;G.0; O 2/ D G.2; 0/ G.3; 0/ D 0:4 > 0). Therefore, condition 1 and 2 are not necessary for real, distinctand non-positive eigenvalues. t u
4 Pathwise Illustration of the Main Result
We recall the notion of strong pathwise duality for Markov processes. Relyingon the fact that Siegmund duality is a strong pathwise duality, we demonstrateProposition 1 from a sample path perspective in Fig. 2.Definition 4 (strong pathwise duality) Let .Xt /t0 and .Yt /t0 be two continuous-time Markov chains on finite state spaces E and F respectively, and H W E F ! Rbe a measurable and bounded function. Suppose that for every T > 0 there arefamilies of processes f.Xsx /s2Œ0;T gx2E and f.Ysy /s2Œ0;T gy2F , defined on a commonprobability space .˝; F ; P/, such that the following holds:1. For all x 2 E and y 2 F, the finite dimensional distributions of .Xsx /s2Œ0;T under P (resp. .Ysy /s2Œ0;T under P) agree with that of .Xt /t2Œ0;T under Px (resp. .Yt /t2Œ0;T under Py ). 534 M.C.H. Choi and P. Patie
2. For all s 2 Œ0; T and all x 2 E, y 2 F, y y H.x; YT / D H.Xsx ; YTs / D H.XTx ; y/ ; P a:s::
Then .Xt /t0 and .Yt /t0 are said to be strongly pathwise dual with respect to H.Theorem 1 (Siegmund duality is strong pathwise duality (Clifford and Sudbury[2])) Let .Xt / and .Yt / be two continuous-time Markov chains on a finite state spaceE, which are dual with respect to the Siegmund kernel HS .i; j/ D 1fijg . Then theyare strongly pathwise dual. We now discuss a procedure to simulate .Xtx / with generator G and its Siegmund ydual .Yt / with generator GO over the time frame Œ0; T . We summarize the procedurein Clifford and Sudbury [2] as follows: since the state space E is finite, both y.Xtx / and .Yt / are uniformizable. Suppose that X0x D x. Let us write c Dmaxfmaxi G.i; i/; maxi G.i; O i/g > 0. Then the inter-arrival time (or holding time).ti /niD1 with t0 D 0 of the transition of .Xtx / is independent and exponentiallydistributed with mean c1 , and the transition follows that of the discrete timeembedded Markov chain with transition matrix P D I C Gc , which is driven byn i.i.d. .Ui /niD1 uniformly distributed numbers on .0; 1/. y To simulate .Yt /, we reverse in both jump directions and jump time of that of y y.Xtx /. Precisely, suppose that Y0 D y. The holding time of .Yt / is .T tnk /nkD0 , Oand its transition is governed by the transition matrix PO D I C Gc driven by .Vi /niD1 , y ywhere Vi D 1 UnC1i . Let YQ t WD YTt . The above procedure is summarized intoAlgorithm 1: To illustrate Algorithm 1, we use it to simulate a continuous time birth and deathprocesses .Xt3 / on E D f0; : : : ; 7g with birth rate and death rate both equal to 0:5for all states apart from 0 and 7, where we have an absorbing boundary at 0,and areflecting boundary at 7 with G.7; 6/ D 0:5. The sample paths are plotted in Fig. 1.
Algorithm 1 Simulate a stochastically monotone CTMC and its Siegmund dualRequire: x; y; T, G, GO c maxfmaxi G.i; i/; maxi G.i; O i/g G GO P I C ; PO IC c c y i 1; t0 0; X0x x; Y0 y repeat Sample ti Exponential(mean D c1 ) i PiC1 until i ti > T n length.ti / 1 Sample .Ui /niD1 Uniform.0; 1/; Set Vi 1 UnC1i for i D 1; : : : ; n Sample n steps of P that starts at x and is driven by .Ui /niD1 , say .Xix /niD1 Sample n steps of PO that starts at y and is driven by .Vi /niD1 , say .Yi /niD1 y y return .ti /niD0 , .Xix /niD0 , .Yi /niD0 Real Spectrum for Skip-Free Markov Chains 535
0 0 5 10 15 20 time t
Fig. 1 Sample paths of birth-death .Xt3 / with an absorbing boundary at 0, G.i; i C 1/ D G.i; i 1/ D 0:5, .YQ4t / and .YQ5t / with T D 20, a D 7
This figure should be compared with Jansen and Kurt [6, Fig. 1 Sec. 4.1]. We can 3observe that strong pathwise duality (cf. Definition 4) holds, that is, X20 y if and yonly if YQ 0 3. Next, to demonstrate our main result Proposition 1, we apply Algorithm 1 tosimulate an upward skip-free process .Xt4 / with generator G in (7). It can be checkedthat G fulfills condition .1/ and .2/ in Proposition 1. Strong pathwise duality holds 4 y if and only if YQ0 4. In essence, while .Xt4 / is upward yin Fig. 2, that is, X20skip-free, its Siegmund dual can be considered as a birth-and-death process beforeit gets absorbed to state 7. 0 1 0 0 0 0 0 0 0 0 B 0:3 0:5 0:2 0 0 0 0 0 C B C B 0:1 0:25 0:65 0:3 0 0 0 0 C B C B C B 0:1 0:2 0:4 1:1 0:4 0 0 0 C GDB C (7) B 0:1 0:2 0:3 0:35 1:45 0:5 0 0 C B C B 0:1 0:2 0:3 0:3 0:6 2:1 0:6 0 C B C @ 0:1 0:2 0:3 0:3 0:5 1:0 3:1 0:7 A 0:1 0:2 0:3 0:3 0:5 0:7 1:0 3:1 536 M.C.H. Choi and P. Patie
0 0 5 10 15 20 time t
Fig. 2 Sample paths of .Xt4 / with G given by (7), .YQ1t / and .YQ2t / with T D 20, a D 7
Acknowledgements The authors would like to thank an anonymous referee for insightfulsuggestions which improved an earlier version of the paper.
References
1. Asmussen, S.: Applied Probability and Queues. Volume 51 of Applications of Mathematics (New York), 2nd edn. Springer, New York (2003). ISBN: 0-387-00211-1. Stochastic Modelling and Applied Probability2. Clifford, P., Sudbury, A.: A sample path proof of the duality for stochastically monotone Markov processes. Ann. Probab. 13(2), 558–565 (1985). ISSN: 0091-17983. Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18(4), 1483–1522 (1990). ISSN: 0091-17984. Fill, J.A.: On hitting times and fastest strong stationary times for skip-free and more general chains. J. Theor. Probab. 22(3), 587–600 (2009). ISSN: 0894-9840, doi:10.1007/s10959-009- 0233-75. Huillet, T., Martinez, S.: Duality and intertwining for discrete Markov kernels: rela- tions and examples. Adv. Appl. Probab. 43(2), 437–460 (2011). ISSN: 0001-8678, doi:10.1239/aap/13086624876. Jansen, S., Kurt, N.: On the notion(s) of duality for Markov processes. Probab. Surv. 11, 59–120 (2014). ISSN: 1549-5787, doi:10.1214/12-PS2067. Kent, J.T., Longford, N.T.: An eigenvalue decomposition for first hitting times in random walks. Z. Wahrsch. Verw. Gebiete 63(1), 71–84 (1983). ISSN: 0044-3719, doi:10.1007/BF005341788. Norris, J.R.: Markov Chains. Volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998). ISBN: 0-521-48181-3. Reprint of 1997 original9. Siegmund, D.: The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Probab. 4(6), 914–924 (1976) Bifurcations in the Solution Structure of MarketEquilibrium Problems
F. Etbaigha and M. Cojocaru
Abstract In this study, the well-known market disequilibrium model with excesssupply and demand is investigated to determine if it exhibits changes in the structureand the number of equilibrium states for specific choices of parameter values. Wepropose to examine the effects of changing separately each price functions andunit transaction cost functions. We study the bifurcation problem (i.e., qualitativechange in equilibrium states) as a parametrized variational inequality problem (VI).We conduct our analysis based on modeling the markets via a projected dynamicalsystem (PDS), which is a type of constraint ordinary differential equations whosecritical points are the market equilibrium states of the economic model. Numericalsimulation for two examples is carried out to see if and when the behavior of thesemarket steady states exhibits any qualitative change.
1 Introduction
The market equilibrium models considered in this work are spatial price equilibriumand market disequilibrium models [11, 14]. The extensive work on the spatialprice equilibrium induced various improvements in both the formulation and thecomputation of this problem. In the formulation, it evolved from the formulation oflinear complementarity [1], to the formulation via variational inequality theory asin [8]; then to PDS models as in [12]. In the computation, numerical methods andapproximation algorithms are utilized to exploit the solutions at different settingsand configurations [1, 12]. It also extended to a market disequilibrium case where theprices have been regulated [16]. The initial market disequilibrium problem appearedin [16]. Nagurney et al. [14] have introduced a new market equilibrium model withexcess supply and demand, which extended from the spatial price equilibrium modelpresented in [11]. In this paper, we examine the possible structural changes of thebehavior of the market equilibrium and disequilibrium problems under a parametervariation. Two frameworks are combined: the theory of PDS, and the theory ofevolutionary variational inequality problems (EVI) (see for example [2, 5, 7] and
F. Etbaigha () • M. CojocaruUniversity of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 537J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_49 538 F. Etbaigha and M. Cojocaru
the references therein). The rest of the paper is organized as follows: in Sect. 2, wegive a brief review of both market equilibrium and disequilibrium models; moreover,their variational inequalities forms and corresponding adjustment dynamics are alsopresented. Section 3 provides a study of bifurcation questions using the conceptof EVI problem. Numerical simulations for two market equilibrium examples arepresented in Sect. 4. The paper is concluded with a discussion and some suggestionsfor future work. We assume the reader is familiar with the definitions of a convex cone, and thoseof the tangent and normal cone to a closed convex nonempty set (see for instance[3]).
2 Brief Review of Market Equilibrium Models
2.1 Market Equilibrium Model with Excess Supply and Demand (MESD)
The disequilibrium market model is presented in [14] and summarized below. Itpresented m supply markets and n demand markets which involved the productionof a hom*ogeneous commodity under perfect competition. The supply at market iis expressed as si and demand at market j as dj . Further, the supply price at eachsupply market i is denoted by i , and j denotes the demand price at demand marketj. All the supplies are grouped into a column vector s 2 Rm , and the supply pricesare grouped in a row vector 2 Rm . In the same context, the demands and demandprices are grouped respectively into a column vector d 2 Rn and a row vector 2Rn . Moreover, the nonnegative commodity shipment attached with a supply anddemand market pair .i; j/ is denoted by Qij . Each trade between the pair .i; j/ isassociated with a unit transaction cost cij . The unit transaction costs which includethe transportation cost and the tax are grouped into a row vector c 2 Rmn , and thecommodity shipments are grouped in a column vector Q 2 Rmn . It is assumed thatthere are excess supply and excess demand at each market, denoted as ui and vj . Allexcess supplies and excess demands are grouped respectively into a column vectoru 2 Rm and a row vector v 2 Rn . Therefore, a feasible model requires the followingconstraints for all supplies, demands and shipments: X X si D Qij C ui ; i D 1; : : : m and dj D Qij C vj ; j D 1; : : : :n: (1) j i
Note that in [11] the equilibrium market is given as above with ui D 0 and vj D 0.Furthermore in [14], it is assumed that each supply price at the supply market isregulated by a fixed minimum supply price i , called the price floor at supply marketi. The fixed maximum demand price at demand market j is denoted by Nj , called theprice ceiling at demand market j. The supply price floors and the demand price Bifurcations in the Solution Structure of Market Equilibrium Problems 539
ceilings are grouped into vectors 2 Rm and N 2 Rn , respectively. The vector,defined as Q 2 Rmn , contains m vectors of fQ i g, and each of these vectors consists ofn components fi g. The vector Q 2 Rmn consists of m vectors fQj g and each of thesevectors has n components f1 ; 2 ; : : : :; n g. In addition to the above restrictions, thesupply, demand, the commodity shipment, excess supply and excess demand, whichconstitute the disequilibrium pattern must also satisfy the following conditions at allsupply and demand markets: ( D j ; if Qij > 0; i C cij (2) j ; if Qij D 0; ( ( D i ; if ui > 0 D Nj ; if vj > 0; i ; j (3) i ; if ui D 0 Nj ; if vj D 0:
The conditions in (2) are known as equilibrium conditions [11]. The conditions (3)are presented in detail in [14]. It is also considered that the supply price at any supplymarket i depends on the supplies at all the markets, that is D .s/. Similarly, thedemand price at any demand market j depends on the demands at all the markets,that is D .d/. In the same context, the unit transaction cost associated with thepair .i; j/ depends on the commodity shipments from i to j, that is c D c.Q/, whereall ; and c are smooth functions. The vectors O D 2 Rm and O D 2 Rnare also provided, where O D .Q; O u/ and O D .Q;O v/: The vector QO 2 Rmn Qconsists of m vectors fO i g and each these vectors contains n components fO i g andthe vector QO 2 Rmn consists of m vectors fQOj g, where each vector has n componentsfO1 ; O2 : : : :; On g.
2.2 Variational Inequality Formulation and Adjustment Dynamics of MESD
The pattern .Q ; u ; v / satisfies the conditions (2) and (3) governing the disequi-librium market problem if and only if it satisfies the VI problem
QO ; u / C c.Q / .Q ..Q QO ; v //:.Q Q / C ..Q O ; u / /:.u u /C O ; v //:.v v / 0 ; 8.Q; u; v/ 2 K and K D Rmn .N .Q C RC RC : m n
(4)Remark: note that in [11] the market equilibrium model without the excess supplyand demand is formulated as a VI as follows:
hF.Q /; Q Q i 0 8 Q 2 K 1 ; K 1 D Rmn C: (5) 540 F. Etbaigha and M. Cojocaru
Thus, according to [12], it is known that the adjustment to spatial price equilibriumstates in a VI problem can be obtained by studying the nonsmooth dynamicalsystem:
dQ.t/ D PTKQ.t/ .F.Q.t///; Q.0/ 2 K 1 ; K 1 D Rmn C; (6) dt
where F W K 1 ! Rmn and Fij .Q/ D i .s/ C cij .Q/ j .d/. In the same manner, we can write the adjustment process of the MESD [7] as
d.Q.t/; u.t/; v.t// D PTK.Q.t/;u.t/;v.t// .F.Q.t/; u.t/; v.t///; K D Rmn C RC RC : (7) m n dt
For simplicity, we will write .Q; u; v/ instead of .Q.t/; u.t/; v.t//. F W K 7! Rmn Rm Rn is defined by F.Q; u; v/ D .A.Q; u; v/; G.Q; u/; D.Q; v//; and A W K 7!Rmn , G W Rmn m m mn n C RC 7! R and D W RC RC 7! R are defined by: Aij D n
QO i .Q; u/ C cij .Q/ QOj .Q; v/; Gi D O i .Q; u/ i ; Dj D Nj Oj .Q; v/. Let the vectorx D .Q; u; v/ 2 K, and F.x/ D F.Q; u; v/ then (7) can be written in the form,
xP D PTKx.t/ .F.x.t///: (8)
3 Bifurcations in MESD
3.1 Formulation of the Bifurcation Problem for Constrained Systems
Bifurcation theory for nonsmooth dynamical systems has not been studied asextensively as smooth dynamical systems (see for instance [15] and the referencestherein). However, the works in [9] and [10] give results for the existence of bifurca-tions in discontinuous Filippov systems and VI. The nonconventional bifurcationsof fixed points and periodic solutions in Filippov systems have been addressed in[10]. In [9], it was shown that if V is the Hilbert space, B W V R ! V and K isclosed convex subset of V then the VI of the form
hu B.u; /; v ui 0 8v 2 K (9)
subject to B.0; / D 0 8 2 R; B.u; / D Bu C R.u/; R.u/ D o.kuk/ as u ! 0has a bifurcation point .0; 0 / for 0 2 R provided there are a sequences .un ; n / ofsolutions of (9) such that kun k ¤ 0 and as n ! 1,n ! 0 and un ! 0, forall n. The above constrain can not be satisfied in our VI formulation, therefore (9)can not be applied here. Since PDS lies outside of the field of classical dynamicalsystems [13], one way to formulate and study the bifurcation problem is by usingthe concept of a parametrized VI problem, sometimes called an EVI. Bifurcations in the Solution Structure of Market Equilibrium Problems 541
Let us assume that an MESD is formulated as in Sect. 2, such that its solution(s) aregiven by equilibrium points of the dynamical system (7). Let 2 R and considerthe system:
d.Q; u; v/ D PTK.Q;u;v/ .F.Q; u; v; //; K D Rmn C RC RC ; m n (10) dt
which is equivalent to xP D PTKx.t/ .F.x.t/; //; where x D .Q; u; v/ 2 K so thatwhenever D 0 we have the initial system (8). We want to study the problem of finding points x 2 K so that [6]
PTK.x / .F.x ; // D 0 , F.x ; / 2 NK .x /; (11)
for some interval of values of 2 Œa; b . From (11), using the definition of a normalcone, we have to:
find points x 2 K s.t. hF.x ; /; y x i 0; 8y 2 K: (12)
In (12), F depends on x and and hence, we can reformulate it as a parametrizedVI problem, or as an EVI problem (see for example [4, 5] for the current use of EVImodels) where the bifurcation parameter replaces the “time” parameter t:
hF.x./; /; y./ x./i 0; 8y./ 2 K./: (13)
According to the EVI theory, all the functions will be in Hilbert spaceL2 .Œa; b ; RnCmCnm /, then F W Œa; b K ! L2 .Œa; b ; RnCmCnm / and the feasible setK can be written as
K D fx 2 L2 .Œa; b ; RnCmCnm / j0 x./ M; a:e in Œa; b g; with x; y 2 K: (14)
Note that: K./ D fw 2 RnCmCnm j 0 w Mg; where M is a large. Essentially, the bifurcation problem becomes an EVI problem where the boundsof the constraint set K./ are fixed, and do not depend on the parameter . In orderto solve such an EVI problem, one can use existing methods in the literature [5].Cojocaru et al. in [5] showed that the linkage between the projected dynamicalsystem and EVI time dependent problem where the critical points of the PDS arethe same as the solutions of EVI problem. Using this approach, the PDS associatedto dependent EVI can be formulated as
dx.; / D PTKx.; / .F.x.; /; //; (15) d
where the time represents the time evolution to the equilibrium x.; :/. Thisformulation is employed to solve the problem in (13). 542 F. Etbaigha and M. Cojocaru
3.2 Bifurcations for MESD
The equilibrium problems are analyzed using different parameterization strategies.Initially, by introducing a parameter that represents the supply price floors, for whichthe system behavior did not change and the equilibrium is still unique. Secondly weimpact the supply prices by introducing a parameter to represent the variation ofthe supplies on the supply price functions, the equilibrium is still unique and minorchanges are observed only in the trajectories convergence for different values of .The same experiments are repeated for the demand prices and the same results areobserved. These results suggest that the bifurcations are not likely to occur withsuch parametrization strategies. Parametrizing the unit transaction cost functionsby introducing to represent the variation of the shipments will likely impact themarket steady states behavior significantly such that c D c.Q; /. Another directionis to introduce into both the supply and demand prices functions such that D.s; /, and D .d; /. It is expected that this change will impact the marketequilibria since changing both the supply and the demand prices at the same timewill dramatically impact the equilibrium and disequilibrium conditions (2) and (3).The next section shows two examples of market equilibrium problems using thisparametrization strategy.It is known that whenever F is strictly monotone or beyond (strongly monotone,etc.) [11] as a function of x, the solution of the MESD problem is unique [13]. Thus,there is only one equilibrium point of (8). From the EVI theory [5, 7], wheneverF.x; / with 2 Œa; b remains strictly monotone, the parametrized MESD (13)still has a unique solution. It is therefore logical to seek values of which willlead to a non-strictmonotonic F.x; /. In these ranges of values we may find distinctbehaviour of the equilibrium.
4 Examples
4.1 Example 1
We consider the example of a market equilibrium problem with two supply and twodemand markets which has a unique equilibrium as shown in [11]. In order to detectif the change on the cost functions impacts the equilibrium, we modify the exampleand introduce in the cost function. At D 0 the equilibrium is still unique.0:8; 2:96; 0; 0/ since the vector field F is still strictly monotone. Additionally, thevariables are adjusted to be dependent where 2 Œa; b .The supply price functions are defined by W R2 ! L2 .Œa; b ; R2 /,1 .s.// D 5s1 ./ C s2 ./ C 2, 2 .s.// D s1 ./ C 2s2 ./ C 3;the demand price functions are defined by W R2 ! L2 .Œa; b ; R2 /,1 .d.// D 2d1 ./ d2 ./ C 28:75, 2 .d.// D d1 ./ 4d2 ./ C 41;and the unit transaction costs are defined by c W Œa; b K ! L2 .Œa; b ; R4 /, Bifurcations in the Solution Structure of Market Equilibrium Problems 543
c11 .Q; / D Q11 ./ C .0:5 C /Q12 ./ C 1, c12 .Q; / D .2 C /Q12 ./ CQ22 ./ C 1:5, c21 .Q; / D .3 /Q21 ./ C 2Q11 ./ C 25, c22 .Q; / D .2 /Q22 ./ C Q12 ./ C 30. The feasible set is K D fQ 2 L2 .Œa; b ; R4 /j0 Q./ 5; a:e in Œa; b g.We consider a vector field F defined as F W Œa; b K ! L2 .Œa; b ; R4 / whereF11 .Q./; / D 1 .s.// C c11 .Q./; / 1 .d.//D 8Q11 ./ C .6:5 C /Q12 ./ C 3Q21 ./ C 2Q22 ./ 25:75.F12 .Q./; / D 1 .s.// C c12 .Q./; / 2 .d.//D 6Q11 ./ C .11 C /Q12 ./ C 2Q21 C 6Q22 ./ 37:5.F21 .Q./; / D 2 .s.// C c21 .Q./; / 1 .d.//D 5Q11 ./ C 2Q12 ./ C .7 /Q21 ./ C 3Q22 ./ 0:75.F22 .Q./; / D 2 .s.// C c22 .Q./; / 2 .d.//D 2Q11 ./ C 6Q12 ./ C 3Q21 ./ C .8 /Q22 ./ 8.Our EVI problem is given by hF.Q ./; /; Q./ Q ./i 0; 8Q./ 2 K; andthe associated PDS can be written as in (15). We examine the effect of changing in the cost functions considering 2 Œ0; 8 to ensure that the costs are always nonnegative. We obtain the equilibrium at eachvalue of by the method in [5] implemented in Matlab. For 0 < 8, for anyarbitrary initial condition, the system shows just one equilibrium corresponding toeach value of . At D 8, the shipments converge to three boundary equilibria,which means that F is non-strictmonotonic. The numerical simulations of the systemat D 8 are presented in the Figs. 1 and 2. Shipments at supply market 1 are
4.5
3.5
2.5
1.5
0.5
Q11 Q12 Q21 Q22
Fig. 1 Values of .Q11 ; Q12 ; Q21 ; Q22 / at equilibrium are presented as a heatmap. The heatmapshows three equilibrium patterns, the equilibria are color-coded such that each repeated pattern isrepresented by the same color 544 F. Etbaigha and M. Cojocaru
a 5
4.5
3.5
3 22
2.5 Q
1.5
0.5
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Q21 b 5
4.5
3.5
3 Q12
2.5
1.5
0.5
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Q 11
Fig. 2 Phase portraits for D 8, both figures show trajectories starting from 40 initial conditions(equilibria shown by square markers). There are three distinct equilibrium points
balanced at: .Q11 ; Q12 / 2 f.0; 1:44/; .1:97; 0/; .0; 1:97/g, and shipments at supplymarket 2 are balanced at: .Q21 ; Q22 / 2 f.5; 0/; .0; 5/; .0; 0/g as shown in Fig. 2.As a result of the change in the number of equilibria, D 8 is a bifurcationvalue. Bifurcations in the Solution Structure of Market Equilibrium Problems 545
4.2 Example 2
Here we modify the above example to paramterize the supply and demand pricefunctions. The example is extended to the case of excess supply and excess demand.Furthermore is introduced into both price functions. Without introducing the system shows one equilibrium .1:6; 2:8; 0; 0; 0; 0; 0; 0/ since F is still strictlymonotone. The supply price functions are defined by1 .s./; / D .5C/s1 ./Cs2 ./C12, 2 .s./; / D .1C/s1 ./C2s2 ./C30.The demand price functions are defined by1 .d./; / D 2d1 ./ .1 /d2 ./ C 45,2 .d./; / D .1 /d1 ./ 4d2 ./ C 55:The unit transaction cost functions are as in the Example 1 with D 0.The feasible set is K D fx 2 L2 .Œa; b ; R8 /j0 x./ 5; a:e in Œa; b g, wherex./ D .Q11 ./; Q12 ./; Q21 ./; Q22 ./; u1 ./; u2 ./; v1 ./; v2 .//. We considerthe supply price floors at the markets are 1 D 10 and 2 D 15 and the demandprice ceilings are N1 D 45 and N2 D 55. Then the vector field F is defined asF W Œa; b K ! L2 .Œa; b ; R8 /; for simplicity we assume Q./ D .x1 ; x2 ; x3 ; x4 /.Then F11 .Q./; u./; v./; / D .8 C /x1 C .6:5 C /x2 C .5 C /u1 C 3x3 C .2 /x4 C 2v1 C u2 C .1 /v2 32.F12 .Q./; u./; v./; / D .6 C /x1 C .11 C /x2 C .5 C /u1 C 6x4 C 4v2 Cu2 C .1 /v1 C .2 /x3 41:5.F21 .Q./; u./; v./; / D .5 C /x1 C .1 C /u1 C .4 C /x2 C .3 /x4 C 2v1 C.1 /v2 C 7x3 C 2u2 .F22 .Q./; u./; v./; / D 2x1 C .5 C /x2 C .3 /x3 C .1 /v1 C 4v2 C 6x4 C.1 C /u1 C 2u2 15,.1 1 ; 2 2 / D ..5 C /x1 C .5 C /x2 C .5 C /u1 C x3 C x4 C u2 C 2; .1 C/x1 C .1 C /x2 C .1 C /u1 C 2x2 C 2x4 C 2u2 C 15/.N1 1 ; N2 2 / D .2x1 C 2x3 C 2v1 C .1 /x2 C .1 /x4 C .1 /v2 ; .1 /x1 C.1 /x3 C.1 /v1 C4x2 C4x4 Cv2 /. Then the EVI problem can be writtenas hF.Q ./; u ./; v ./; /; .Q./; u./; v.// .Q ./; u ./; v .//i 0. We set 2 Œ7; 8 to ensure the prices are nonnegative. The experiments iscarried out as discussed in Example 1. For each value of there exist only oneequilibrium except for the value of D 6 at which two equilibria occur. Then at D 6, F is non-strictmonotonic, and D 6 is a bifurcation value. Figure 3shows the two equilibria. Shipments at supply market 1 .Q11 ; Q12 / 2 f.0; 5/; .5; 0/gand shipments at supply market 2 .Q21 ; Q22 / 2 f.0; 0:83/; .0; 2:4/g are shown in 546 F. Etbaigha and M. Cojocaru
4.5
3.5
2.5
1.5
0.5
Q11 Q12 Q21 Q22 u1 u2 v1 v2
Fig. 3 The heatmap shows the equilibrium pattern .Q11 ; Q12 ; Q21 ; Q22 ; u1 ; u2 ; v1 ; v2 /. Each valueis represented by singular color, the number of equilibria in this case two which is indicated byswitch in color
(a) 5 (b) 4 4.5 3.5 4 3 3.5 2.5 Q 22
3Q12
2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5
Q11 Q21
Fig. 4 Phase portraits for D 6 show the convergence of the shipments at each market. (a).Q11 ; Q12 / convergence. (b) .Q21 ; Q22 / convergence
Fig. 4. The excess supply and demand at market 1 .u1 ; v1 / 2 f.0; 0/; .4:9; 0/g andthe excess supply and demand at market 2 .u2 ; v2 / 2 f.4:18; 0/; .5; 0/g are presentedin Fig. 5. Bifurcations in the Solution Structure of Market Equilibrium Problems 547
(a) 5 (b) 5 4.5 4.5 4 4 3.5 3.5 3 3v1
v2 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 u1 u2
Fig. 5 Phase portraits for D 6 show the convergence of the excess supply and excess demandat each market. (a) .u1 ; v1 / convergence. (b) .u2 ; v2 / convergence
5 Conclusion
In this study we considered the bifurcation problem for the market equilibriummodel as an EVI problem. Furthermore, we studied the impact of changing thesupply price, demand price and the cost functions on the market equilibrium states.With both the cost functions and price functions, the effect of the variations ofthe parameter is seen on the number of equilibria occurring at specific values.The equilibrium states were obtained using trajectories of the associated projecteddynamics. The empirical results on two examples showed that bifurcations occurin such systems. Further investigation is required to extend the applicability of theproposed method to other equilibrium models. Moreover, additional future workis to study the usage of nonlinear parameter dependencies to explore existence ofbifurcations in the solution structure of market equilibrium problems.
References
1. Asmuth, R., Eaves, B.C., Peterson, E.L.: Computing economic equilibria on affine networks with lemke’s algorithm. Math. Oper. Res. 4(3), 209–214 (1979) 2. Barbagallo, A., Mauro, P.: Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses. J. Optim. Theory Appl. 155(1), 288–314 (2012) 3. Cojocaru, M.-G.: Monotonicity and existence of periodic orbits for projected dynamical systems on hilbert spaces. Proc. Am. Math. Soc. 134(3), 793–804 (2006) 4. Cojocaru, M.-G.: Double-layer dynamics theory and human migration after catastrophic events. In: Cojocaru, M.-G. (ed.) Nonlinear Analysis with Applications in Economics, Energy and Transportation, pp. 65–86. Bergamo University Press, Bergamo (2007) 5. Cojocaru, M.-G., Daniele, P., Nagurney, A.: Projected dynamical systems and evolutionary variational inequalities via hilbert spaces with applications1. J. Optim. Theory Appl. 127(3), 549–563 (2005) 548 F. Etbaigha and M. Cojocaru
6. Cojocaru, M.-G., Jonker, L.: Existence of solutions to projected differential equations in hilbert spaces. Proc. Am. Math. Soc. 132(1), 183–193 (2004) 7. Daniele, P.: Dynamic Networks and Evolutionary Variational Inequalities. Edward Elgar Publishing, Cheltenham/Northampton (2006) 8. Florian, M., Los, M.: A new look at static spatial price equilibrium models. Reg. Sci. Urban Econ. 12(4), 579–597 (1982) 9. Le, V.K., Schmitt, K.: Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems, vol. 123. Springer Science & Business Media, Berlin (1997)10. Leine, R., Van Campen, D., Van de Vrande, B.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23(2), 105–164 (2000)11. Nagurney, A.: Network Economics: A Variational Inequality Approach, vol. 10. Springer Science & Business Media, Berlin (1998)12. Nagurney, A., Takayama, T., Zhang, D.: Massively parallel computation of spatial price equilibrium problems as dynamical systems. J. Econ. Dyn. Control 19(1), 3–37 (1995)13. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications, vol. 2. Springer Science & Business Media, Berlin (2012)14. Nagurney, A., Zhao, L.: Disequilibrium and variational inequalities. J. Comput. Appl. Math. 33(2), 181–198 (1990)15. Perko, L.: Differential Equations and Dynamical Systems, vol. 7. Springer Science & Business Media, Berlin (2013)16. Thore, S.: Spatial disequilibrium. J. Reg. Sci. 26(4), 661–675 (1986) Pricing Options with Hybrid StochasticVolatility Models
Glynis Jones and Roman Makarov
Abstract We introduce a hybrid stochastic volatility model where the asset priceprocess follows the Heston model and interest rates are governed by a two-factorstochastic model. Two cases are considered. First, it is assumed that interest ratesand asset prices are uncorrelated. The characteristic function method is used toderive semi-analytical pricing formulae for plain vanilla options. In the second casewe introduce a correlation between the asset price process and the short rate processand use Monte Carlo simulations for pricing options. To reduce the stochastic error,we implement the control variate method where an estimator of the option valuefor the uncorrelated case is used as a control variate. The options are priced with avarying correlation coefficient. We observe that the control variate method allowsus to speed up Monte Carlo computations by a factor with the magnitude of severalhundreds. The efficiency of the method is higher for smaller values of the correlationcoefficient. We then study the impact a correlation between the two processes has onoption prices. It has been noticed that the call option price is an increasing functionof the correlation coefficient.
1 Introduction
In 1973, Fischer Black, Myron Scholes and Robert Merton [2] introduced atechnique for finding a closed-form solution for the price of a plain European option.The price, V D V.t; S/, of a European-style option is a function of time t and assetprice S (at time t) that satisfies the partial differential equation
@V 1 @2 V @V C 2 S2 2 C rS rV D 0 (1) @t 2 @S @S
subject to the terminal condition V.T; S/ D f .S/, where r is the risk-free interestrate, is the volatility of the stock, and f .S/ is the payoff at maturity T. Here we
G. Jones • R. Makarov ()Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo,ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 549J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_50 550 G. Jones and R. Makarov
assume no dividend on the stock. The solution to (1) is given by the discountedexpectation of the terminal payoff:
V.t; S/ D er.Tt/ EQ Πf .ST / j St D S ; (2)
where Q is the risk-neutral probability measure with the bank account Bt D ert takenas a numeraire. The Black–Scholes–Merton formula (2) assumes that the price of an underlyingasset such as a stock, fSt gt0 , follows geometric Brownian motion (under the risk-neutral measure Q):
dSt D rSt dt C St dWt :
The main drawback of this model is that the volatility of the underlying riskyasset is assumed to be constant. This assumption does not fit many factors observedon financial markets such as volatility smiles and skews. Option pricing theory hasevolved since 1973. Other models that take into account many market phenomenahave been developed. Examples of such models include models with stochasticvolatility and stochastic interest rates. In this paper, we consider a hybrid modelthat combines a stochastic volatility model for asset prices with a stochastic shortrate process (see [1, 8, 14] and references wherein). In particular, we develop amodification of the Heston model [9] with stochastic interest rates that are governedby a two-factor model (see [4]). We consider a two-factor modification of theVasicek model [15] but other models such the Hull–White model [10] and the Cox–Ingersoll–Ross (CIR) model [5] can also be used. To specify the hybrid model, we combine two systems of stochastic differentialequations. The first one defines the price of a risky asset and its volatility. The secondsystem defines the short rate process. As a result, we obtain a four-factor stochasticvolatility model named here as the Heston–Vasicek model. This hybrid model fitsin the class of affine diffusion processes (see [6]) for which a closed-form solutionof the characteristic function exists. However, the derivation of the characteristicfunction for a hybrid multi-factor model with a full matrix of correlations is achallenging problem. In this paper we study a simpler approach. First, we assumethat the stock price process and the interest rate process are uncorrelated. Fouriertechniques are used to obtain no-arbitrage prices of European-style options. Second,we introduce a correlation between asset prices and interest rates and use the MonteCarlo method to simulate the coupled system of SDEs and to price options. Toreduce the stochastic error, we use the control variate method (see, e.g., [7]), wherean estimator of the option value for the uncorrelated case is used as a control variate. The rest of this paper is organised as follows. In Sect. 2, we formulate the hybridmodel and obtain characteristic functions for the uncorrelated case. A correlationbetween the asset price and interest rate processes is then introduced and options arepriced using the Monte Carlo method. In Sect. 3, we discuss and analyse numericalresults, and our conclusions are drawn in Sect. 4. Pricing Options with Hybrid Stochastic Volatility Models 551
2 The Hybrid Model
In this section, we construct a four-factor hybrid stochastic volatility model where astochastic interest rate is introduced into a stochastic volatility model. We define ourmodel such that the log-price process, fXt D ln St gt0 , follows the Heston modeland the short rate process, frt gt0 , is governed by the two-factor Vasicek model.Therefore, it is called the Heston–Vasicek model. The hybrid model consists of fourcoupled SDEs: t p dXt D .rt /dt C t dWts ; (3) 2 p dt D . t /dt C t dWt ; (4) drt D . C ut aN rt /dt C 1 dWtr ; (5) N t dt C 2 dWtu ; dut D bu (6)
where Wts , Wt , Wtr , Wtu are standard Brownian motions such that dWts dWt D s dtand dWtr dWtu D ru dt. The two-factor Vasicek model (5)–(6) can be written as atwo-factor Gaussian model (the equivalence is proved in [4]):
dxt D axt dt C dWtx ; (7) y dyt D byt dt C dWt ; (8) rt D xt C yt C .t/ ; (9)
where xt and yt are Gaussian processes, .t/ is a deterministic function given by.t/ D r0 eaT C a .eat 1/, and W x and W y are correlated standard Brownianmotions with correlation coefficient xy . The parameters of (5)–(6) and (7)–(8) arerelated as follows:
aN D a; bN D b; p 1 D 2 C 2 C 2xy ; 2 D .a b/; xy C ru D p : 2 C2 C2xy
First, we assume that W x and W y are not correlated with W s and W . Thus, thecorrelation matrix for the four Brownian motions is given by 2 3 1 s 0 0 6s 1 0 07 6 7: (10) 40 0 1 xy 5 0 0 xy 1 552 G. Jones and R. Makarov
The option pricing formula for a standard European call takes the following form: h RT i Q C.t; S/ D Et;r;S e t ru du .ST K/C : (11)
QHere, the expectation Et;r;S Πcomputed under the risk-neutral measure Q isconditional on frt D r; St D Sg. The discounting factor cannot be pulled out sinceit is stochastic, so the mathematical expectation is computed as follows: h RT i Q C.t; S/ D Et;r;S e t ru du .ST K/C h RT i Q D Et;r;S e t ru du .ST K/ fST >Kg (12) h RT i h RT i Q D Et;r;S e t ru du ST fST >Kg Et;r;S Q e t ru du K fST >Kg :
To calculate the conditional expectations in the r.h.s. of (12), we apply the changeof numeraire technique. The first expectation can be simplified by using the assetprice as a numeraire and the second expectation can be simplified by using the zero-coupon bond as a numeraire. So, we have Q1 BT =Bt RtT ru du C.t; S/ D Et;r;S e ST fST >Kg ST =St RT BT =Bt Q2 Et;r;S e t ru du K fST >Kg Z.T; T/=Z.t; T/ (13) Q1 Q2 D SEt;S fST >Kg KZ.t; T/Et;S fST >Kg
D SQ1 .ST > K j St D S/ KZ.t; T/Q2 .ST > K j St D S/ D SF1 KZ.t; T/F2 ; Rt h RT i Qwhere Bt D e 0 ru du is the bank account, Z.t; T/ D Et;r e t ru du is the time-t priceof a unit zero-coupon bond with maturity T, Q1 is the EMM (equivalent martingalemeasure) with the stock St used as a numeraire asset, and Q2 is the T-forward EMMwith the zero-coupon bond Z.t; T/ used as a numeraire. As in [9], the characteristic functions need to be found for computing theprobabilities F1 D Q1 .ST > K j St D S/ and F2 D Q2 .ST > K j St D S/. Since theinterest rate in this model is no longer constant, the characteristic functions for theHeston model have to be modified to include the stochastic interest rate component.To determine the form of the new characteristic functions we start with the log-priceprocess for the SDE (3) re-writing it as follows [11]: t p dXt D rt dt C . dt C t dWts / D rt dt C dXtH ; (14) 2 Pricing Options with Hybrid Stochastic Volatility Models 553
where fXtH gt0 is the log-price process for the Heston model with interest r D 0.Integrating both sides gives us the following: Z T Z T Z T dXs D rs ds C dXsH H) XT D RT C XTH ; t t t
RTwhere RT D t ru du. Then, we have the characteristic functions fHVj ./, j D 1; 2 ofthe hybrid model as follows:
Œei.RT CXT / D Et;rj ŒeiRT Et;Xj ŒeiXT ; Q Q H Q Q H fHVj ./ D Et;r;X j ŒeiXT D Et;r;X j (15)
where we used the fact that RT and XTH are independent. Therefore, the newcharacteristic functions admit the following form:
fHVj ./ D fVj ./fj ./; j D 1; 2 ;
where fVj is the characteristic function for the Vasicek model and fj is the character-istic function for the Heston model (under Qj ). The functions fj , j D 1; 2 are already known [9, 13, 16] so we only need todetermine the characteristic functions of the integrated short rate process, RT , underEMMs Q1 and Q2 , respectively,
fV1 ./ D Et;r Œe Q1 iRT ; fV2 ./ D Et;r Œe Q2 iRT : (16)
If the short rate rt is a Gaussian process, then the integrated process Rt is Gaussian 2 z2as well. Hence, we can use the identity EŒeiZ D eiz 2 for a normal randomvariable Z with mean z and variance z2 . The probability distribution of the shortrate process does not change under the EMM Q1 . Therefore, Eqs. (7) and (8) can beused to find the mean and the variance of RT under Q1 :
r0 aT r0 eaT T R D EŒRT D e C 2C 2 C ; a a a a a 2 2 1 3 VR D Var.RT / D T C eaT e2aT a a 2a 2a 2 2 1 3 C T C ebT e2bT b b 2b 2b 2xy eaT 1 ebT 1 e.aCb/T 1 C TC C : ab a b aCb 554 G. Jones and R. Makarov
Under the T-forward EMM Q2 the short rate process, rt D xt C yt C .t/, isdefined by the following SDEs [4]: h 2 i dxt D axt 2 .1 ea.Tt/ / xy b .1 eb.Tt/ / dt C dWtx ; (17) h 2 i dyt D byt 2 .1 eb.Tt/ / xy a .1 ea.Tt/ / dt C dWt : y (18)
Therefore, the mean and variance of RT under the EMM Q2 are, respectively,
EŒRT D xint C yint C int ; 2 2 aT 1 2aT 3 Var.RT / D TC e e a a 2a 2a 2 2 bT 1 2bT 3 C TC e e b b 2b 2b 2xy eaT 1 ebT 1 e.aCb/T 1 C TC C ; ab a b aCb
where Z T 2 T2 s T 2 xint D E x.s/ds D .1 eaT / .1 ebT / ; 0 a 2 b 2 Z T 2 T 2 s T 2 yint DE y.s/ds D .1 ebT / .1 eaT / ; 0 b 2 a 2 Z T r0 r0 eaT T int D .s/ds D eaT C 2 C 2 C : 0 a a a a a
Substituting these into Eq. (16) gives us the new characteristic function fV2 . Theprice of the call option under the new model is as follows:
C.t; S/ D SF1 KZ.t; T/F2 ; where (19) Z 1 1 1 exp.i ln K/ Fj D C Re fHVj ./ d; j D 1; 2 ; (20) 2 0 i
and Z.t; T/ is the price of a zero-coupon bond. The integral in (20) can be evaluatednumerically using a quadrature rule or the fast Fourier transform (FFT) method[13, 16]. Note that the price of the put option can be calculated using the put-callparity. Clearly, we are able to derived the pricing formula for European options inclosed form (19)–(20) due to the assumption that the asset price process and theshort rate process are uncorrelated. Pricing Options with Hybrid Stochastic Volatility Models 555
We then move on to the correlated case. Let us assume that W x is correlatedwith W s , i.e. dWtx dWts D dt O with O 2 .1; 1/. Alternatively, we can introduce thecorrelation between W y and W s or between W x and W . We would like to see howO affects the option prices as it changes. The correlation matrix for the system ofSDEs (3), (4), (7), (8) is as follows: 2 3 1 s O O xy 6 s 1 O xy s 7 O s 6 7: (21) 4 O O s 1 xy 5 O xy O xy s xy 1
One can easily show that this matrix is positive definite for all s ; xy ; O 2 .1; 1/.The Cholesky decomposition method can then be applied to generate correlatednormal random variables. These correlated random variables are used in MonteCarlo simulations of the hybrid model. As is seen from (21), the correlation coefficients Corr.W s ; W y /, Corr.W ; W x /,Corr.W ; W y / cannot approach 1 or 1 as jj O ! 1 since xy ; s 2 .1; 1/. So, theselection of two processes between which the correlation is introduced affects therange for the correlation coefficient between another pair.
3 Computational Results and Analysis
We would like to calibrate the model using historical data and then use theparameters obtained for option pricing. First, the two-factor Gaussian model iscalibrated using yield rates obtained from the US treasury website (see [3, 4]). Theparameters obtained for the short rate model are as follows:
a D 0:2071; b D 2:6414; D 0:0100; D 0:0117; xy D 0:8317; D 0:0113:
Using these parameters, the Heston model has been calibrated using the least squaremethod [13]. The historical prices of options on SPDR S&P 500 ETF (SPY) forJune 10, 2010 were used with different strike prices and different maturities (optionprices were obtained from the BloombergTM database). The initial asset value isS0 D 108:88. The least squares method is used to minimize the difference betweenmarket prices and model prices of options. This method minimizes the sum of thesquared errors (the market price minus the model price) made in every iteration. Toachieve the best possible results, the same routine has been run multiple times withdifferent starting points to get an optimal solution. The parameters that provide thebest fit to market prices are
0 D 0:0648; D 0:0166; D 0:3534; D 0:1223; s D 0:8380: 556 G. Jones and R. Makarov
Monte Carlo estimate of the option price Confidence interval for the option price 10.7 11.5 MC estimate 10.6 Exact value 11Option price
Option price 10.5 10.5
10.4 10
10.3 9.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 M (the number of simulations) x 105 M (the number of simulations) x 105
Fig. 1 Monte Carlo simulations for the at-the-money call option with strike price K D 108:88
Substituting these values into the option pricing formula (19), we can calculate no-arbitrage prices of call options for the uncorrelated case. To verify the pricing formula (19), the exact option prices were comparedwith prices obtained using Monte Carlo simulations. The Monte Carlo pricesare approaching the respective exact price in all our tests. For example, Fig. 1demonstrates the convergence of Monte Carlo prices for the at-the-money calloption (with K D S0 ). To construct Monte Carlo estimates for option prices, the Euler method (see,e.g., [12]) is used. Discrete-time approximations of sample paths, .XO i ; O i ; xO i ; yO i / .Xi t ; i t ; xi t ; yi t / with i D 0; 1; : : : ; n 1 and t D Tn , are generated using thefollowing scheme: p p XO iC1 XO i C t .Ori 0:5maxf0; O i g/ C maxf0; O i g tzs ; p p O iC1 O i C t.v O i / C maxf0; O i g tz ; p xO iC1 xO i aOxi t C tzx ; p yO iC1 yO i bOyi t C tzy ; a.iC1/ t rOiC1 xO iC1 C yO iC1 C r0 ea.iC1/ t e : a
The initial conditions are xO 0 D 0, yO 0 D 0, rO0 D r0 , and XO 0 D ln S0 . For everytime step, four correlated normal random variables, .zs ; z ; zx ; zy /, are required; theyare obtained using the Cholesky decomposition of the correlation matrix given by RT C(10) or (21). A sample value of the discounted payoff, e 0 ru du eXT K , is thencomputed. The trapezoidal rule is used to approximated the discounting factor. Pricing Options with Hybrid Stochastic Volatility Models 557
Monte Carlo simulations usually have a large stochastic error associated withthem. To reduce the variance of our Monte Carlo estimator of the option value weuse the control variate method. The uncorrelated case (for which the option pricingformula is available in closed form) is used to construct the control variate. Thesame i.i.d. normal samples are used to calculate the estimates for both correlatedand uncorrelated cases. In Table 1 we examine how the option prices vary with . O As O approaches 0,the option price gets closer to the analytical price calculated for O D 0. We findthat the option price increases as O increases for at-the-money (K D 108:88), in-the-money (K 2 f100; 106g), and out-of-the-money (K 2 f110; 115g) options. Thereasoning behind this is that as interest rates go up, the return on stocks increasesthereby increasing the value of a call option. We can say the call option priceand the interest rates are positively correlated. Therefore, increasing the correlationcoefficient between the interest rates and log-price processes increases the price ofa call option as is seen from our simulations. The ratio of the standard deviations is calculated (the standard deviation of thecrude estimator over the standard deviation of the controlled estimator). This givesus the factor by which the control variate method improves the crude Monte Carlomethod. This ratio ranges from 238 to 3846 depending on the values of O and K.As is seen from Table 1 and Fig. 2, the control variate method gives a significantimprovement over the standard estimator when the correlation coefficient is close tozero.
4 Conclusion
In this paper we have successfully derived a semi-analytical formula for no-arbitragepricing a standard European option under the hybrid four-factor Heston–Vasicekmodel, where the interest rate process and the log-asset price process are uncorre-lated. We then studied the impact a correlation between the two processes had onoption pricing using Monte Carlo simulations. It has been noticed that the price ofa European call is an increasing function of the correlation coefficient. The use ofa variance reduction method greatly improved our results for the correlated case aswe can see from the ratios of standard deviations presented in Table 1. For futurework, it will be interesting to study a hybrid model that combines the Heston modeland the two-factor Hull–White model and compare its performance with the modelstudied in this paper. 558
Table 1 The impact of O on option pricesO K D 100 Std ratio K D 106 Std ratio K D 108:88 Std ratio K D 110 Std ratio K D 115 Std ratio0.9 13.0576 314.4654 9.3820 284.3563 7.8718 273.9726 7.3296 268.0965 5.2128 240.96390.8 13.0597 389.1051 9.3843 357.1429 7.8742 340.1361 7.3321 332.2259 5.2153 298.50750.7 13.0616 476.1905 9.3866 436.6812 7.8765 414.9378 7.3344 406.5041 5.2176 363.63640.6 13.0636 477.0095 9.3889 537.6344 7.8789 510.2041 7.3368 500.0000 5.2200 448.43050.5 13.0656 729.9270 9.3912 671.1409 7.8813 641.0256 7.3392 625.0000 5.2224 558.65920.4 13.0676 934.5794 9.3935 862.0690 7.8837 826.4463 7.3416 813.0081 5.2248 719.42450.3 13.0695 1282.0513 9.3958 1176.4706 7.8861 1136.3636 7.3440 1098.9011 5.2272 980.39220.2 13.0715 1960.7843 9.3981 1785.7143 7.8885 1724.1379 7.3464 1666.6677 5.2296 1492.53730.1 13.0735 3846.1538 9.4004 3571.4286 7.8909 3448.2759 7.3489 3448.2759 5.2320 2941.17650 13.0755 – 9.4027 – 7.8932 – 7.3513 – 5.2344 –0.1 13.0775 3846.1538 9.4050 3571.4286 7.8956 3448.2759 7.3537 3448.2759 5.2368 2941.17650.2 13.0794 1923.0769 9.4073 1818.1818 7.8980 1724.1379 7.3560 1694.9153 5.2392 1492.53730.3 13.0814 1282.0513 9.4096 1176.4706 7.9004 1123.5955 7.3585 1098.9011 5.2416 980.39220.4 13.0834 934.5794 9.4119 862.0690 7.9028 819.6721 7.3608 806.4516 5.2440 735.29410.5 13.0854 724.6377 9.4142 666.6667 7.9051 636.9427 7.3632 625.0000 5.2464 555.55560.6 13.0873 584.7953 9.4165 534.7594 7.9075 510.2041 7.3656 500.0000 5.2487 444.44440.7 13.0893 476.1905 9.4187 434.7826 7.9099 414.9378 7.3680 404.8583 5.2511 361.01080.8 13.0913 389.1051 9.4210 355.8719 7.9123 340.1361 7.3704 332.2259 5.2535 296.73590.9 13.0933 312.5000 9.4233 286.5330 7.9146 272.4796 7.3728 266.6667 5.2559 238.6635Calculations were done using the maturity time T D 0:5, the time step t D 0:001, and the number of sample paths M D 1;000;000 G. Jones and R. Makarov Pricing Options with Hybrid Stochastic Volatility Models 559
Fig. 2 The impact of the correlation coefficient O on the performance of the control variatemethods
Acknowledgements R. Makarov wishes to acknowledge the generous support of the NSERCDiscovery Grant program.
References
1. Ahlip, R., Rutkowski, M.: Pricing of foreign exchange options under the Heston stochastic volatility model and CIR interest rates. Quant. Financ. 13(6), 955–966 (2013) 2. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Pol. Econ. 81(3), 637–654 (1973) 3. Blanchard, A.: The two-factor Hull-White model: pricing and calibration of interest rates derivatives. Accessed 28 Mar 2014 4. Brigo, D., Mercurio, F.: Interest Rate Models-Theory and Practice: With Smile, Inflation and Credit. Springer, Berlin/New York (2007) 5. Cox, J.C., Ingersoll, J.E., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53(2), 385–407 (1985) 6. Duffie, D., Pan, J.: Transform analysis and asset pricing for affine jump-diffusions. Economet- rica 68(6), 1343–1376 (2000) 7. Glynn, P.W., Szechtman, R.: Some new perspectives on the method of control variates. In: Monte Carlo and Quasi-Monte Carlo methods, 2000 (Hong Kong), pp. 27–49. Springer, Berlin (2002) 8. Grzelak, L.A., Oosterlee, C.W.: On the Heston model with stochastic interest rates. SIAM J. Financ. Math. 2(1), 255–286 (2011) 9. Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327–343 (1993)10. Hull, J., White, A.: Pricing interest-rate-derivative securities. Rev. Financ. Stud. 3(4), 573–592 (1990)11. Kienitz, J., Kammeyer, H.: An implementation of the Hybrid-Heston-Hull-White model. Available at SSRN 1399389 (2009)12. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Volume 23 of Applications of Mathematics (New York). Springer, Berlin (1992)13. Rouah, F.D.: The Heston Model and Its Extensions in Matlab and C. Wiley, Hoboken (2013)14. Van Haastrecht, A., Lord, R., Pelsser, A., Schrager, D.: Pricing long-maturity equity and FX derivatives with stochastic interest rates and stochastic volatility. Insur.: Math. Econ. 45(3), 436–448 (2009) 560 G. Jones and R. Makarov
15. Vasicek, O.: An equilibrium characterization of the term structure [reprint of J. Financ. Econ. 5(2), 177–188 (1977)]. In: Financial Risk Measurement and Management. Volume 267 of International Library of Critical Writings in Economics, pp. 724–735. Edward Elgar, Cheltenham (2012)16. Zhu, J.: Applications of Fourier Transform to Smile Modeling, 2nd edn. Springer Finance. Springer, Berlin (2010) Theory and implementation Delay Stochastic Models in Finance
Anatoliy Swishchuk
Abstract The paper is devoted to an overview of delay stochastic models in financeand their applications to modeling and pricing of swaps. The volatility processis an important concept in financial modeling. This process can be stochasticor deterministic. In quantitative finance, we consider the volatility process to bestochastic as it allows to fit the observed market prices under consideration, aswell as to model the risk linked with the future evolution of the volatility, whichdeterministic model cannot. Most stochastic dynamical systems (SDS) in financedescribe the state of security (asset or volatility) as a value at time t;-instantaneousor current value at time t: We would like to take into account not only the currentvalue at time t; but also values over some time interval Œt ; t ; where is apositive constant and is called the delay. In this way, we incorporate path-dependenthistory of the volatility under consideration. In this paper we will mainly focuson newly developed so-called delayed Heston model that significantly improveclassical Heston model with respect to the market volatility surface fitting by 44 %.Review of some other delay stochastic models in finance will be given as well.
1 Introduction: Variance and Volatility Swaps
A security is a tradable financial asset of any kind, e.g., stock. A stock is a portionof ownership in a corporation. If S.t/ is a value of stock at time t; then S.t /is a delayed stock price. Volatility is a measure for variation of price of a financialinstrument (stock, etc.) over time. If V.t/ is the volatility at time t; then V.t / isa delayed volatility. One of the main problems in finance is pricing of volatility and variance (squareof volatility) swaps. A forward contract or simply a forward is a contract betweentwo parties to buy or sell an asset at a specified future time at a price agreed upontoday. Volatility swaps are defined as forward contracts on future realized stockvolatility. Variance swaps are similar contracts on variance, the square of the futurevolatility.
A. Swishchuk ()University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 561J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_51 562 A. Swishchuk
Volatility swaps allow investors to profit from the risks of an increase or decreasein future volatility of an index of securities or to hedge against these risks. If youthink current volatility is low, for the right price you might want to take a positionthat profits if volatility increases. Payoff of variance swap is defined as
N.VR Kvar /;
where VR is the realized stock variance (quoted in annual terms) over the life of thecontract, Kvar is the strike price, N is the notional amount, Z 1 T VR WD Vs ds: T 0
Price of variance swap is then:
Pvar D EQ ferT .VR Kvar /g;
where r is the risk-free discount rate corresponding to the expiration date T; and EQdenotes the risk-neutral expectation. Payoff of volatility swap is defined as p N. VR Kvol /; pwhere VR is the realized stock volatility (quoted in annual terms) over the life ofthe contract, and s p Z 1 T VR WD Vs ds: T 0
Price of volatility swap is then: p Pvol .x/ D EQ ferT . VR Kvol /g:
If dS.t/ is a change of stock price at time t; then usually
dS.t/ D a.t; S.t//dt C V.t; S.t//dW.t/ C b.t; S.t//dL.t/ (1) D “drift” C “noise” C “random jumps”;
where a.t; S.t// is a drift coefficient, V.t; S.t// is a diffusion coefficient, b.t; S.t// isa jump size coefficient, and W.t/ and L.t/ is a Brownian motion and a jump process(e.g., Lévy process, see [12]) (independent or correlated). Delay Stochastic Models in Finance 563
There are many papers (see, e.g., [1, 4, 5, 7, 8, 10, 11, 13]) in finance that considera a.t; S.t //; V V.t; S.t // or both parameters t and S depending on delay : In this paper, we will focus on volatility or variance models with delay in finance. The volatility process V.t; S.t// in (1) is an important concept in financialmodeling. This process can be stochastic or deterministic. In quantitative finance,we consider the volatility process V.t; S.t// to be stochastic as it allows to fit theobserved market prices under consideration, as well as to model the risk linked withthe future evolution of the volatility, which deterministic model cannot. If V.t; S.t// is the volatility at time t; then we consider some models for thechange of V.t/ at time t; dV.t/:
dV.t; S.t// D c.t; V.t; S.t///dt C d.t; V.t; S.t///dW.t/ C e.t; V.t; S.t///dL.t/ D “drift” C “noise” C “random jumps”: (2)
In delayed stochastic volatility models the coefficients c; d; e depend on delay or contain path-dependent history of volatility.
2 Some Delayed Stochastic Models in Finance: An Overview
Significance of path-dependency (a.k.a. delay) may be seen from the Fig. 1 below:the price of variance swap crucially depends on delay, where we used S&P60Canada Index as our real data (vertical line stands for delivery price, and two planelines stand for jump intensity (left) and for delay (right)).1. Continuous-time GARCH model for Stochastic Volatility with Delay (see [9]). We assume in this paper the following model for volatility V.t; S.t//: Z p 2 dV.t; S.t// ˛ t D C V.s; S.s//dW.s/ .˛ C /V.t; S.t//: dt t
Here, all the parameters ˛; ; ; are positive constants and 0 < ˛ C < 1: All the parameters of this model were inherited from its discrete-time analogue: ˛ 2 Vn D C ln .Sn1 =Sn1l / C .1 ˛ /Vn1 ; lD ; l which, in the special case l D 1, is a well-known GARCH(1,1) model for stochastic volatility without conditional mean of log-return (see [2, 3]). Here, Sn is a stock price at time n: We used this model to find the prices of variance and volatility swaps (see [14]).2. Multi-Factor Stochastic Volatility with Delay (see [15]). 564 A. Swishchuk
Dependence of Delivery Price on Delay and Jump Intensity
−4 x 10
5 Delivery Price
1 1 20 15 0.5 10 5 0 Jump Intensity Delay
Fig. 1 Dependence of delivery price on delay and jump intensity (S&P60 Canada Index)
The main idea is to consider continuous-time GARCH model for stochastic volatility with coefficient above that itself is stochastic and depends on drift and noise .t; chance/: In [15], CAMQ, we considered three two-factor and one three-factor stochas- tic volatility models with delay and we calculated the prices of variance swap for each model. One of the examples is: 8 hR p i2 ˆ < dV.t/ D t C ˛ t V.s/dW.s/ .˛ C /V.t/: dt t
:̂ dt =t D dt C ˇdW1 .t/;
where S.t/ is defined from the equation p dS.t/ D S.t/dt C V.t/St dW.t/;
St WD S.t / and Wiener processes W.t/ and W1 .t/ may be correlated.3. Stochastic Volatility with Delay and Jumps (see [16]). Delay Stochastic Models in Finance 565
In this paper, the volatility satisfies the following equation: Z Z dV.t; S.t// ˛h t p t i2 D C V.s; S.s//dW.s/C ys dN.s/ .˛ C /V.t; S.t//; dt t t
where W.t/ is a Brownian motion, N.t/ is a Poisson process with intensity and yt is the jump size at time t. We assume that EŒ yt D A.t/, EŒ ys yt D C.s; t/; s < t and EŒ y2t D B.t/ D C.t; t/, where A.t/; B.t/; C.s; t/ are all deterministic functions. We calculated the price of variance swap in this case.4. Lévy-based Stochastic Volatility with Delay (see [17]). The stochastic volatility in this paper satisfies the following Lévy-driven equation: hR p i2 dV.t;S.t// ˛ t dt D C t V.u; S.u/dL.u/ .˛ C /V.t; S.t//
where L.t/ is a Lévy process independent of W.t/: Here, > 0 is a mean- reverting level (or long-term equilibrium of V.t; St /), ˛; > 0; and ˛ C < 1: We also calculated the price of variance swap in this case.
3 Delayed Heston Model
Classical Heston model (see [6]) is one of the most popular stochastic volatilitymodels in the industry as semi-closed formulas for vanilla option prices areavailable, few (five) parameters need to be calibrated, and it accounts for the mean-reverting feature of the volatility. In this section we will focus on newly developedso-called delayed Heston model that significantly improve classical Heston modelwith respect to the market volatility surface fitting by 44 %. In this model, we takeinto account not only current state of volatility at time t but also its past history oversome interval Œt ; t ; where > 0 is a constant and is called the delay. In this way,our model incorporates path-dependent history for volatility. We will show how tomodel and price variance and volatility swaps (forward contracts on variance andvolatility) for the delayed Heston model and how to hedge volatility swaps usingvariance swaps (see [18]). The main points of this section are motivation, advantage and goal:• Motivation: to include past history (a.k.a. delay) of the variance (over some delayed time interval Œt ; t );• Advantage: Improvement of the Volatility Surface Fitting (44 % reduction of the calibration error) compare with Classical Heston model;• Goal: to price and hedge volatility swaps. We consider in a first approach adjusting the Heston drift by a deterministicfunction of time so that the expected value of the variance under our new delayedHeston model is equal to the one under ‘delayed vol’. Our approach can therefore be 566 A. Swishchuk
seen as a variance 1st moment correction of the Heston model, in order to accountfor the delay. It is important to note that our model is a generalization of the classicalHeston model (the latter corresponding to the zero delay case D 0 of our model).We performed numerical tests to validate our approach. With recent market data(Sept. 30th 2011, underlying EURUSD), we performed the model calibration on thewhole market vanilla option price surface (14 maturities from 1M to 10Y, 5 strikesATM, 25 Delta Call/Put, 10 Delta Call/Put). The results show a significant (44 %)reduction of the average absolute calibration error compared to the Heston model(i.e. average of the absolute differences between market and model prices). Further, we consider variance and volatility swaps hedging and pricing in ourdelayed Heston framework. These contracts are widely used in the financial industryand therefore it is relevant to know their price processes (how much they worth ateach time t) and how we can hedge a position on them, i.e. theoretically cancel therisk inherent to holding one unit of them. Using the fact that every continuous local martingale can be represented as atime-changed Brownian motion, as well as the Brockhaus and Long approximation(that allows to approximate the expected value of the square-root of an almostsurely non negative random variable using a 2nd order Taylor expansion approach),we were able to derive closed formulas for variance and volatility swaps priceprocesses. In addition, as variance swaps are relatively liquid instruments in themarket (i.e. they can be easily bought and sold), we considered the question ofhedging a position on a volatility swap using variance swaps in our framework. We are able to derive a closed formula for the dynamic hedge ratio, i.e. thenumber of units of variance swaps to hold at each time in order to hedge a positionon a volatility swap. In the following subsections we give a list of main steps in pricing of varianceand volatility swaps and hedging of volatility swaps for delayed Heston model.
3.1 Delayed Heston Model for Variance
As we mentioned before, our main motivation is to take into account past history ofthe varinace in its diffusion (over some delayed time interval Œt ; t ). The Heston model is one of the most popular stochastic volatility models in theindustry, as semi-closed formulas for vanilla option prices are available, few (five)parameters need to be calibrated, and it accounts for the mean-reverting feature ofthe volatility: p dSt D rSt dt C Vt St dWtQ p dVt D Œ. 2 Vt / dt C ı Vt dWtQ ;
where St is a stock price, Vt is the stochastic variance (just square of volatility),WtQ is the Wiener process with respect to the risk-neutral probability Q and r is theinterest rate. Delay Stochastic Models in Finance 567
We would like to take into account not only its current state (as it is the case inthe Heston model) but also its past history over some interval Œt ; t ; where isa positive constant and is called the delay. Namely, at each time t; the immediatefuture volatility at time t C will not only depend on its value at time t but also onall its history over Œt ; t : Namely, at each time t, the immediate future volatilityat time t C will not only depend on its value at time t but also on all its historyover Œt ; t : We would like to mention that the non-Markovian continuous-timeGARCH model (see [14]) Z t p dVt 1 D . 2 Vt / C ˛ . Vs dZsQ . r/ /2 Vt dt t
is not suitable in this case, as there is no closed-form solution available for thismodel. Therefore, we proposed the following Markovian delayed Heston model forvariance (see [18]): 2 p Q dVt D Œ. Vt / 2C 1.t/ dt R t CQ ı Vt dWt Q (3) .t/ WD ˛ . r/ C t E .Vs /ds E .Vt / :
We can notice that the classical Heston model and ‘delayed variance’ in (3) arevery similar in the sense that the expected values of the variances are the same –when we make the delay tends to 0 in ‘delayed variance’. As mentioned before, theHeston framework is very convenient for practitioners, and therefore it is naturallytempting to adjust the Heston dynamics in order to incorporate – in some way – thedelay introduced in ‘delayed variance’. We note, that lim !0 supt2RC j .t/j D 0:
3.2 Variance & Volatility Swaps Pricing Techniques
We show in this section how to calculate the variance and volatility swaps. RT Consider the realized variance: VR WD T1 0 Vs ds; where Vt is defined in (3), andsuppose that p Kvar D EQ ŒVR ; Kvol D EQ Œ VR :
To calculate Kvar we need only EQ ŒVR ; but for calculating Kvol we need more,due to Brockhaus and Long approximation:
p p VarŒZ EŒ Z EŒZ ; (4) 8EŒZ 3=2
namely, VarŒVR : 568 A. Swishchuk
Using time-changed Brownian motion representation for continuous local mar-tingales: if xt WD .V0 2 /e t C e t .Vt 2 /; and R dxt D f .t; xt /dWtQ ; xt D W O Tt ; Tt D< x >t D t f 2 .s; xs /ds; where 0 ˛ .r/2 ˛ 2 WD 2 C ; WD ˛ C C .1 e /; then
Vt D 2 C .V0 2 /e t C e t W O Tt D EQ ŒVt C e t W O Tt ; (5)
we get closed formula for Variance Swap and Volatility Swap fair strikes. Formula(5) for the variance is the time-change representation for the variance in the delayedHeston model and our main object in finding the variance and volatility swaps. The parameter 2 in (5) can be interpreted as the delayed-adjusted long-rangevariance. We note, that 2 ! 2 as ! 0: The parameter in (5) can be interpreted as the delayed-adjusted mean-revertingspeed. We note, that ! as ! 0:
3.3 Volatility Swap Hedging
Consider the Price Processes: p Variance Swap: Xt .T/ WD EtQ ŒVR ; and Volatility Swap: Yt .T/ WD EtQ Œ VR ;where RT VR WD T1 0 Vs ds: Portfolio containing 1 Volatility Swap and ˇt Variance Swaps has the followingform:
˘t D er.Tt/ ŒYt .T/ Kvol C ˇt .Xt .T/ Kvar / Rt If It WD 0 Vs ds is the accumulated variance at time t; then:
1 RT Xt .T/ D EtQ Œq It T C T t Vs ds WD g.t; It ; Vt / 1 RT Yt .T/ D EtQ Œ It T C T t Vs ds WD h.t; It ; Vt /
We compute the infinitesimal variations of these processes (using the fact thatXt .T/ and Yt .T/ are martingales):
@g p dXt .T/ D @V ı Vt dWtQ ; @h p t dYt .T/ D @Vt ı Vt dWtQ ; (6) @h @g p d˘t D r˘t dt C er.Tt/ Œ @V t C ˇt @V t ı Vt dWtQ ; Delay Stochastic Models in Finance 569
0.19
0.18 Naive Kvol Kvol 0.17
0.16
0.15
0.14
0.13 0 1 2 3 4 5 6 7 8 9 10 T (year)
Fig. 2 Naive Volatility Swap vs. adjusted Volatility Swap strikes
hence, to keep our portfolio ˘t risk-neutral, we have to eliminate the coefficient atthe random part in (6): )
@h @Yt .T/ @Vt @V ˇt D @g t D @X .T/ (7) t @Vt @Vt
-hedge ratio. We take the parameters that have been calibrated in above-mentioned section(vanilla options on September 30th 2011 for underlying EURUSD, maturities from1M to 10Y, strikes ATM, 25D Put/Call, 10D Put/Call), namely, they are: .v0 ; ; 2 ; ı; c; ˛; / D .0:0343; 3:9037; 108; 0:808; p 0:5057; 71:35; 0:7821/: We plot below the naive Volatility Swap strike Kvar and the adjusted Volatility pSwap strike Kvar Var .V R/ Q 3 along the maturity dimension, see Fig. 2, as well as the 2 8Kvar VarQ .VR /convexity adjustment 3 ; see Fig. 3, respectively. Also, we plot initial hedge 2 8Kvarratio ˇ0 .T/ with respect to the formula (7) with t D 0; see Fig. 4.
4 Discussion and Some Open Problems
In this paper is we gave an overview of delay stochastic models in finance andtheir applications to modeling and pricing of swaps. Most stochastic dynamicalsystems in finance describe the state of security (asset or volatility) as a value attime t;-instantaneous or current value at time t: We took into account not only the 570 A. Swishchuk
0.025
Convexity Adjustment 0.02
0.015
0.01
0.005
0 0 1 2 3 4 5 6 7 8 9 10 T (years)
Fig. 3 Convexity adjustment (see (4))
–2.4
–2.6 Initial Hedge Ratio
–2.8
–3
–3.2
–3.4
–3.6
–3.8 0 1 2 3 4 5 6 7 8 9 10 T (years)
Fig. 4 Initial Hedge ratio ˇ0 .T/
current value at time t; but also values over some time interval Œt ; t ; where is a positive constant and is called the delay. In this way, we incorporated path-dependent history of the security (asset or volatility) under consideration. In thispaper we mainly focused on newly developed so-called delayed Heston model thatsignificantly improve classical Heston model with respect to the market volatilitysurface fitting by 44 %. Review of some other delay stochastic models in financehas been given as well. Delay Stochastic Models in Finance 571
There are some very interesting open problems arising from the delay stochasticmodels in finance. One of them is modeling and pricing of covariance andcorrelation swaps in the case of two underlying assets with delayed Heston modelfor variances. Another problem is a comparison of volatility swap or other volatilityderivatives pricing for delayed Heston model and Levy-based stochastic volatilitymodel Levy-based stochastic volatility model. Also, there are some financial modelsthat contain fractional Brownian motion as an uncertainty. It would be good tocompare our delay stochastic models with fractional models. We shall leave all theseproblems for our future research papers.
References
1. Arriojas, M., Hu, Y., Mohammed, S-E.A., Pap, G.: A delayed Black and Scholes formula. Stoch. Anal. Appl. 25, 471–492 (2007) 2. Bollerslev, T.: Generalized autoregressive conditional heteroskedasicity. J. Econom. 31, 307– 327 (1986) 3. Bollerslev, T., Chou, R., Kroner, K.: ARCH modeling in finance: a review of the theory and empirical evidence. J. Econom. 52, 5–59 (1992) 4. Dokuchaev, N.: Modelling possibility of short-term forecasting of market parameters. Ann. Econ. Finance 16(1), 43–161 (2015) 5. Dibeh, G., Harmanani, H.M.: A stochastic chartist fundamentalist model with time delays. Comput. Econ. 40(2), 105–113 (2012) 6. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), (1993), 327–343 7. Ivanov, A., Kazmerchuk, Yu., Swishchuk, A.: Theory, stochastic stability and applications of stochastic differential delay equations. A survey of recent results. Differ. Equ. Dyn. Syst. 11 (1–2), 55–115 (2003) 8. Kazmerchuk, Y.I., Swishchuk, A.V., Wu, J.H.: Black-Scholes Formula Revisited: Security Markets with Delayed Response, Bachelier Finance Society 2nd World Congress, Crete (2002) 9. Kazmerchuk, Y., Swishchuk, A., Wu, J.-H.: A continuous-time GARCH model for stochastic volatility with delay. Can. Appl. Math. Q. 13(2), 123–148 (2005)10. Kazmerchuk, Yu., Swishchuk, A., Wu, J.-H.: The pricing of options for security markets with delayed response. Math. Comput. Simul. 75, 69–79 (2006)11. Li, J.-C., Mei, D.-C.: The influences of delay time on the stability of a market model with stochastic volatility. Physica A: Stat. Mech. Appl. 392(4), 763–772 (2013)12. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)13. Stoica, G.: A stochastic delay financial model. Proc. Am. Math. Soc. 133(6) 1837–1841 (2004)14. Swishchuk, A.: Modeling and pricing of variance swaps for stochastic volatilities with delay. WILMOTT Mag. (19), 63–73 (Sept. 2005)15. Swishchuk, A.: Modeling and pricing of variance swaps for multi-factor stochastic volatilities with delay. Can. Appl. Math. Q. 14(4), 439–467 (2006)16. Swishchuk, A., Li, X.: Pricing of variance and volatility swaps for stochastic volatilities with delay and jumps. Int. J. Stoch. Anal. 27 (2011). http://dx.doi.org/10.1155/2011/43514517. Swishchuk, A., Malenfant, K.: Variance swaps for local Levy based stochastic volatility with delay. Int. Rev. Appl. Financ. Issues Econ. 3(2), 432–441 (2011)18. Swishchuk, A., Vadori, N.: Smiling for the delayed volatility swaps. Wilmott Mag (November 2014) Semi-parametric Time Series Modellingwith Autocopulas
Antony Ware and Ilnaz Asadzadeh
Abstract In this paper we present an application of the use of autocopulas formodelling financial time series showing serial dependencies that are not necessarilylinear. The approach presented here is semi-parametric in that it is characterizedby a non-parametric autocopula and parametric marginals. One advantage of usingautocopulas is that they provide a general representation of the auto-dependencyof the time series, in particular making it possible to study the interdependence ofvalues of the series at different extremes separately. The specific time series that isstudied here comes from daily cash flows involving the product of daily natural gasprice and daily temperature deviations from normal levels. Seasonality is capturedby using a time dependent normal inverse Gaussian (NIG) distribution fitted to theraw values.
1 Introduction
In this study, autocopulas are used to characterise the joint distribution betweensuccessive observations of a scalar Markov chain. A copula joins a multivariatedistribution to its marginals, and its existence is guaranteed by Sklar’s theorem[8]. In particular, a Markov chain of first order with any given univariate margincan be constructed from a bivariate copula. A theoretical framework for the useof copulas for simulating time series was given by [3], who presented necessaryand sufficient conditions for a copula-based time series to be a Markov process,but not necessarily a stationary one. They presented theorems specifying whentime series generated using time varying marginal distributions and copulas areMarkov processes. Joe [4] proposed a class of parametric stationary Markovmodels based on parametric copulas and parametric marginal distributions. Chenand Fan [2] studied the estimation of semiparametric stationary Markov models,
This work was supported by MITACS, Direct Energy and by an NSERC Discovery Grant.A. Ware () • I. AsadzadehUniversity of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 573J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_52 574 A. Ware and I. Asadzadeh
using non-parametric marginal distributions with parametric copulas to generatestationary Markov processes. The term ‘autocopula’ was first used to describe the unit lag self dependencestructure of a univariate time series in [7], and we adopt the terminology here.We make use of the framework presented in [3] to produce Markov processessuch that the marginal distribution changes over time. The main benefit of usingautocopulas for univariate time series modelling is that the researcher is ableto specify the unconditional (marginal) distribution of Xt separately from thetime series dependence of Xt [6]. We apply a semiparametic method which ischaracterized by an empirical autocopula and a parametric time varying marginaldistribution. This allows us to capture seasonal variations in a natural way. This isan important feature of our model, motivated by the fact that many financial timeseries exhibit seasonality, particularly those arising from energy and commoditymarkets. The remainder of this paper is organized as follows. In Sect. 2, we introduce thedata; Sect. 3 describes the model, including a review of copulas, and of the NormalInverse Gaussian distribution. This section also includes details of the calibrationand simulation procedures, and the final section presents some results.
2 The Data
The motivation for this project came from the desire to develop a parsimoniousmodel that could help to capture so-called load-following (or swing) risk. This isone of the main sources of financial uncertainty for an energy retailer, and arisesfrom the combination of retail customer consumption (volume) uncertainty andprice uncertainty. Both volume (V) and price (P) are driven to a large extent byweather. In particular, average daily temperature is one of the main drivers of dailynatural gas consumption in various North American markets: this in turn drivesmarket prices through a supply and demand process. Some of the load-following risk exposure can be hedged using gas forwards andtemperature derivatives. A significant part of the risk exposure that cannot be easilyhedged and is linked to the daily product between the weather deviation from normal(žF) and the daily price deviation from the expected value of the ex-ante forwardprice. To make this more specific, let P denote the last-traded forward monthlyindex price, and V the expected monthly average volume. Cash flows for the retailerdepend on the product PV, and the uncertainty in this quantity can be written
.P C P/.V C V/ PV D P V C V P C P V:
We can capture the first-order dependence between volume and weather deviationsby writing V D ˇ W C , where represents higher-order dependence on Was well as exogenous risk factors; modelling these other contributions was beyondthe scope of this study. Forward instruments in weather and natural gas markets Semi-parametric Time Series Modelling with Autocopulas 575
Fig. 1 Product of weather and gas price deviations ( P W) in Algonquin over 2003–2014.Spikes correspond to combinations of high weather deviation from normal and high spot pricedeviation from next forward month
can then be used to hedge risks corresponding to the terms P V and V P. It canbe seen that the term P W is an important component of unhedged risk in thesecashflows. One approach to modelling this component would be to develop separatemodels for weather and natural gas prices (both daily and forward prices). However,because of the desire for parsimony, we instead seek a model that allows us to studythe time-series Xt D . P W/t , in order to estimate the range and probabilities ofpossible outcomes at the level of a complex portfolio of retail load obligations. Here we focus on Algonquin Citygate gas prices and the Boston Logan stationfor weather data. The data cover the period 1 January 2003–31 June 2014 on adaily basis, and are shown in Fig. 1. The most dramatic feature of the graph is thepresence of intermittent clusters of spikes, during which the gas prices rise fromtheir approximate average daily value and at the same time temperature rises orfalls drastically. These mostly occur during winter, although large deviations alsooccur at other times of the year. It is also clear that the marginal densities of theseobservations will not be well-represented by normal distributions.
3 The Model
Here we introduce the simulation model in more detail, providing a brief reviewof copulas, and the normal inverse Gaussian distribution, which we use for themarginal densities. 576 A. Ware and I. Asadzadeh
3.1 Copulas and Autocopulas
A copula1 is a multivariate distribution function defined on a unit cube Œ0; 1 n ,with uniformly distributed marginals. In the following, we use copulas for theinterdependence structure of time series and, for simplicity and the fact that weare interested in the first order lag interdependence, we focus on the bivariate case,although the approach can be used to capture dependence on higher order lags. Let F12 .x; y/ be the joint distribution function of random variables X and Ywhose marginal distribution functions, denoted as F1 and F2 respectively, arecontinuous. Sklar’s theorem specifies that there exists a unique copula functionC.u; v/ D F12 .F11 .u/; F21 .v// that connects F12 .x; y/ to F1 .x/ and F2 .y/ viaF12 .x; y/ D C.F1 .x/; F2 .y//. The information in the joint distribution F12 .x; y/ isdecomposed into that in the marginal distributions and that in the copula function,where the copula captures the dependence structure between X and Y. Variousfamilies of parametric copulas are widely used (Gaussian, Clayton, Joe, Gumbelcopulas, for example). In a time series setting, we use a copula (or autocopula) to capture thedependence structure between successive observations. More generally, we have thefollowing definition [7].Definition 1 (Autocopula) Given a time series Xt and L D fli 2 ZC ; i D1; : : : ; dg a set of lags, the autocopula CX;L is defined as the copula of the d C 1dimensional random vector .Xt ; Xtl1 ; : : : ; Xtld /. If a times series Xt is modelled with an autocopula model with unit lag, withautocopula function C.u; v/ D CX;1 .u; v/, and (time-dependent) marginal CDFFt .x/, then, for each t, the CDF of the conditional density of Xt given Xt1 canbe expressed
@C FXt jXt1 .x/ D Ft1 .Xt1 /; Ft .x/ : (1) @uWe will discuss issues related to calibration and simulation below. Autocopula models include many familiar time series as special cases. Forexample, it is straightforward to show that an AR(1) process, yt D ˛yt1 CˇC.t/,can be modelled using theautocopula framework using the marginal distribution yˇ=.1˛/F1 .y/ D ˚ p (where ˚ denotes the standard normal CDF) and a 2 =.1˛ 2 / 2 1˛Gaussian copula with mean D ˇ=.1 ˛/ and covariance 1˛2 . ˛1 Part of the motivation for the use of autocopulas in time series modelling isthat, while correlation coefficients measure the general strength of dependence, theyprovide no information about how the strength of dependence may change across
1 For more discussion on the theory of copulas and specific examples, see [5]. Semi-parametric Time Series Modelling with Autocopulas 577
Fig. 2 Estimated values ofthe quantities C.u; u/=.u/ andC.u; u/=.1 u/ showinglower and upper taildependence in the observedvalues of P W
the distribution. For instance, in the dataset we consider here there is evidence oftail dependence, whereby correlation is higher near the tails of the distribution. Wecan quantify this using the following definition ([4], Section 2.1.10).Definition 2 (Upper and Lower Tail Dependence) If a bivariate copula C issuch that limu!1 C.u; u/=.1 u/ D U exists, where C.u; u/ D 1 C.1; u/ C.u; 1/ C C.u; u/, then C has upper tail dependence if U 2 .0; 1 and no upper taildependence if U D 0. Similarly, if limu!0 C.u; u/=.u/ D L exists, C has lowertail dependence if L 2 .0; 1 and no lower tail dependence if L D 0 In Fig. 2 we show estimates of the quantities C.u; u/=.u/ and C.u; u/=.1 u/,where here we use the order statistics of the time series Xt D . P W/t to generatea preliminary empirical proxy for the copula function C. It is clear from the figurethat neither set of values tends towards zero in the limit u ! 0 or u ! 1, and weconclude that the data exhibit nonzero tail dependence.
3.2 Time Varying Marginal Distribution
As noted above, the marginal densities for our time series will not be normal.We found that the normal inverse Gaussian (NIG) distribution provided a moresatisfactory fit. More information about this distribution and its applications canbe found in [1]. Here we review its definition and properties.
3.2.1 Definition and Properties of the NIG Distribution
A non-negative random variable Y has an inverse Gaussian distribution withparameters ˛ > 0 and ˇ > 0 if its density function is of the form
˛ .˛ ˇy/2 fIG .yI ˛; ˇ/ D p y3=2 exp ; for y > 0: 2ˇ 2ˇy 578 A. Ware and I. Asadzadeh
A random variable X has an NIG distribution with parameters ˛, ˇ, and ı if
XjY D y N. C ˇy; y/ and Y IG.ı; 2 /; pwith WD ˛ 2 ˇ 2 , 0 jˇj < ˛ and ı > 0. We then write X NIG.˛; ˇ; ; ı/.Denoting by K1 the modified Bessel function of the second kind, the density is givenby ı˛ exp ı C ˇ.x / p fNIG .xI ˛; ˇ; ; ı/ D p K1 ˛ ı 2 C .x /2 : ı 2 C .x /2
There is a one-to-one map between the parameters of the NIG distribution and themean, variance, skewness and kurtosis of the data. We first use moment matching todetermine initial estimates for the parameters, which we use as starting values fora MLE search method. The corresponding fit to the data is shown in Fig. 3, wherethe best fitting normal density is also shown. It can be seen that the NIG fit is quitegood. However, it is evident from Fig. 1 that the time series is strongly seasonal.We seek to capture this seasonality through the marginal densities by making one ofthe parameters of the NIG model time-dependent. This was achieved by assumingthe parameter to be constant in each month, and maximizing the resulting jointlikelihood across the entire data set. Of the four possible choices, ı gave the greatestimprovement to the AIC and BIC, as shown in Table 1. The estimated values of ıare shown in Fig. 4, and the seasonal pattern that is evident in the original data isevident again here.
Fig. 3 Histogram of observed data ( P W), with fitted normal distribution and NIG distribution.The estimated NIG parameters are: ˛ D 0:0980, ˇ D 0:0131, D 0:0122 and ı D 2:3799
Table 1 Calibrated values of Model AIC BIC Log likelihoodAIC, BIC, and Log likelihood ˛ be time varying 34;622:85 35;793:11 17;133:42 ˇ be time varying 36;280:04 37;450:30 17;962:02 be time varying 35;850:14 37;020:40 17;747:07 ı be time varying 33;495:75 34;666:02 16;569:87 Semi-parametric Time Series Modelling with Autocopulas 579
Fig. 4 Calibrated monthly values of ı from the combined NIG likelihood, ˛ D 0:0293, ˇ D0:0205, and D 0:0323
Fig. 5 Calibrated monthly values of ı, together with an example of a simulated path, as well asa colour contour plot of the quantiles from a large number of simulated paths. Seasonal meanand variance in (2) are linear combination of a constant and sin 2nt and cos 2nt for increasingn. The estimated parameters are as follows: a D 0:31, b D .0:93; 0:30; 0:09; 0:22; 0:12,0:032; 0:01; 0:03; 0:01/, and D .0:15; 0:09; 0:01/
As can be seen in Fig. 4, the value of ı tends to be higher in winter and lower insummer. The time series of values appears to be mean reverting with seasonal meanand variance. p We model the time series using a seasonal mean reverting process fort D ı t :
tC1 D at C b.t/ C .t/zt ; (2)
where the zt are independent standard normal samples. The mean and variance areestimated using periodic functions with periods from 1 year down to 3 months.Simulated and estimated values of ıt are shown in Fig. 5. Twenty thousand paths 580 A. Ware and I. Asadzadeh
were simulated using (2), and for each month the set of values was used to determinequantiles, which were then used to create the coloured patches shown in the figure.The darker patches correspond to quantiles nearer to the centre of the distribution,and the lighter patches to quantiles nearer the extremes. Once we have values of ıt , we can obtain the time varying cumulative distributionfunction and time varying density function. The NIG cumulative distributionfunction does not have a closed form solution, so we can compute the CDF usingGaussian quadrature to evaluate the following integral. Z xt F.xt I ˛; ˇ; ; ı/ D fNIG .Xt I ˛; ˇ; ; ıt /dXt (3) 1
In next section we explain the procedure to calculate the empirical autocopulas andsimulate cash flows.
3.3 Estimating the Empirical Autocopula
Having estimated the time-dependent NIG densities, we use these to produce a timeseries of values Vt D Ft .Xt / 2 Œ0; 1 . If the marginal densities were exact, thesewould be uniformly distributed on Œ0; 1 . In practice, they will only be approximatelyuniform, and we generate an additional empirical marginal density and an empirical(auto)copula to capture the joint density of .Vt ; Vt1 /. The empirical autocopula C is estimated by first estimating an empirical jointdensity for .Vt ; Vt1 / in the form of a strictly increasing continuous function ˚.; /that is piecewise bilinear. The domain Œ0; 1 2 is partitioned into rectangles containingapproximately similar numbers of samples .Vt1 ; Vt /, and taking ˚ to be thecumulative integral of the sum of indicator functions for these rectangles, scaled bythe number of samples in each rectangle. ˚ is then used to create strictly increasingpiecewise linear marginal densities ˚1 and ˚2 . The inverses of these densitiesare therefore also piecewise linear, and when composed with ˚ they generate apiecewise bilinear copula function C.u1 ; u2 / D ˚ ˚11 .u1 /; ˚21 .u2 / . This process is illustrated in Fig. 6. In Fig. 6a we plot the pairs of transformedvalues ˚1 .Vt1 /; ˚2 .Vt / , together with the outlines of rectangles used to generatethe piecewise bilinear function C. As mentioned, these rectangles contain roughlyequal numbers of points; constructing the empirical autocopula in this way ensuresthat it is strictly increasing, and well-suited to enable the computations involvedin time series simulation (see below) to be carried out efficiently. The resultingempirical autocopula C is shown in Fig. 6b. This function is binlinear on each of therectangles shown in Fig. 6a, but is less regular than it looks. The corresponding joint 2Cdensity, @u@1 @u2 .u1 ; u2 /, is shown in Fig. 6c. It can be seen that the density is highernear .0; 0/ and near .1; 1/, which is consistent with the tail dependency observedearlier. Semi-parametric Time Series Modelling with Autocopulas 581
Fig. 6 Generation of the empirical autocopula. (a) Scatter plot of ˚1 .Vt1 / against ˚2 .Vt /. Eachrectangle contains about 100 points (note that rectangles with around 25 points were used in thesimulation). (b) Empirical autocopula C.u1 ; u2 / defined to be bilinear on each of the rectanglesshown in (a). (c) The empirical density @2 C=@u1 @u2 , which is constant on each of the rectanglesshown in (a)
3.4 Simulation of Time Series Using Autocopula
Armed with the time-dependent NIG densities Ft ./, the empirical marginal densi-ties FV;i ./ and the empirical autocopula C.; /, we can generate simulated values xtas follows.1. Given an initial value x0 , generate v0 D F0 .x0 /.2. For t D 0; 1; : : : , given vt , generate vtC1 : a. Set u1 D ˚1 .vt /. b. Given u1 , create the piecewise linear function C.u/ WD C.u1 ; u/=u1 . c. Set u2 D C1 .U/, where U is an independent uniform random draw. d. Set vtC1 D ˚21 .u2 /.3. For each t > 0, set xt D Ft1 .vt /.Here we have used the fact (already alluded to in (1)) that, if U1 and U2 are uniformrandom variables whose joint distribution is the copula C.u1 ; u2 /, then, for u1 > 0,the cumulative density function for U2 , conditional on U1 D u1 , is
@C C.u1 ; u2 / PŒU2 < u2 jU1 D u1 D .u1 ; u2 / D : @u2 u1
The proof of this can be found in, for example, [3]. The fact that C is a piecewisebilinear function means that C will be piecewise linear. Moreover, the constructionof the empirical copula as described in Sect. 3.3 ensures that it is an increasingfunction with a limited number of corners. Its inverse can then be constructedreadily, and will also be an increasing piecewise linear function with a limitednumber of corners, and so can be evaluated with little computational effort. Indeed,in practice the computation of the final step in the above algorithm, the inversionof the time-dependent NIG densities, took more time than the copula-relatedcomputations. 582 A. Ware and I. Asadzadeh
4 Results
In Fig. 7 we show a 12-year sample time series for P W computed as describedin Sect. 3.4. In addition, we simulated around 700 independent time series andcomputed, for each month, the 99th percentile of values produced in that monthacross all simulations. What can be seen in the sample path is the same mixtureof quiescent periods and periods with large deviations from zero. There is someevidence of ‘clumps’ of large deviations occurring in winter months, although thisis less clear than in the original data (see Fig. 1). There is, nevertheless, an increasedoccurrence of large deviations in winter months, as can be seen from the plot of the99th percentiles superimposed on the sample simulation, shown in Fig. 7. Figure 8 reproduces the data from Fig. 2, with error bars corresponding to the 5thand 95th percentiles of the values obtained from the simulations, showing that thesimulations have reproduced the observed tail dependence.
Fig. 7 One example of simulated daily values of P W, together with the 99th percentiles ofcollected monthly values from around 700 simulations
Fig. 8 Estimated values of the quantities C.u; u/=.u/ and C.u; u/=.1 u/ for the originalobservations of P W. Also shown are error bars corresponding to the 5th and 95th percentilesof the values obtained from around 700 simulations Semi-parametric Time Series Modelling with Autocopulas 583
References
1. Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I.: Lévy Processes: Theory and Applications. Springer Science & Business Media, New York (2001)2. Chen, X., Fan, Y.: Estimation of copula-based semiparametric time series models. J. Econom. 130(2), 307–335 (2006)3. Darsow, W.F., Nguyen, B., Olsen, E.T., et al.: Copulas and Markov processes. Ill. J. Math. 36(4), 600–642 (1992)4. Joe, H.: Multivariate Models and Multivariate Dependence Concepts. CRC Press/Taylor & Francis Group, Boca Raton/London, New York (1997)5. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer Science & Business Media, New York (2007)6. Patton, A.J.: Copula–based models for financial time series. In: Handbook of Financial Time Series, pp. 767–785. Springer, Berlin/Heidelberg (2009)7. Rakonczai, P., Márkus, L., Zempléni, A.: Autocopulas: investigating the interdependence struc- ture of stationary time series. Methodol. Comput. Appl. Probab. 14(1), 149–167 (2012)8. Sklar, M.: Fonctions de répartition à n dimensions et leurs marges. Université Paris 8, (1959) Optimal Robust Designs of Step-StressAccelerated Life Testing Experimentsfor Proportional Hazards Models
Xaiojian Xu and Wan Yi Huang
Abstract Accelerated life testing (ALT) is broadly used to obtain reliabilityinformation of a product in a timely manner. The Cox’s proportional hazards (PH)model is often utilized for reliability analysis. In this paper, we focus on designingALT experiments for hazard rate prediction when a PH model is adopted. Due tothe nature of prediction made from ALT experimental data, attained under the stresslevels higher than the normal condition, extrapolation is encountered. In such case,the assumed model can not be tested. For possible imprecision in an assumed PHmodel, the method of construction for robust designs is explored. As an example,we investigate the robust designs for the situation where baseline hazard functionin the fitted PH model is simple linear but the true baseline hazard function isactually quadratic. The optimal stress-changing times are determined by minimizingthe asymptotic variance and asymptotic squared bias.
1 Introduction
Failure data are needed in order to quantify the life characteristics of a product.However, under normal design conditions such failure life data are very difficultto obtain for a product with high reliability within a reasonable time period. Toovercome this problem, accelerate life testing (ALT) has been well-developed toshorten the lifetime of a product and quickly obtain failures. In general, test units in accelerated life testing experiments are subjected tohigher than normal design level of stresses. The use of such accelerating stresses fora particular material is established by engineering practice. In an ALT experiment,both censoring and time-dependent loading plan are often used. In this paper,we consider time-censoring and time-dependent step-stress loading plan. Time-censoring occurs when the experiment stops at a predefined censoring time.In time-dependent step-stress ALT, all test units are subjected to stress levelsincreasing by steps: first, all test units are subjected to a lower stress level for a
X. Xu () • W.Y. HuangDepartment of Mathematics and Statistics, Brock University, St. Catharines, ON L2S 3A1,Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 585J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_53 586 X. Xu and W.Y. Huang
specified length of time, then the stress level increases and the test continues for allunfailed test units, and so on. To avoid invalid estimation, the range of stress levels should be restrictedin a stress loading plan. The failure data obtained at accelerated conditions areextrapolated to estimate the characteristic of life distribution at normal designconditions. In the literature of designing a step-stress ALT experiment, most work has beendone for the cases when a simple step-stress plan is adopted, when the model isfully parametric, or when model assumed is exactly correct. To mention a few:Miller and Nelson [9] have first presented Q-optimal designs for simple step-stresstests with complete data from an exponential distribution. Their designs minimizethe asymptotic variance (AVAR) of the maximum likelihood estimator (MLE) ofthe mean life at a normal design stress. Bai et al. [2] have extended the theory ofMiller and Nelson [9] to censored data. They have constructed the optimal simplestep-stress ALT with time-censoring. Bai and Kim [1] have presented an optimalsimple step-stress ALT for a Weibull distribution under time-censoring. The optimallow stress level and stress change time are obtained by minimizing the AVAR ofthe MLE of a specified percentile at normal design stress. Fard and Li [5] havealso presented a step-stress ALT for a Weibull distribution under time-censoring;however, their optimal stress-changing time is obtained by minimizing the AVAR ofthe MLE for reliability instead. Recent work on optimal designs for step-stress ALThas been reviewed by Hunt and Xu [6]. They have adopted a generalized Khamis-Higgins model for the effect of changing stress levels and also assumed that thelifetime of a test unit follows a Weibull distribution; however, their designs are alsooptimal when both the shape and scale parameters are considered being functions ofthe stress levels. The resulting optimal design chooses the stress-changing time inorder to minimize AVAR of the MLE of reliability at the normal design stress leveland at a pre-specified time. Ma and Meeker [8] have extended the results of Bai and Kim [1] to providea general method for multiple step-stress ALTs for a log location-scale familyof distributions. They have presented an approach to calculate the large-sampleapproximate variance of the MLE, computed from step-stress ALT data, for aquantile of the failure time distribution at normal design conditions. Jiao [7] hasfirst developed the simple step-stress ALT plan when a PH model is utilized forreliability prediction. Their optimal stress level is obtained by minimizing thevariance of the MLE of hazard rate at the normal design stress level and over apre-specified time period. Elsayed and Zhang [3] have presented an optimal simplestep-stress ALT plan based on a PH model to obtain the most accurate reliabilityfunction estimates at normal design stress. They have also formulated a nonlinearprogramming problem to minimize AVAR of the hazard rate estimator over aprespecified time period at normal design stress. The general theory for the asymptotic distribution of MLEs under modelmisspecification has been derived by White [11]. He has examined the consequencesand detection of model misspecification when the maximum likelihood techniqueis used for estimation. The proposed quasi-MLE converges to a well-defined Optimal Robust Design of ALT 587
limit. In addition, the properties of their quasi-MLE and the information matrixhave been exploited to yield several useful tests when model misspecification issuspected. Pascual [10] has presented the methodology for deriving the asymptoticdistribution of MLEs of model parameters with constant stress ALT when the stress-life relationship is misspecified. When possible departures from an assumed ALTmodel are suspected, his ALT plans can provide protection against potential biaswithout much loss in efficiency. However, the methodology developed there can beused to derive robust designs only for constant stress ALTs. In a complete general setting, robust designs for one-point extrapolation whichis the case for ALT have been discussed in Wiens and Xu [12], for least squaresestimation of a mean response. As Fang and Wiens [4] pointed out, “Extrapolationto regions outside of that in which observations are taken is, of course, an inherentlyrisky procedure and is made even more so by an over-reliance on stringent modelassumptions.” The classical optimal designs minimize the variance alone. However,when the fitted models are incorrect, the estimation is biased. A robust design shouldbe obtained in an optimal way so that even when the fitted model was not exactlycorrect, the designs can still be relatively efficient with a small bias. The present paper extends previous work to construct optimal robust ALTdesigns by taking into account multiple step-stress, semi-parametric model assump-tion, and possible imprecision in the assumed model. Due to the nature of theprediction made from ALT experimental data, attained under the stress levelshigher than the normal design condition, extrapolation is encountered. For possibleimprecision in an assumed PH model, the method of construction for robust designsbecomes significantly important. Therefore, we consider the situation where a PHmodel with a simple linear baseline hazard function is fitted; however, the truebaseline hazard function is a quadratic function. Optimal robust designs will beobtained in order to protect against possible departure from the assumed model witha minimum loss of efficiency. We propose a two-stage design procedure, in Sect. 4,where the optimal stress-changing times are derived by minimizing the asymptoticsquared bias (ABIAS2 ) at the first stage and then minimizing the AVAR at the secondstage.
2 Model Misspecification and Asymptotic Distribution of MLE Attained
The Cox’s proportional hazards (PH) model is one of the most important meansfor predicting the lifetime of a product. This model provides a flexible method foridentifying the effects of the covariates on failure rate. A PH model with a covariate(applied or transformed stress) independent of time is generally expressed as
.tI s/ D 0 .t/exp.s/; (1) 588 X. Xu and W.Y. Huang
where 0 .t/ is the baseline hazard rate function at the time of t, s is a stress levelused in an ALT test, and is a unknown parameter. It is assumed that the true model is Model (1) with a quadratic baseline hazardfunction, denoted by MT , i.e., Model (1) with 0 .t/ D 0 C 1 t C 2 t2 and D˛. However, the fitting model is Model (1) with a linear baseline hazard function,denoted by MF , i.e., Model (1) with 0 .t/ D 0 C 1 t and D ˇ. The maximum likelihood method has been used for estimating the modelparameters. Define D Œ0 ; 1 ; 2 ; ˛ T and D Œ0 ; 1 ; ˇ T . Let ` .I / be thelog-likelihood function under MT , and ` .I / be the log-likelihood function underMF , both with the same design, . The expected log-likelihood ratio under MT overMF is
I. W / D EMT Œ`.I / `.I / : (2)
In this paper, we use a 3-step-stress design to demonstrate our design method;however, the method developed can be applied to any m-step-stress ALT .m 2/. We let fF .tI s/ and GF .tI s/ be the probability density function and the cumulativedistribution function at failure time t and stress level s under MF , respectively. Wealso let c be the censoring time, and Fi .t/ D GF .tI si /, i D1, 2, 3. In addition,we define a such that the cumulative distribution function under stress level s2 ,evaluated at a is the same as that under stress level s1 , evaluated at 1 ; and we defineb such that the cumulative distribution function under stress level s3 , evaluated at bis the same as that under stress level s2 , evaluated at a C 2 1 . For a 3-step-stressALT design, the likelihood function from observed lifetimes ti , i D 1; : : : ; n, is P n 3iD1 ni Y n1 Y n2 Y n3 Y L .I / D fF .ti I s1 / fF .xi I s2 / fF .yi I s3 / .1 GF .zI s3 // ; (3) iD1 iD1 iD1 iD1
where a D F21 .F1 . 1 //, b D F31 .F2 .a C 2 1 //, x D a Ct 1 , y D b Ct 2 ,and z D b C c 2 . Given Model (1) with 0 .t/ D 0 C 1 t and D ˇ, we have h 1 i fF .tI s/ D .0 C 1 t/ exp.ˇs/ exp 0 t C t2 exp.ˇs/ ; 2 h 1 2 i GF .tI s/ D 1 exp 0 t C t exp.ˇs/ : 2The negative log-likelihood function is
X n1 1 ` .I / D n1 ˇs1 ln .0 C 1 ti / 0 ti C ti2 exp .ˇs1 / iD1 2 n2 h
X 1 i n2 ˇs2 ln .0 C 1 xi / 0 xi C x2i exp .ˇs2 / iD1 2 Optimal Robust Design of ALT 589
n3 h
X 1 i n3 ˇs3 ln .0 C 1 yi / 0 yi C y2i exp .ˇs3 / iD1 2 ! X3 h 1 i C n ni 0 z C z2 exp .ˇs3 / : (4) iD1 2
Taking the derivative of (4) with respect to 0 ; 1 and ˇ, we have
Xn1 @` .I / 1 D ti exp .ˇs1 / @0 iD1 .0 C 1 ti / n2 X 1 xi exp .ˇs2 / iD1 .0 C 1 xi / n3 X 1 yi exp .ˇs3 / iD1 . 0 C 1 yi / 3 ! X C n ni Œz exp .ˇs3 / ; (5) iD1
Xn1 @` .I / ti ti2 D exp .ˇs1 / @1 iD1 .0 C 1 ti / 2 n2 X xi x2 i exp .ˇs2 / iD1 .0 C 1 xi / 2 n3 X yi y2 i exp .ˇs3 / iD1 .0 C 1 yi / 2 3 ! 2 X z C n ni exp .ˇs3 / ; (6) iD1 2
and 1 h
X 1 i n @` .I / D n1 s1 C s1 0 ti C ti2 exp .ˇs1 / @ˇ iD1 2 n2 h
X 1 i n2 s2 C s2 0 xi C x2i exp .ˇs2 / iD1 2 590 X. Xu and W.Y. Huang
n3 h
X 1 i n3 s3 C s3 0 yi C y2i exp .ˇs3 / iD1 2 ! X3 h 1 i C n ni s3 0 z C z2 exp .ˇs3 / : (7) iD1 2
T Let D 0 ; 1 ; ˇ be the value of that minimizes (2), which is thesolutions of 0 ; 1 ; ˇ by setting all equations in (5), (6), and (7) equal to 0: Supposethat the data are collected under design, , with sample size n: The experimenterfits Model MF to the data by using MLE method. We adopt quasi-MLE approachby taking ni ’s to be independent although they are p actually not. We let b denotethe quasi-MLE of . By Theorem 3.2 of White [11], n .b / is asymptoticallynormal with mean 0 and covariance matrix C .I / which is defined as
C .I / D ŒA .I / 1 B .I / ŒA .I / 1 ; (8)
where 2 @ ` .; / A .I / D EMT (9) @@T
and @` .; / @` .; / B .I / D EMT : (10) @ @T
3 Optimality Criteria
We consider to estimate the hazard rate over a given time period at the normal designstress level. We determine the optimal stress-changing times 1 and 2 in order tominimize the ABIAS2 and AVAR of the MLE of a hazard function average overa specific period of time, .0; T , under normal design stress level sD . For a givendesign, the MLE estimator of the hazard rate at sD can be obtained by:
b 0 C b1 t/ exp b̌sD : MF .tI sD / D .b (11) Optimal Robust Design of ALT 591
The ABIAS2 of (11) average over .0; T is Z ABIAS 2 T
b MF .tI sD / jMT dt 0 Z 2 D T h b i EMT MF .tI sD / MT .tI sD / dt 0 Z T 2 0 C 1 t exp2 .ˇ sD / dt ; (12) 0 0 C 1 t C 2 t exp .˛sD /
and the AVAR of (11) average over .0; T is Z T h b i AVAR MF .tI sD / jMT dt 0
1 Z T h iˇ h @b iT ˇˇ @b @b @b ˇ @b @b ˇ D ˇ C WD ˇ dt. (13) n @b 1 @b̌ 0 @b b @b 1 @b̌ 0 @b 0 D b D
4 Optimal Robust Designs
In ALT practice, a certain number of failures under each test stress level is oftenrequired. Such requirement is made to avoid that the stress changing times occurtoo soon to provide a reasonable step-stress ALT having the same number of stresslevels as planned. Please see Elsayed and Zhang [3] for an example of simple step-stress ALT. We also take such practical requirement into our design consideration.These requirements on the minimum number of failures at each stress level are asfollows:(a) The expected number of failures at stress level s1 has a minimum value MNF1 :
n Pr Œt 1 js1 MNF1 I (14)
(b) The expected number of failures at stress level s2 has a minimum, called MNF2 :
.n n1 / Pr Œa C t 1 2 js2 MNF2 I (15)
(c) The expected number of failures at stress level s3 has a minimum, named MNF3 :
.n n1 n2 / Pr Œb C t 2 c js3 MNF3 ; (16)
where ni is the number of failures under si , i D 1; 2; 3: 592 X. Xu and W.Y. Huang
Under the constraints of minimum number of failures at each stress level, wepropose a two-stage optimal robust design procedure. At the first stage, we constructa benchmark design so that the estimation bias can be reduced to a minimal possible.At the second stage, we update the benchmark design to an optimal robust design inorder to minimize the AVAR. We illustrate our method of designing a 3-step-stress ALT experiment byrevisiting an example discussed in Elsayed and Zhang [3] with initial values:0 D 0:0001, 1 D 0:5, 2 D 0:0015, ˛ D 3800, and T D 300: We take s2be 197.5, the average of low (s1 D 145) and high (s3 D 250) stress levels. All othervalues of the parameters in the example remain the same. We first determine the parameters needed for (12). The value of 0 ; 1 can bedetermined by minimizing the absolute distance between quadratic baseline hazardfunction and linear baseline hazard function over the whole testing period, namely Z T 0 ; 1 D arg min 0 C 1 t C 2 t2 .0 C 1 t/ dt: (17) 0
We have considered five scenarios of the constraints which are listed in Table 1. At the first stage, the optimal 1 and 2 for our benchmark design can bedetermined in order to minimize (12) under each constraint. We denote B Ck asthe optimal design obtained in benchmark under the given constraint Ck , wherek D 1; ; 5. At the second stage, we adopt the benchmark design parameters.We set the derivatives in (5), (6), and (7) to be zero, and solve for 0 ; 1 ; ˇ. Thisis a constrained nonlinear problem. We solve it in an iterative way. First, we fix ˇat 3800, and search for f f 0 and 1 in positive ranges of 0 and 1 within their95 % confidence intervals in order to maximize the log-likelihood function. Then,B Ck can be obtained for B D Œf f 0 ; 1 ; 3800 by minimizing (12). Second, with0 and 1 obtained from the first step, we update our estimate of ˇ to ˇ . By Titerating these two steps, we have D 0 ; 1 ; ˇ . Then, the robust design canbe obtained for . We iterate these two steps, until either 0 ; 1 or ˇ remainsunchanged or the difference between its values obtained from the current step andthat from the previous step is sufficiently small within a pre-specified range. Finally,we can obtain the optimal 1 and 2 based on the most updated estimates for .
Table 1 Constraint cases for 3-step-stress ALTNames of the Expected numberconstraints Constraint parameters of total failuresC1 MNF1 D 40; MNF2 D 20; MNF3 D 10: 70C2 MNF1 D 40; MNF2 D 15; MNF3 D 15: 70C3 MNF1 D 30; MNF2 D 30; MNF3 D 10: 70C4 MNF1 D 30; MNF2 D 20; MNF3 D 20: 70C5 MNF1 D 40; MNF2 D 10; MNF3 D 10: 60 Optimal Robust Design of ALT 593
Table 2 The optimal First stage Second stage Efficiencystress-changing times for Design 1 2 Design 1 2 effk .R; B/both stages B C1 160 201 R C1 160 201 2:0758 B C2 161 189 R C2 91 119 2:9840 B C3 148 198 R C3 78 128 3:6900 B C4 147 177 R C4 77 107 12:4564 B C5 177 203 R C5 177 203 1:6514
We denote R Ck as the robust design obtained for three-step-stress ALT under thegiven constraint Ck , where k D 1; ; 5. We define the efficiency of R Ck relative toB Ck in terms of AVAR as AVAR B Ck ; B effk .R; B/ D : (18) AVAR .R Ck ; /
Our resulting robust designs, R Ck , for both stages and their relative efficiencies arepresented in Table 2.
5 Discussion and Future Research
The example above indicates that our two-stage design procedure has providedsignificant efficiency gains. The average efficiency of the optimal robust designs isas 4.57 times as that of their corresponding benchmarks. The minimum gain is 65 %for Scenario C5 . At a maximum, our resulting two-stage design for Scenario C4 canbe as 12.46 times efficient as its benchmark (Please see the value of effk .R; B/ inthe second last row of Table 2). Compared C1 to C5 , there only 10 required averagefailures less, but in result, the efficiency gain has increased from 65 % to 107 %. Among the five scenarios considered in the example, the optimal robust designsobtained at the second stage for both C1 and C5 remains the same as theircorresponding benchmarks attained at the first stage. We note that when theMNF3 is much lower than MNF1 , the constraint has limited the design space forfurther minimizing the estimation variance at the second stage. For other scenarios,the optimal robust designs obtained after two stages are far different from theirbenchmarks. Both stress-changing times are much shorter for the optimal robustdesigns than its benchmarks. Consequently, the testing stress levels are increasedmore quickly and more failures are observed under higher stress levels, so that thevariability in resulting estimation shall be reduced. In general, we would recommend that for designing a multiple step-stress ALT,the experimenters should consider as less restriction on the total number of failuresas possible, at the same time, keep the number of expected failures at each stresslevel as even as they can so that the estimation quality can be further improved. 594 X. Xu and W.Y. Huang
In this paper, we focus on designing a multiple step-stress ALT experiment forhazard rate estimation; however, the proposed design procedure can be appliedfor estimation or prediction of any other quantity of interest, such as reliabilityof a product at a given time under the normal design conditions. Furthermore,the proposed method can also be extended with possible modification to an ALTexperiment involved multiple stress factors. For this situation, a more generalversion of (1) can be considered as .tI s/ D 0 .t/ exp T s ; (19)
where s is a column vector of multiple (applied or transformed) stresses used inALT, and is a column vector of unknown parameters. Optimal and robust designsfor step-stress ALT, when (19) is adopted, will be investigated in our future research.
References
1. Bai, D.S., Kim, M.S.: Optimum simple step-stress accelerated life tests for Weibull distribution and type-I censoring. Nav. Res. Logist. 40, 193–210 (1993) 2. Bai, D.S., Kim, M.S., Lee, S.H.: Optimum simple step-stress accelerated life tests with censoring. IEEE Trans. Reliab. 38, 528–532 (1989) 3. Elsayed, E.A., Zhang, H.: Design of optimum simple step-stress accelerated life testing plans. In: Proceedings of the 2nd International IE Conference, Riyadh, Kingdom of Soudi Arabia (2005) 4. Fang, Z., Wiens, D.P.: Robust extrapolation designs and weights for biased regression models with heteroscedastic errors. Can. J. Stat. 27, 751–770 (1999) 5. Fard, N., Li, C.: Optimal simple step stress accelerated life test design for reliability prediction. J. Stat. Plan. Inference 139, 1799–1808 (2009) 6. Hunt, S., Xu, X.: Optimal design for accelerated life testing with simple step-stress plans. Int. J. Performability Eng. 8, 575–579 (2012) 7. Jiao, L.: Optimal Allocations of Stress Levels and Test Units in Accelerated Life Tests. Rutgers University, New Brunswick (2001) 8. Ma, H., Meeker, W.Q.: Optimum step-stress accelerated life test plans for log-location-scale distributions. Nav. Res. Logist. 55, 551–562 (2008) 9. Miller, R., Nelson, W.: Optimum simple step-stress plans for accelerated life testing. IEEE Trans. Reliab. 32, 59–65 (1983)10. Pascual, F.G.: Accelerated life test plans robust to misspecification of the stress-life relation- ship. Technometrics 48, 11–25 (2006)11. White, H.: Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25 (1982)12. Wiens, D.P., Xu, X.: Robust designs for one-point extrapolation. J. Stat. Plan. Inference 138, 1339–1357 (2008) Detecting Coalition Fraudsin Online-Advertising
Qinglei Zhang and Wenying Feng
Abstract Online advertising becomes to play a major role in the global advertisingindustry. Meanwhile, since publishers have strong incentives to maximize thenumber of views, clicks, and conversions on advertisem*nts, the publisher fraudis a severer problem for advertisers and worth the endeavor to detect and preventthem. By reviewing the literature of the frauds in online advertising, the fraudscan be categorized as non-coalition attacks and coalition attacks in general. In thispaper, we attempt to mitigate the problem of coalition frauds by proposing a newhybrid detecting approach that identifies the coalition frauds from both economicand traffic perspectives. Moreover, we propose an algorithm to detect the coalitionfrauds efficiently with an inductive style and greedy strategy.
1 Introduction
With accordance to a recent report [21], online advertisem*nts in 2014 account fornearly one-quarter of the global advertising spend. Figure 1 depicts the generalprocess of the online advertising. The Internet publisher, the advertiser, and thead intermediaries (i.e. ad exchangers) are the three key participants in the onlineadvertising setting. The publishers make money through hosting websites withadvertisem*nts, and advertisers pay for having their ads displayed on publishers’websites. Ad-exchangers such as Google’s DoubleClick [4] and Yahoo!’s Right-Media [18] are involved as brokers who pair the publishers’ ad requests with themost profitable advertiser bid for the request. Since publishers earn revenue basedon the number of views, clicks, and actions that are generated by the advertisem*nt,publishers have strong incentives to maximize those numbers. While publisherscan apply legitimate methods to attract more traffic to their websites, dishonestpublishers attempt to generate invalid traffic to make more money. In particular,invalid (fraudulent) traffic is identified as impressions, clicks, or even actions that arenot the result of genuine user interest [1]. The publisher fraud is a severer problemfor advertiser and worth the endeavor to detect and prevent them.
Q. Zhang • W. Feng ()Trent University, Peterborough, ON, K9J 7B8, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 595J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_54 596 Q. Zhang and W. Feng
Fig. 1 The process of online Users Brandsadvertising
Advertisers Publishers
Intermediaries
One of the challenges for detecting the online advertising frauds is to determinecharacteristics that signal the potential of fraudulent traffics. Currently, neitherpractical systems nor academic research offer a standard mechanism for the frauddetection in online advertising. Moreover, although using data analysis techniques isan evident option, existing solutions are far from mature and there are many researchgaps need to be tackled. Among various attacks in online advertising, coalitionfrauds are a type of attacks that become more and more prevalent nowadays. Sincesuch attacks can be launched with lower cost while still difficult to be identified. Inthis paper, we attempt to mitigate the problem of coalition frauds by proposing anew detecting algorithm [7]. The paper is structured as follows. Section 2 reviews the related works offrauds in online advertising. In Sect. 3, we propose a hybrid detection approach forcoalition frauds. We also discuss the effectiveness and efficiency of the proposedapproach. We conclude and discuss the future work in Sect. 4.
2 Literature Review: Frauds in Online Advertising
With an extensive investigation in both academic and industrial context, we identifya collection of known attacks in online advertising. In particular, fraudulent publish-ers use two types of resources, machines, and web pages, to deploy an attack. Theattacks can be performed either by the machines, or by the web pages, or by both.From the machine side, the attacks can be performed either by real human or bysoftware. The fraudsters hire malicious users to constantly reload a page and click onads. In order to make such attacks harder to trace and detect, some fraudsters abuselegitimate services to hide identifying information like cookies, IP address, etc.,from HTTP requests. Fraudsters also resort to bot to simulate impressions, clicks, orconversions. Initially, those fraudulent bots sit on static machines and issue HTTPrequests to certain URLs, which makes the identification of them relatively easy.Later on, more sophisticated techniques use anonymous proxies and malware-typebots to achieve the IP diversity of the traffic. On the side of web pages, the fraudulent Detecting Coalition Frauds in Online-Advertising 597
publishers can use various of different mechanisms to commit frauds. For example,impression stuffing and keywords stuffing attacks inflate the number of impressions.Fraudulent publishers put excessive numbers of banners on their pages to get a largenumber of impressions for each page view. In particular, half a dozen or even moreof different ad banners could present on a single page [5]. Sometimes, those bannersare even stacked in front of each other, which make those back banners invisible.Moreover, to drive higher traffic to the page than it deserves, fraudsters invisiblyuse some high-value advertising keywords (e.g. in hidden HTML tags, or by textwith the same color as the backgrounds). Furthermore, the publisher can also inflatethe number of clicks with coercion attacks and force browser attacks. With coercionattacks, the content of ads are made invisible or modified to something more relevantto the user, and the users are instruct to perform clicks unknowingly on ads. Thefraudulent publisher inserts additional HTML code that force the browser to clickon the ads. With such client-side script, the users generate request implicitly andclick on the ads stealthily when they loading the webpages.
2.1 The Classification of the Online Advertising Frauds
The classical way to detect publisher frauds is achieved by monitoring the perfor-mance of advertisem*nts on sites. In other words, fraudulent traffic are identifiedas low quality traffic. However, such approach can be too aggressive that evenlegitimate traffic can be discarded. In addition, the malicious intents of fraudstersare not identified directly. Since the fraudsters can control the performance ofadvertisem*nts loaded on their sites, they can easily fool the detection tools bymimicking validity metrics to avoid being detected. To complement the classical traffic monitoring approach, more advanced tech-niques are proposed and applied to identify the pattern of fraudulent behaviors.More specifically, a classification of the fraudulent behaviours help to recognizea more generic pattern of frauds. A related work with a detailed discussion on theclassification of online advertising frauds can be found in [14]. As aforementioned,fraudulent publishers use machines and sites to deploy an attack. Regardlessof different attacking mechanisms (e.g. coercion, robots, etc.), the premise fordata analysis approaches to detect publisher frauds is to identify the correlationsbetween the attacking machines and the corresponding sites. There are two types ofrelationships between fraudulent machines and sites. In an one-to-one relationship,fraudsters only control their own machines and sites. On the other hand, in a many-to-many relationship, a group of fraudsters share their resources to perform theattacks. Consequently, the online advertising frauds can be categorized as two types:the former type of attacks is called the non-coalition attack, while the latter type iscalled the coalition attack. With respect to the above classification, the key to identify fraudulent trafficis to identify the association between fraudulent websites and machines. Moresophisticated attacks blur the strong correlation between the fraudulent sites and 598 Q. Zhang and W. Feng
machines, which makes them difficult to be detected. In the case of non-coalitionattacks, fraudsters only control their own machines and sites. An elementary attackwith repeated views, clicks, or actions from one machine on one site can be easilydetected and blocked. Various data analysis techniques have been proposed todetect such attacks, such as finding duplicates in data streams [12, 22], identifyingabnormal source IPs that are shared by large number of users [19], and etc. Tocompromise those data analysis detection techniques, fraudsters need to developsophisticated techniques (e.g., simulate fake visitors, or frequently change theidentification of users) to avoid being detected. In other terms, to increase thedifficulty of being detected, the cost of launching non-coalition attacks will alsoincrease significantly. On the other hand, since the resources are shared by severalfraudsters, coalition frauds are difficult to be identified while the cost of launchingthem is still relatively low.
2.2 Countermeasures for Frauds in Online Advertising
In the literature, various countermeasures have been explored to mitigate the fraudsproblem in online advertising. In [3], an alternative pricing model is proposed toremove the incentive of fraud. However, the approach modifies current pricingmodel, which requires a global effort and is not likely to happen in the near future. Preventing and detection are two general measurements to mitigate fraudulentproblems. In the context of online advertising, it is important to terminate theidentified publishers from re-join the ad-networks. A typical preventing approachis to create a block list which specifies the signals that indicate fraudulent activities.By looking at different characteristics of HTTP requests (e.g. IP address, publisherID, Customers ID), a potential fraudulent traffic might be filtered based on themaintained block list. One key challenge of the preventing approach is to determinecharacteristics that signal the potential of fraudulent traffics. The signal is consid-ered as confidential information for the companies that run the ad network. Detection techniques found in both practice and academic research can begenerally categorized as positive and active detection. The active detection meansthat additional interactions are conducted with the web and/or users to verify thevalidity of the traffic. Y. Peng et al. [16] proposed an active detection technique,which combats malicious scripts clickbots by creating and validating an impression-click identifier. Another example of active detection is using bluff ads [6]. Thestrategy of bluff ads is to serve some ads that are purposely uninviting, and test thelegitimacy of the individual click. However, the active detection is usually limited toaddress a specific type of fraud, which leads to an arms race between the fraudulentpublishers and the detectors. On the other hand, passive detection mainly dependson applying data analysis techniques to the aggregate number of traffic logs. Data mining techniques can be used for both signature-based detection (e.g. [9]),and anomaly-base detection (e.g. [19]). While signature-based detection works wellfor known malicious patterns, anomaly-based detection is more effective to identify Detecting Coalition Frauds in Online-Advertising 599
Table 1 Summary of the fraud detection techniques in the literature Signature-based Anomaly-basedActive [6, 16] Need interactions with the web and/or usersPositive [9] [8, 10, 12, 13, 15, 17, 19] Use data analysis and our new techniques to the traffic hybrid-approach logs Work well for More effective to identify known malicious new fraudulent pattern patterns
fraudsters with updated types of attacks. Related research about passive frauddetection mainly focuses on developing different algorithms to recognize a specificbehaviour pattern of attacks. For example, the pattern of duplicate clicks within ashort period of time from the same visitor is recognized as one elementary attack.Algorithms about finding duplicates in data streams such as [12, 19] are applied todetect such an attack and its variation. Moreover, different features are proposed todistinguish the fraudulent behaviour patterns from normal behaviours. In [19], thenumber of users sharing the source IP is used as a metric to train the normal patternof behaviors. Attacks are detected as an anomalous deviation from the expectedpublishers’ IP size distribution. Recent efforts [9, 10, 17] have also been done byusing classification algorithms based on multiple features to classify the valid andinvalid traffic. For the coalition attacks, few published work [8, 13, 15, 20] haveproposed algorithms to combat them. The techniques in [13, 15, 20] mainly basedon the similarity among sites to identify the coalitions,while [8] aims to identify thecoalitions from the economic incentive of attackers. We articulate the similarity-based and the gain-based metrics in detail later and adopt both of them in ourdetection technique. A summary of existing detection techniques in the literatureis given in Table 1.
3 Detecting Coalition Frauds in Online Advertising
Unlike non-coalition attacks, the main challenge of detecting coalition attacks is toidentify the correlations of multiple publishers instead of merely focusing on thepattern of a single publisher.
3.1 Identifying the Coalition Frauds
In order to identify the coalition groups, we can consider the problem from theeconomic perspective. In particular, there are two observations for the coalition 600 Q. Zhang and W. Feng
groups: (1) the coalition groups inherently have higher ROI (return of investment)than normal, (2) the expect gain to each member increases while more fraudstersare involved. The ratio of gain and cost can be used as an estimator of ROI. Therefore, themetric relates to the first observation can be expressed by
W.g/ GPR.g/ D GPR ; (1) R.g/
where W.g/ is the gain derived by the members of the coalition g, R.g/ denotes theamount of the resources of the coalition, and GPR is a predefined threshold of theminimum GPR (Gain per Resource) for coalition groups. A coalition has a high value of GPR, but not vice versa. In other words, a coalitiongroup with some normal publishers or visitors may still have a high value of GPR.To capture the property of the second observation, we use the definition of GPRgroup. Formally, we define g is a GPR-GPOUP iff 8.g j g0 g W GPR.g0 / <GPR.g/ /, where g0 denotes any subgroup of g. Therefore, a coalition group can bedefined in terms of GPR as follows:Definition 1 (gain-based) A group of publishers and visitors, denoted as g, iscalled a coalition iff GPR.g/ GPR and g is GPR group. Moreover, a coalitionis a maximal coalition iff : 9.g00 j g g00 W g00 is a coalition /. As another key character of coalition attacks, the traffic to an attacker’s sites arefrom a relatively larger set of attacking machines that shared by many attackers. Tocapture the correlations between publishers, we can measure the traffic similarityamong them. Several general similarity measurements can be applied in our contextto capture the similarity metric. Since the number of repeat visitors is an important indicator for detecting fraudsin online advertising, we use a bag model to denote the visitors to a publisher. Unlikesets, elements can be duplicated in bags. Let Bp1 and Bp2 denote the bags of visitorsto the publishers p1 and p2 , respectively. The pairs of sites p1 and p2 have similartraffic iff
jBp1 u Bp2 j Similarity. p1 ; p1 / D SIM ; (2) jBp1 t Bp2 j
where u and t denote the intersection and union of bags. We use a pair .n; e/ torepresent an element in a bag, where n indicates how many times the e repeats inthe bag. The definitions of u and t are given as follows:
8..n; e/ j .n; e/ 2 A u B W 9.m1 ; m2 j .m1 ; e/ 2 A ^ .m2 ; e/ 2 B W n D min.m1 ; m2 / / / 8..n; e/ j .n; e/ 2 A t B W ..n; e/ 2 A ^ : 9.k j k 2 N W .k; e/ 2 B // _..n; e/ 2 B ^ : 9.k j k 2 N W .k; e/ 2 A // _. 9.m1; m2 j .m1 ; e/ 2 A ^ .m2 ; e/ 2 B W n D max.m1; m2 / / / Detecting Coalition Frauds in Online-Advertising 601
The lhs of the Equation (2) is a variant of Jaccard coefficient [2] capturingthe traffic similarity of Bp1 and Bp2 , and SIM on the rhs denotes the minimumthreshold of similarity for legitimate pairs of sites. Since two legitimate sites onlyhave negligible similarity, it is unlikely that every pair of sites in a random group issimilar. In other terms, the coalition attackers form a group that is pair-wise similar.Formally, the definition of coalition groups with regard to the traffic similarity isgiven as follows:Definition 2 (similarity-based) A group of publishers g is a coalition group iff8.p1 ; p2 j .p1 2 g/ ^ .p2 2 g/ ^ .p1 ¤ p2 / W Similarity.p1 ; p2 / SIM /The aforementioned two approaches capture the key characters of coalition attacksfrom different perspectives. The gain-based approach calculates the economic gainof each group (see Definition 1), while the similarity based approach considers thetraffic similarity within each group (see Definition 2). The reader can refer to [8]and [15] respectively for more details of these two approaches. However, a mainchallenge for both approaches is to define the proper threshold (see Equations 1and 2). For example, some legitimate popular sites may have equal or even greaterGPR than the coalition groups, and legitimate sites with similar traffic may be falselyidentified as fraudsters. Therefore, to increase the confidence for detecting coalitionattacks, it would be beneficial to consider the characters from both perspectivesand combine both measurements. While the GPR is a measurement for a group ofentities, the similarity is a measurement only for a pair of entities. In order to getmuch more confidence on the signal of coalitions, we further give the definition ofcoalitions as follows:Definition 3 (hybrid) Let g denote a subset of all publishers, we say g is a coalitiongroup iff
8.g0 j g0 < g W GPR.g0 / < GPR.g/ / ^ GPR.g/ GPR ^ 8.p1 ; p2 j .p1 2 g/ ^ .p2 2 g/ ^ .p1 ¤ p2 / W similarity.p1 ; p2 / SIM /
3.2 Detecting the Coalition Frauds
With accordance to the above definition of coalition frauds, we have three strategiesto detect such coalitions. The first strategy is to detect the GPR groups that satisfythe GPR properties and their GPR value are greater than GPR . Then we verify thepair-wise similarity property for each identified group. The second strategy is tocalculate the similarity for all pairs and identify all pair-wised similar groups. Foreach identified group, we further verify the GPR properties of the group and itsGPR value. The third strategy is to dynamically check the two properties whileidentifying the coalition groups. Such strategy will be the most efficient way, whichis critical when dealing with big data. Common advanced analytics disciplinessuch as statistical analysis, data mining, and predictive analytics all can be used 602 Q. Zhang and W. Feng
to deal with big data. Machine learning algorithms (e.g. neural network, decisiontree) is especially useful for the analysis of big data, since it can explore thehidden characters with less reliance on human direction. On the other hand, the keyneck-bottle of using machine learning algorithms in our context is the demand offirst extracting related features for all potential groups, which is a computationallyexpensive process. However, the adaption of the third strategy is not straightforward.We should validate that the derived detection algorithm can correctly identify thefrauds corresponding to the definition given in the previous section. Given a group of publishers and visitors g, let Gclick .g/ and Gsimilarity .g/ denotethe click graph and the similarity graph associated with g, respectively. Beforegiving the detection algorithm, we first define two key properties used in thealgorithm as follows:• GPR-Core property: g is a GPR group H) 8.g0 j Gclick .g0 / is a connected subgraph of Gclick .g/ W g’ is a GPR group /• similar-clique property: g is a similar clique H) 8.g0 j Gsimilarity .g0 / is a subgraph of Gsimilarity .g/ W g’ is a similar clique / With accordance to the definition of coalition frauds given in Definition 3, wehave the following proposition:Proposition 1 A group g is a coalition group iff GPR.g/ GPR and g satisfiesboth the CPR-Core property and the similar-clique property. Due to the limit of space, we omit the proof of the proposition. Algorithm 1depicts the pseudocode to detect coalition frauds with the hybrid approach. Tocombine the aforementioned two criteria dynamically, we apply an inductive styleand greedy strategy in the algorithm, which can dramatically prune the search spaceand improve the efficiency of the detection technique. It can be proved that theGPR-Core and the similar-clique properties are anti-monotone. In other words, theGPR-Core and the similar-clique properties (used in Line 5 and 6, respectively) holdwith the induction. Consequently, Proposition 1 indicates that the given algorithmcan correctly detect the coalition frauds in our context.
Algorithm 1 Detecting coalition frauds 1: G calculate the base case value for gpr measurement 2: E calculate the pair similarity for the similarity measurement 3: for size in range(2, len(V)) do 4: potential INDUCTIVE_ KEY _ GENERATOR (previous,V) 5: current1 check the potential group with the GPR-Core property 6: current2 check the potential group with the similarity-clique property 7: current current1 \ current2 8: coalition[size] append if it satisfies the GPR group threshold 9: previous coalitions[size]10: end for11: max_coalitions MAXIUM.coaltions/ Detecting Coalition Frauds in Online-Advertising 603
Another key challenge for both gain-based and similarity-based approaches is thecomplexity of the algorithm. Given the set of all publishers as P and visitors as V, theGPR approach require calculating the GPR for all potential subgroups of G, whichcomplexity is exponential to jPj C jVj. On the other hand, the similarity approachrequires to identify the pairwise similar group which is equivalent to discovering themaximal cliques in the sites’ similarity graph. In general, finding maximal cliquesin a graph also has exponential complexity. The problem of detecting coalitions isNP-hard in general. In practice, the problem of detection frauds in online advertisingusually involves processing large sets of data. To further improve the efficiency ofthe detection algorithm, we adopt a MapReduce [11] computing model which canfacilitate the parallel, distributed computing on multiple clusters. The lines of 1, 2, 4,5, 6 and 11 in the pseudocode of Algorithm 1 are implemented with the MapReduceparadigm. However, the implementation of the algorithm is out the scope of thispaper.
3.3 Preliminary Result
To evaluate the effectiveness of the proposed detecting techniques, we apply ourdetection technique to various traffic patterns in this section. As we mentionedearlier, the value of the threshold is critical when deciding whether or not a groupis a coalition. To avoid the negative false alarms, both the parameters SIM and GPRshould set to low values. However, the positive false alarms could increase when thevalues are too low. Existing techniques for detecting coalition approach (e.g. [8, 15])usually determine the value of the threshold by making a trade-off between thepositive and negative false alarms. Unlike those techniques with a singe metric, thehybrid approach adapted in this paper could minimize the rate of negative falsealarms by setting the threshold to relatively low values. On the other hand, the rateof positive false alarms is reduced by combing the decisions from two differentperspectives, Fig. 2 illustrates click graphs of three simplified traffic patterns. Thenodes of p1 , p2 , and p3 denote the publishers, while the nodes of v1 , v2 , v3 , and
a b c
V1 P1 2 P1 P1 V1 V1
V2 P2 V2 P2 2 P2 V4 V2
P3 V3 2 P3 V3 P3 V3
Fig. 2 Different traffic patterns with/without coalition Attacks 604 Q. Zhang and W. Feng
Table 2 The results of the three different detection techniques Similarity-based [15] GPR-based [8] Hybrid (our approach)Case (a) Fraud (✔) Fraud (✔) Fraud (✔)Case (b) Frauds (✖) Normal (✔) Normal (✔)Case (c) Normal (✔) Frauds (✖) Normal (✔)
v4 denote the visitors. The case (a) is an example of coalition attacks that the threepublishers share the resources. The case (b) and case (c) are two examples that theresources are not really shared by those publishers. However, the two latter casesare easy to be mis-identified since they show similar traffic metrics as coalitionattacks. Table 2 shows the results of checking the three cases with different detectiontechniques. To identify the coalition attack of case (a), the parameter GPR shouldbe at least 1, and the parameter SIM should be equal to or less than 13 . However,the case (b) is falsely identified as coalition frauds with SIM D 13 , while the case(c) is falsely identified with GPR D 1. But with the proposed hybrid approach,we can avoid such positive false alarms while remaining the precision for detectingcoalition frauds.
4 Conclusions and Future Work
In this paper, we adapt a hybrid approach to identify the coalition frauds from twodistinguish perspectives. To detect the coalition groups, we proposed an algorithmwith inductive styles and greedy strategy, and discuss the correctness and efficiencyof the algorithm. Finally, we evaluate the effectiveness of the proposed detectiontechnique by applying it to several trial cases. As further work, we can apply the proposed technique to different data sets tofurther evaluate the algorithm. Since there are no benchmark data sets for coalitionfrauds in online advertising, we can generate synthetic data sets by simulating thetraffic of both normal and fraudulent traffics. We also can apply the algorithm toindustrial data sets and validate the performance of the algorithm by using thirdparty services that can label the traffic logs. Moreover, we can improve the precisionof our hybrid detection technique by exploring the coalition frauds problems frommore other perspectives and integrating them to our detection algorithm.
Acknowledgements The authors thank the referee for some helpful comments. The project wassupported by the Mathematics of Information Technology and Complex Systems (MITACS) ofCanada and the EQ Advertising Group Ltd. Detecting Coalition Frauds in Online-Advertising 605
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B. Boreland and H. Kunze
Abstract In 2000, Frame and Cogevina (Comput Graph 24(5):797–804, 2000)introduced a method for constructing fractals using circle inversion maps. Theliterature (Clancy and Frame, Fractals 3:689–699, 1995; Frame et al., A fractalrepresentation of grammatical complexity in time series. Manuscript in Preparation,1996; Frame and Cogevina, Comput Graph 24(5):797–804, 2000; Leys, ComputGraph 29(3):463–466, 2005) focuses on the graphical aspect of such fractals, with-out presenting a careful development of the underlying mathematical framework.In this paper, we present such a framework, making a strong connection to iteratedfunction systems (IFS) theory. Our final result establishes that the set valued systemof circle inversion maps induced by a collection of non-touching circles in the planehas a unique set attractor. We present a graphical example using the chaos game.
1 Introduction
Circle inversion was introduced in Apollonius of Pregas’ book, Plane Loci, and hasdrawn interest in geometry. Mandelbrot introduced the early concepts of fractals inthe 1970s and, in The Fractal Geometry of Nature, Mandelbrot discusses successiveinversion with respect to a family of M circles. The literature applies the chaos gameto circle inversion maps [2, 4] to explore the graphical aspect, while casually statingthat there are contraction maps involved. As we will see in Sect. 2, a circle inversionmap is only contractive on part of its domain, making clear the need for care. In thispaper, we build the needed rigorous mathematical framework. In Sect. 2, we presentsome background concepts on circle inversion maps. In Sect. 3, we establish somecontractivity results for a system of two circle inversion maps, which we extend inSect. 4 to a general system of non-touching circles. In Sect. 4, we also establish theexistence of a unique set attractor to the system of set-valued circle inversion maps.Approximations of this fractal set can be drawn using the chaos game; we providean example in Sect. 5. We can use the already established content to now build a
B. Boreland () • H. KunzeDepartment of Mathematics and Statistics, University of Guelph, Guelph, ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 609J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_55 610 B. Boreland and H. Kunze
framework to show why this random inversion algorithm arrives at a fractal, or inother words, converges to a set attractor.
2 Inversion in a Circle
Let C 2 R2 be a solid circle with centre o and boundary @C; then the followingproperty is true. e
Property 1 @C divides R2 into three pieces• the interior of the circle, Cint• the boundary of the circle, @C• the exterior of the circle, Cext where C D Cint [ @CDefinition 1 Given C in R2 with fixed centre o and radius r we define the inversionmap T on R2 n fog by e e jjT.x/ ojj jjx ojj D jj˘.x/ ojj2 D r2 ; 8x 2 R2 n fog (1) e e e e e e e ewhere jj jj is the Euclidean norm and ˘.x/ is the projection of x onto @C along the e write the inversion eradial ray from o to x. Alternatively, we can map as e e d.T.x/; o/d.x; o/ D d 2 .˘.x/; o/ D r2 ; 8x 2 R2 n fog (2) e e ee e e e ewhere d.x; y/ D jjx yjj. ee e eLemma 1 Given a circle C with inversion map T we have the following: i. T W Cint n fog ! Cext ii. T W Cext ! eC intiii. T W @C ! @C
Proof Follows from the definition of T and Property 1. When a point y D T.x/ satisfies (2), y is called the inverse of x where T.y/ D x e e eand T.b/ D b, foreb 2 @C. e e e e eDefinition 2 Given a collection of non touching circles Ci , i D 1; : : : ; N, withcentres oi and radii ri we define the inversion map Ti on R2 n foi g by e e jjTi .x/ oi jj jjx oi jj D jj˘i .x/ oi jj2 D ri2 ; 8x 2 R2 ; x ¤ oi (3) e e e e e e e ewhere ˘i .x/ is the projection of x onto @Ci along the radial ray from oi to x. e e e e Circle Inversion Fractals 611
We can build (3) when x and Ti .x/ satisfy the expression e e jj˘i .x/ oi jj2 Ti .x/ D oi C e e .x oi / e e jjx oi jj2 e e e e Any point x 2 R2 can be represented in terms of the parametrization for the circleCi : e
x D ari w.t/ C oi for some a 0; t 2 Œ0; 2 ; (4) e e ewhere w.t/ D .cos.t/; sin.t//. If a < 1 then x 2 Ci;int , if a D 1 then x 2 @Ci and if e x2C .a > 1 then e e i;ext e 2 For x 2 R n foi g, we have a > 0, and we find that with some easy simplifications ewe obtain e
jjri w.t/jj2 1 Ti .x/ D Ti .ari w.t/ C oi / D oi C e 2 .ari w.t// D oi C ri w.t/ (5) e e e e jjari w.t/jj e e a e eThus, we see clearly why Ti is called a circle inversion map: under the action of Ti ,the radial scaling factor ai > 0 becomes a radial scaling factor of a1i .
3 Mathematical Framework for Two Non-touching Circles
We now develop a mathematical framework for a system of two non-touchingcircles. We begin by observing properties along the radial rays. We define somenotation for use in upcoming results. Given a circle C1 in R2 with centre o1 , let e Ro1 .o1 ; b1 / D semi infinite ray from o1 through b1 2 @C1 ; with endpoint o1 e e e e e e Ro1 ;1 .o1 ; b1 / D infinite line through o1 and b1 e e e e e Ro1 ;int .o1 ; b1 / D Ro1 .o1 ; b1 / \ C1 e e e e e e Ro1 ;ext .o1 ; b1 / D Ro1 .o1 ; b1 / n fRo1 ;int .o1 ; b1 /; b1 g e e e e e e e e e eLemma 2 For C1 2 R2 with centre o1 , b 2 @C1 , b1 2 @C1 , and T1 the inversionmap for C1 , we have the following: e e e
i. T1 W Ro1 ;int .o1 ; b1 / n fo1 g ! Ro1 ;ext .o1 ; b1 / T1 W Re e e e e e ii. o1 ;ext .o1 ; b1 / ! Ro1 ;int .o1 ; b1 /iii. e T1 .b/ D b e e e e e 2e e iv. T1 W Ro1 ;int .o1 ; b1 / n fo1 g ! Ro1 ;int .o1 ; b1 / e e e e e e v. T12 W Ro1 ;ext .o1 ; b1 / ! Ro1 ;ext .o1 ; b1 / e e e evi. T12 .a/eD a e e e 612 B. Boreland and H. Kunze
Proof (i), (ii) and (iii) follow from Lemma 1. (iv), (v) and (vi) follow by iteratingLemma 1 a second time. Now, given two non-touching circles C1 and C2 in R2 with centres o1 and o2 ,respectively, define .Ci /int , .Ci /ext , @Ci , Ti and projection maps ˘i . Let e e
Ro1 o2 .x; y/ D portion of the open line segment between o1 and o2 from x to y ee e e e e e e Ro1 o2 .oi ; bi / D Ro1 o2 .o1 ; o2 / \ Ci ee e e ee e e Ro1 o2 .b1 ; b2 / D Ro1 o2 .o1 ; o2 /nfRo1 o2 .oi ; bi /; i D 1; 2g ee e e ee e e ee e eLemma 3 For i; j 2 1; 2 with i ¤ j, and the preceding set up, we have i. Ti W Roi oj .oj ; bi / ! Roi oj .oi ; bi /ii. Ti W Reee e ee e e oi oj .oi ; bi / ! Roi oj .oj ; bi / ee e e ee e eProof Both (i) and (ii) are easily proved using Lemmas 2(ii) and 2(i), respectively. It is worth noting that in Lemma 3, the centres of the circles need to specialconsideration.Theorem 1 Let C1 and C2 be two circles in R2 with centres o1 , o2 , respectively.Then there exists a c 2 Œ0; 1/ such that, e e
d.T1 .x1 /; T1 .x2 // c d.x1 ; x2 /; 8x1 ; x2 2 Ro1 o2 .o2 ; b2 / e e e e e e ee e e
Proof Introduce the x-axis so that Ro1 o2 .o2 ; b2 / lies along it. This means that x1 , b1and o1 correspond to numbers alongethe e e we think of T as mapping numbers e x-axis; e e 1 eto numbers on the axis. Since x1 > o1 ” T1 .x1 / > o1 , the circle inversion mapalong our x-axis is
.b1 o1 /2 T1 .x1 / D o1 C .x1 o1 /
Now,
0 .b1 o1 /2 0 .b1 o1 /2 T1 .x1 / D 2 H) T1 .x1 / .x1 o1 / .b2 o1 /2
By the Mean Value Theorem we get 0 jT1 .x1 / T1 .x2 /j D jT1 ./jjx1 x2 j, 2 R .b1 o1 /2 jx1 x2 j .b2 o1 /2 D c jx1 x2 j, c < 1 Circle Inversion Fractals 613
Returning to the vector notation, we have proved there exists a c 2 Œ0; 1/ such that,
d.T1 .x1 /; T1 .x2 // c d.x1 ; x2 /; 8x1 ; x2 2 Ro1 o2 .o2 ; b2 / e e e e e e ee e e Theorem 1 leads to the following significant corollary, which follows from twoapplications of Theorem 1.Corollary 1 There exists a c 2 Œ0; 1/ such that,
d..T1 ı T2 /.x1 /; .T1 ı T2 /.x2 // c d.x1 ; x2 /; 8x1 ; x2 2 Ro1 o2 .o1 ; b1 / e e e e e e ee e e
In Corollary 1, .Ro1 o2 .o1 ; b1 /; j j/ is a complete metric space and T1 ı T2 is acontraction map fromee .Roe e 1 o2 .o1 ; b1 /; j j/ to itself. Banach’s Fixed Point Theoremapplies: there exists a uniqueexN on e e e the radial ray in C such that .T ı T /.Nx/ D xN . 1 1 2 eA similar statement can be made for T2 ı T1 and the radial ray in C2 . Beginning e eat any point on the ray from o1 to o2 , repeated alternating application of T1 and T2 e e of these two fixed points.approaches the two cycle consisting
4 Collections of Non-touching Circles
We present the following result in R2 with the understanding that this may be a twodimensional subspace of a higher dimensional setting. In this result we consider acollection of non-touching circles in the plane.Lemma 4 Given a circle C1 with centre o1 , radii r1 and inversion map T1 , e 1 d.T1 .x1 /; T1 .x2 // D d.x1 ; x2 /; 8x1 ; x2 2 R2 n fo1 g; e e a1 a2 e e e e e
where, for j D 1; 2, xj D aj r1 w1 .tj / C o1 for some tj 2 Œ0; 2 , aj > 0. e e eProof We consider two cases.Case 1: x1 ; x2 and o1 lie on the same radial ray. Without loss of generality, we e caneplace thee three points on the x-axis at locations x D .a r ; 0/, x D 1 1 1 2 .a2 r1 ; 0/ and o1 D .0; 0/. Then, e e e d.T1 .x1 /; T1 .x2 // D jjT1 .x1 / T1 .x2 /jj e e e e jj˘1 .x1 /jj2 jj˘1 .x2 /jj2 D e kx k2 e1 x e x kx2 k2 e 2 1 2e e r1 r 2 D 2 2 1 1.a r ; 0/ 1 .a r ; 0/ r1 a1 2 2 2 1 r1 a2 614 B. Boreland and H. Kunze
ˇ ˇ ˇ 1 1 ˇ 1 ˇ Dˇ r1 ˇˇ D r1 ja2 a1 j a1 a2 a1 a2 1 D d.x1 ; x2 / a1 a2 e e
Case 2: x1 ; x2 and o1 are not on the same radial ray. Once again the boundary @C1 canebe parametrized e e as in (4). There exists t1 ; t2 2 Œ0; 2 , t2 > t1 , without loss of generality, such that
˘1 .xj / D r1 w1 .tj / C o1 and xj D aj r1 .cos.tj /; sin.tj // C o1 ; j D 1; 2: e e e e e Note that this case includes the choice t2 D t1 C , two points on the same diameter but on opposite sides of the centre. After straightforward calculations, we arrive at
d2 .x1 ; x2 / D a21 r12 C a22 r12 2a1 a2 r12 cos.t2 t1 / (6) e e By (5),
1 T1 .xj / D o1 C r1 w.tj / e e aj e
Thus by using (5), (6) and some simplifying we have,
1 1 2 2 d2 .T1 .x1 /; T1 .x2 // D 2 r12 C 2 r12 r cos.t2 t1 / e e a1 a2 a1 a2 1 1 D d2 .x1 ; x2 /; a21 a22 e e
which upon taking the square root gives the desired result. We will use Lemma 4 and the following definition when establishing thesubsequent contractivity result.Definition 3 Given a collection of non-touching circles Ci , i D 1; : : : ; N, withcentres oi and radii ri , for x D ai ri wi .t/ C oi 2 R2 , ai > 0, t 2 Œ0; 2 , we define e e e e ai;min D min fai g > 1 x2Cj ;j¤i ecorresponding to the radial scaling of the closest point to oi in all other circles Cj ,j ¤ i. e Circle Inversion Fractals 615
Theorem 2 Ti : Cj ! Ci , i ¤ j, is contractive. That is, there exists a c 2 Œ0; 1/such that,
d.Ti .x1 /; Ti .x2 // c d.x1 ; x2 /; 8x1 ; x2 2 Cj ; i ¤ j e e e e e e
Proof By Lemma 4 we have that
1 1 d.Ti .x1 /; Ti .x2 // D d.x1 ; x2 / 2 d.x1 ; x2 / D c d.x1 ; x2 / e e a1 a2 e e ai;min e e e e
Since Ti : Ci;int ! Ci;ext is expansive, we redefine the map as follows.Definition 4 Given a collection of non-touching circles Ci , i D 1; : : : ; N, withcentres oi , define e [ N XD Ci iD1
and, relative to each Ci , define
Ti W X ! X
by ( Ti .x/ if x 2 Cj ; j ¤ i T i .x/ D e e e x if x 2 Ci e eTheorem 3 T i W X ! X satisfies
.i/ d.T i .x1 /; T i .x2 // c d.x1 ; x2 / for some c 2 Œ0; 1/; 8x1 2 Cj ; x2 2 Ck ; j; k ¤ i e e e e e e .ii/ d.T i .x1 /; T i .x2 // D d.x1 ; x2 / 8x1 ; x2 2 Ci e e e e e e.iii/ d.T i .x1 /; T i .x2 // c d.x1 ; x2 / for some c 2 Œ0; 1/; 8x1 2 Ci ; 8x2 2 Cj ; j ¤ i e e e e e e
Proof (i) follows from Theorem 2, (ii) follows immediately since T i is the identitymap in this case. For (iii), with x1 2 Ci and x2 2 Cj , we have a2 > 1 > a1 0, and,for values t1 ; t2 2 Œ0; 2 , e e
d2 .x1 ; x2 / D ri2 a21 C a22 2a1 a2 cos.t2 t1 / ; (7) e e 616 B. Boreland and H. Kunze
using (6), and with some expanding and factoring we are left with
r2 d2 .T i .x1 /; T i .x2 // D i2 a22 a21 C 1 2a1 a2 cos.t2 t1 / (8) e e a2
The bracketed expression in (8) is less than or equal to the bracketed expressionin (7) provided that
a22 a21 C 1 a21 C a22 ; where a2 > 1 > a1 0: (9)
Hence, letting y D a21 and z D a22 , we consider the function
f .y; z/ D y yz C z:
If we show that f .y; z/ 1 for y 2 Œ0; 1/ and z > 1, then (9) holds. We write
f .y; z/ D y yz C z D y.z 1/ C .z 1/ C 1 D .z 1/.1 y/ C 1
and we see that f .y; z/ > 1 for y < 1 and z > 1, proving the result. This means thatthe statement in case (iii) of our theorem holds with c D a12 . Now, using Theorem 3 we can prove the following contractivity result involvingcompositions of (non-touching) circle inversion maps.Theorem 4 For i ¤ j, T i ı T j : X ! X is contractive:
d..T i ı T j /.x1 /; .T i ı T j /.x2 // c d.x1 ; x2 / for some c 2 Œ0; 1/; 8x1 ; x2 2 X e e e e e e
Proof If x1 ; x2 2 Ck ; k ¤ j, then the result follows by applying Theorem 3(i) twice.If x1 ; x2 e 2 Cej , then the result follows by applying Theorem 3(ii) and 3(i). If x1 2 eCj ; x2 2 Ck ; k ¤ j, then the result follows by applying Theorem 3(iii) and 3(i). e e e By Theorem 4 and Banach’s Fixed Point Theorem, for each i we can say thatthere exists a unique xN j in Ci such that .Ti ı Tj /.Nxj / D xN j for each choice of j. In fact,by Corollary 1 we know e lieseon the appropriate radial ray, e that each such fixed pointRoi oj .oi ; bi /. ee Wee now e formulate this multi-circle framework in the context of sets. For a set 2A R , define
TO i .A/ D fT i .x/; 8x 2 Ag e e Circle Inversion Fractals 617
Define the distance from a set A to a set B by
d.A; B/ D sup d.x; B/ D sup inf d.x; y/ x2A x2A y2B
The Hausdorff distance between the non-empty compact sets A and B is definedby
dH .A; B/ D maxfd.A; B/; d.B; A/g
Let H .X/ denote the set of all non-empty compact subsets of X. It is well knownthat .H .X/; dH / is complete.Theorem 5 For i ¤ j,
1. TO i ı TO j W H .X/ ! H .X/2. TO i ı TO j is contractive on .H .X/; dH /
Proof The first claim follows because T i ı T j W H .X/ ! H .X/. The proof relieson the continuity of the map ([1], Lemma 2, page 80). By Theorem 4, we can denoteby cij the contractivity factor with respect to d of T i ı T j . For A; B 2 H .X/,
dH ..TO i ı TO j /.A/; .TO i ı TO j /.B// 8 9 < = D max sup inf d.x1 ; x2 /; sup inf d.x1 ; x2 / : O O e e x 2.TO ıTO /.B/ x2 2.TOi ıTOj /.A/ e e ; x1 2.TOi ıTOj /.A/ x2 2.T i ıT j /.B/ 1 e e e i j e ( D max sup inf d..T i ı T j /.x1 /; .T i ı T j /.x2 //; x1 2A x2 2B e e e e ) sup inf d..T i ı T j /.x1 /; .T i ı T j /.x2 // x1 2B x2 2A e e e( e ) cij max sup inf d.x1 ; x2 /; sup inf d.x1 ; x2 / x1 2A x2 2B e e x1 2B x2 2A e e e e e e D cij dH .A; B/
We reach our final theorem, which establishes the contractivity of the union ofthe maps in Theorem 5. 618 B. Boreland and H. Kunze
O S NTheorem 6 For A 2 H .X/, define T.A/ D .TOi ı TO j /.A/. Then i;jD1 i¤j
1. TO W H .X/ ! H .X/2. TO is contractive on .H .X/; dH /
Proof The first claim follows from Theorem 5 because the union is finite. For thesecond claim we have 0 1 B[N [ N C O dH .T.A/; O T.B// D dH B @ . O ı TO /.A/; T i j .TO i ı TO j /.B/C A i;jD1 i;jD1 i¤j i¤j
max dH ..TO i ı TOj /.A/; .TO i ı TO j /.B//; property of dH 1i;jN
max cij dH .A; B/; by Theorem 5 1i;jN
D c dH .A; B/
Finally, by Theorem 6 and Banach’s Fixed Point Theorem we can conclude that O satisfyingthere exists a unique fixed point A 2 H .X/, the set attractor of T,
O /DA T.A
5 The Chaos Game
Following Sect. 3, we can use the Chaos Game to plot approximations of the setattractor A of T. O Choose an initial point x 2 X. With equal probability select one 0 eof the composition maps T i ı T j , i ¤ j. Apply the map to x0 to produce x1 2 X (in Ci ,of course). Continue randomly selecting composition maps e to produceethe sequence 1fxn gnD0 . If we start with a point on A , all points are on A . In practice, we stop after e large number of iterations, dependent upon the resolution of the plot we aresomeproducing. An example of a fractal generated by a five-circle system is given in Fig. 1. Circle Inversion Fractals 619
Fig. 1 Five circles used to generate a set attractor with 10,000 iterations of the chaos game
References
1. Barnsley, M.F.: Fractals Everywhere. Academic, London (1988)2. Clancy, C., Frame, M.: Fractal geometry of restricted sets of circle inversion. Fractals 3, 689–699 (1995)3. Frame, M., Clancy, C., Peak, D.: A fractal representation of grammatical complexity in time series. Manuscript in Preparation (1996)4. Frame, M., Cogevina, T.: An infinite circle inversion limit set fractal. Comput. Graph. 24(5), 797–804 (2000)5. Leys, J.: Sphere inversion fractals. Comput. Graph. 29(3), 463–466 (2005) Computation of Galois Groups in magma
Andreas-Stephan Elsenhans
Abstract We give an outline of a degree independent algorithm for the computationof Galois groups, as it is implemented in magma (van der Waerden BM, Algebra 1,Springer, Berlin 1960) (version 2.21). Further, we summarize several performancetests and list hard examples.
1 Introduction
Given a polynomial f with rational coefficients, one can ask for the Galois groupof the splitting field (viewed as an extension of the rationals). This is a classicalproblem in algorithmic algebra. The first algorithm to solve it was described by vander Waerden in his famous book on algebra [12]. Later, more practical algorithmswere given [2, Sec. 6.3]. A crucial point on the above question is the way one wants to present theresult. One could give the Galois group as an abstract group. But this would bean incomplete answer, as this does not give the action on the splitting field. On theother hand, describing the action on the splitting field would require to construct it.And this would be impractically large in many cases. A good compromise is to give complex or p-adic root approximations and theaction of the Galois group on them. This is the starting point of the Stauduharmethod [11]. The approach generalizes to any global field.
1.1 Previous Implementations
Several people implemented Galois group algorithms that used precomputed tableswith all the combinatorial data necessary for the computation. Most notable is theimplementation of K. Geißler that covered irreducible polynomials up to degree23 [7].
A.-S. Elsenhans ()Mathematisches Institut, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germanye-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 621J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_56 622 A.-S. Elsenhans
The first degree independent implementation that could also handle reduciblepolynomials was given by C. Fieker and J. Klüners [6]. However, this implementa-tion ran out of memory or needed hours of CPU time with several polynomials ofdegree 30. Recently, the author worked on bottlenecks of this package. The aim of thisarticle is to give a summary of what is now possible.
2 Stauduhar’s Step
Definition 1 Let U G Sn be permutation groups. They act on ŒX1 ; : : : ; Xn by permutation of the variables. A polynomial I 2 ŒX1 ; : : : ; Xn is called a U-invariant if I D I for all 2 U.It is called a relative invariant for U G if its stabilizer in G is equal to U.Lemma 1 Relative invariants exist of all pairs of subgroups U G Sn .Proof Choose a monomial P with trivial stabilizer in Sn . E.g., m WD X1 X22 Xn1 n1 .Then the orbit sum 2U m is a relative invariant. t uDefinition 2 Let U G be two groups. We denote by G==U a set of cosetrepresentative of G=U. With this preparation, we can describe the Stauduhar step [11].Theorem 1 Let f 2 ŒT be a degree n polynomial with distinct roots r1 ; : : : ; rnin an arbitrary splitting field. Further, let G Sn be a permutation group thatcontains the Galois group of f , U be any subgroup of G and I 2 ŒX1 ; : : : ; Xn arelative invariant for U G. Finally, we assume that the values .I/.r1 ; : : : ; rn / for 2 G==U are pairwise distinct. Then the Galois group of f is contained in U if and only if .I/.r1 ; : : : ; rn / isrational. The above theorem leads to the following algorithm for the computation ofGalois groups.1. Choose a starting group (e.g., G D Sn ).2. Compute the conjugacy classes of the maximal subgroups of G.3. For each subgroup class representative U G do the following: a. Compute a relative invariant I. b. Determine .I/.r1 ; : : : ; rn / for 2 G==U. c. Use the theorem above to find all the conjugates of U that contain the Galois group.4. If no subgroup of G is found that contains the Galois group, return G as the Galois group. Galois Groups 623
5. Intersect all the subgroups found that contain the Galois group.6. Redo all the steps above with the intersection as starting group.Remark 1 There are various problems that have to be solved to make the aboveapproach practical.1. First, we need a good strategy to pick an initial group that is as small as possible. One approach for this is the use of subfields [7, Chap. 5.1]. In case of a reducible polynomial, one can compute the Galois group of each factor and take the direct product as a starting group [6].2. In magma, the computation of maximal subgroups is done using the Cannon- Holt-algorithm [1].3. How can we prove the rationality of .I/.r1 ; : : : ; rn / when we only work with approximations of the roots? One way to do this is described in [7, Chap. 3.3]. This requires to work with p-adic precisions that are proportional to the index of the subgroup. This is practical only when the index is small. However, it is possible to work with a moderate p-adic precision and derive a heuristic result that has to be proven later.
3 The Use of the Frobenius
3.1 Using Cycle Types
Let a monic polynomial f 2 ŒT be given and a prime p that does not divide itsdiscriminant. It is well known that the local Galois group of the p-adic splitting fieldof f is a subgroup of the Galois group of the global splitting field. The local Galois group is generated by the Frobenius element. When we take thedegrees of the irreducible factors of the reduction of f modulo p, we get the cycletype of the Frobenius element. Doing this for several primes, we derive some information about the Galois groupthat can be used as follows:1. If one gets sufficiently many different cycle types then one can prove that the Galois group is the full symmetric group Sn . More precisely, in the case that f 2 ŒT is irreducible, n is at least 8 and one has a cycle of prime length l with n 2 < l < n 2, the alternating group is contained in the Galois group. If, in addition, an odd permutation is detected then the Galois group is Sn .2. Before we apply Stauduhar’s step to a subgroup, we can check that it contains all the cycle types found. 624 A.-S. Elsenhans
3.2 Short Cosets
A more sophisticated way to use the local Galois group is provided by the so calledshort cosets [7, Chap. 5.2]. The idea behind them is that, when working with p-adicroot approximations, the Frobenius Frobp is known as an explicit permutation of theroots. Thus, a necessary condition for U to contain the Galois group is that it containsthe Frobenius permutation. We call the remaining coset representatives
G==Frobp U WD f 2 G==U j Frobp 2 U g
the short cosets. They can be computed without listing all the representatives G==U([7, Algorithmus 5.12], [4]). In many cases, the number of short cosets is very small. A naive explanation forthis is the following: 1 / Frobp 2 U ” Frob. p 2U
1Assuming the conjugates Frob. p / to be equidistributed in G, the probability to hit 1U is ŒGWU . Thus, for large index subgroups this number is very small. Of course, thisis just a very coarse heuristic.
4 The Invariants
To make the Stauduhar method run, we need a relative invariant for each pair ofgroups, the algorithm may run into. As there are 25,000 transitive permutationgroups in degree 24, it is not practical to store them in a database. We have tocompute them at run time. A more detailed description of the construction of theinvariants if given in [3, 5–7]. The main tool for this is the usage of block systems.Definition 3 A partition B1 ; : : : ; Bk of f1; : : : ; ng is called a system of blocks for atransitive subgroup G Sn if
Bi 2 fB1 ; : : : ; Bk g
for all 2 G.Theorem 2 Let U be maximal in G Sn and B D fB1 ; : : : ; Bk g be a block systemfor U with B1 D f1; : : : ; jg. P P1. If B if not a block system for G then B2B . i2B Xi /2 is a relative invariant.2. If B is a block system of G, denote by 'B the action of G on the blocks. Galois Groups 625
P P • If 'B .U/ 6D 'B .G/ then I0 . i2B1 Xi ; : : : ; i2Bk Xi / is a relative invariant. Here, I0 is a relative invariant for 'B .U/ P'B .G/. • If StabU .B1 /jB1 ¤ StabG .B1 /jB1 then 2U== StabU .B1 / I0 .X1 ; ::; Xj / is a relative invariant. Here, I0 is a relative invariant for StabU .B1 /jB1 StabG .B1 /jB1 .A proof of this is given in [7, Satz 6.14, 6.16]. If these constructions do not apply then we have to do a deeper inspection of thestructure of the permutation groups. Here are a few examples:Example 11. Consider the groups
H D f.1 ; : : : ; 10 / 2 A10 5367 3 j 1 10 D idg Ì S10 D T30 5407 A3 o S10 D T30 D G:
A relative invariant is given by
10 Y I WD X3i2 C 3 X3i1 C 32 X3i : iD1
5396 54212. The above example has the extension T30 T30 by adding the generator D .1 2/.4 5/ .28 29/ to both groups. A relative invariant is given by I C I . In invariant theory, this construction of an invariant out of an invariant of a subgroup is known as the Reynolds operator.3. Consider the groups 4831 G D T30 D .=2/15 Ì GL4 .2 / D S2 o GL4 .2 / S2 o A15
and H D T30 3819 D N Ì GL4 .2 / G. Here, N is the Hamming code in 15 2 . Recall the combinatorial structure of P3 .2 / and the Hamming code: • P3 .2 / has 15 points and 15 planes. • The stabilizer of one plane is a subgroup U0 GL4 .2 / of index 15. • The subgroup U0 decomposes the points into two orbits of size 7 and 8. • The Hamming code N is given as the subspace of 15 2 . To construct it, we label the coordinates with the points of P3 .2 /. The linear equations that define N are X Ci D 0 i2P3 . 2 /nE
for all the planes E P3 .2 /. • The automorphism group of the Hamming code is GL4 .2 /. 626 A.-S. Elsenhans
• We have #G D 660;602;880; ŒG W H D 16. • The important property of the subgroup U0 is that it stabilizes one of the linear relations that define the Hamming code N. The interested reader may consult [10, Sec. 3.3] for a more general introduction to the Hamming code. To construct an invariant, we start with the preimage U of U0 in G. In a first step, we construct an H \ U-invariant that is not a U-invariant by putting Y I0 WD .X2 X1 / : 2U==StabU f1;2g
This means, we turn the linear relation of N that is stabilized by U0 into an invariant. Now, a direct application of the Reynolds operator gives us the invariant X I WD I0 2H==H\U
as a sum of 15 products.Experiment 1 To get an overview of the complexity of the invariants that may beused by our code, we use the database of transitive groups up to degree 30 [8] andapply our invariant constructions to all the maximal subgroups. We use the notationTkn for the k-th group of degree n in the database. The construction of all theseinvariants took about 1 h. We use the cost function
.Number of multiplications in one evaluation/ deg.IG;U / min.45; ŒG W U /
to identify hard cases. They are listed in Table 1.Example 2 When we compute the Galois group of a polynomial with Galois groupT1153 30 , we run into the hardest case listed above. To get the group of a test polynomial took about 10 s. About 1 s was used in theStauduhar step that involves the hard invariant. The computation was done on one core of an Intel i7-3770 processor running at3.40 GHz.
Table 1 Pairs of groups with costly invariantsGroups Index deg.I/ Evaluation costsP L2 . 8 / D T932 A9 120 6 387 multiplicationsM24 D T24;680 24 A24 1,267,136,462,592,000 6 759 powersT1153 30 T30 4863 20,160 8 5221 multiplications Galois Groups 627
Experiment 2 Using the database of polynomials [9], we can pick 1954 polynomi-als in degree 16 and 1117 polynomials in degree 20. One for each transitive groupof degree 16 (resp. 20). The total running time to get all the Galois groups for the degree 16 polynomialsis 1094 s. The time for all the degree 20 polynomials is 907 s. The slowest examplesare polynomials with large coefficients. They took 5 (resp. 13) s. This is explainedby the larger p-adic precision that has to be used. A slow example with small coefficients is P455 WD x16 11x8 C 9. The Galoisgroup of this polynomial is T45516 . It takes 2.2 s to compute it. This is explained bythe fact that our method has to inspect 219 conjugacy classes of subgroups. The computation was done on one core of an Intel i7-3770 processor running at3.40 GHz.
5 Proving the Galois Group
The uses of heuristic p-adic precision and the short cosets result in an unprovenGalois group. The first proof algorithm was described in [7, Chap. 5.3]. Our newproof algorithm is a slight variation. The idea to confirm the results is that the Galois group predicts the factorizationsof resolvent polynomials. We can compute these resolvents and check that theprediction is in fact correct. This proves that the Galois group is contained in thestabilizer of the factorization pattern. These set stabilizers can be computed inmagma. This results in a subgroup of Sn that is proven to contain the Galois group.We repeat this with several resolvent polynomials until we run into a contradictionor confirm the heuristic result.
6 Conclusion
The Galois group package of magma (version 2.21) provides a degree independentimplementation of the Stauduhar method. It can compute the Galois group of arational degree 30 polynomial with moderate coefficient size in a few seconds on amodern standard PC. The initial result is heuristic. But, a proof algorithm is availableas well.
References
1. Cannon, J., Holt, D.: Computing maximal subgroups of finite groups. J. Symb. Comput. 37, 589–609 (2004) 2. Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Berlin/New York (1993) 628 A.-S. Elsenhans
3. Elsenhans, A.-S.: Invariants for the computation of intransitive and transitive Galois groups. J. Symb. Comput. 47, 315–326 (2012) 4. Elsenhans, A.-S.: A note on short cosets. Exp. Math. 23, 411–413 (2014) 5. Elsenhans, A.-S.: Improved methods for the construction of relative invariants for permutation groups (preprint, 2015) 6. Fieker, C., Klüners J.: Computation of Galois groups of rational polynomials. LMS J. Comput. Math. 17, 141–158 (2014) 7. Geißler, K.: Berechnung von Galoisgruppen über Zahl- und Funktionenkörpern. Dissertation, Berlin (2003) 8. Hulpke, A.: Constructing transitive permutation groups. J. Symb. Comput. 39(1), 1–30 (2005) 9. Klüners J., Malle G.: A database for field extensions of the rationals. LMS J. Comput. Math. 4, 182–196 (2001)10. van Lint, J.H.: Introduction to Coding Theory. Springer, Berlin/New York (1992)11. Stauduhar, R.P.: The determination of Galois groups. Math. Comput. 27, 981–996 (1973)12. van der Waerden, B.L.: Algebra I. Springer, Berlin (1960) Global Dynamics and Periodic Solutionsin a Singular Differential Delay Equation
Anatoli F. Ivanov and Zari A. Dzalilov
Abstract Differential delay equation " Œx 0 .t/ C cx 0 .t 1/ C x.t/ D f .x.t 1//is considered where " > 0 and c 2 R are parameters, and f W R ! R is piece-wise continuous. For small values of the parameter " a connection is made to thecontinuous time difference equation x.t/ D f .x.t 1//; which is further linked tothe one-dimensional dynamical system x 7! f .x/. Two cases of the nonlinearityf are treated: when it is continuous and of the negative feedback with respect toa unique equilibrium, and when it is of the so-called Farrey-type with a singlejump-discontinuity. Several properties are studied, such as continuous dependenceof solutions on the singular parameter " and the existence of periodic solutions.Open problems and conjectures are stated for the case of genuinely neutral equation,when c ¤ 0.
1 Introduction
Differential delay equations (DDEs) of the form " x 0 .t/ C cx 0 .t 1/ C x.t/ D f .x.t 1//; t0 (1)
appear as mathematical models of various real world phenomena. They were firstmentioned, to the best of our knowledge, in paper [7] as exact reductions ofnonlinear boundary value problems for one-dimensional wave equations modellingviolin string oscillations. The very same idea of reduction was later used in paper[4] describing complex oscillations in electrical circuits with tunnel diodes. Notethat in both papers the reduction is to the neutral type DDEs (1), when c ¤ 0.This is a well known idea and approach that certain boundary value problemsfor hyperbolic partial differential equations can be exactly reduced to differential
A.F. Ivanov ()Department of Mathematics, Pennsylvania State University, Lehman, PA 18627, USAe-mail: [emailprotected]. DzalilovFederation University Australia, Mt Helen Campus, Ballarat, VIC 3350, Australiae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 629J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_57 630 A.F. Ivanov and Z.A. Dzalilov
difference equations of various types [5]. The retarded type DDEs (1), when c D 0,appear in numerous other applications, a partial list of which can be found e.g.in [1, 3, 6]. This case is more studied compared with the neutral case of c ¤ 0.Very few results are available in the latter case. This paper is an attempt to initiatea comprehensive study of neutral differential delay equations (1) in one particularapproach as a singular perturbation problem. When " D 0 Eq. (1) becomes a continuous time difference equation (DE) of theform
x.t/ D f .x.t 1//; t 0: (2)
The dynamics of the latter as t ! 1 are largely determined by the interval map f ,or the equivalent scalar difference equation
xnC1 D f .xn /; n 2 N0 WD N [ f0g: (3)
In this paper we attempt to relate several properties of solutions of DDEs (1) tothe well-studied and understood properties of solutions of difference equations (2)and (3). In Sect. 2 we list basic facts and recall necessary knowledge about differentialdelay equations (1), difference equations with continuous time (2), and intervalmaps associated with scalar difference equation (3). They all can be readily foundin e.g. monographs [1, 2, 5, 6]. Section 3 contains our main results and is made of two subsections. Section 3.1deals with the continuous dependence on the singular parameter " > 0 between thesolutions of DDE (1) and its limiting case " D 0, the continuous time differenceequation (2). Since the solutions to Eq. (2) are typically discontinuous at tk D k 2N0 the closeness between the solutions of equations (1) and (2) can be provedeverywhere on finite intervals Œ0; T except a small vicinity of the integer pointstk . In the case when the solutions to (2) are continuous the closeness is uniformon the entire interval Œ0; T . The precise statements are given by Theorem 1 and itsCorollary. Section 3.2 deals with the existence of periodic solutions to DDE (1) andtheir asymptotic shape as " ! 0C. The case of continuous negative feedback andone-sided bounded f is considered in Sect. 3.2.1. The other case of this paper, whenf is a Farey-type nonlinearity, is treated in Sect. 3.2.2. We show the existence ofthe so-called square-wave periodic solutions for small " > 0 in both cases whichcorrespond to cycles of the map f . The proof is similar in both cases, and it isoutlined for the case of retarded DDEs (1), when c D 0. Section 4 concerns some open questions and conjectures about the generalneutral DDEs (1) when c ¤ 0. It particular, it is a natural expectation that theperiodicity results of Sects. 3.2.1 and 3.2.2 can be extended to this case. Thisconjecture is confirmed by preliminary numerical simulations. At present we donot have a rigorous proof of the periodicity in the neutral case. This paper is a result of authors’ presentation at the AMMCS-CAIMS Congress2015 held at the WLU during 7–12 June 2015. It contains some recently obtained Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation 631
mathematical results which ideas and proofs are only outlined here, due to theirlength and complexity as well as to the space limitation of the proceedings. A fullscale paper with all the mathematical details and proofs included is a forthcomingwork.
2 Preliminaries
For arbitrary '.t/ 2 C1 .Œ1; 0 ; R/ WD X1 there exists unique solution x"' .t/to Eq. (1) defined and continuous for all t 1. The solution is differentiableeverywhere for t 1 except at the integer values tk D k 2 N0 ; where it istypically continuous only. The solution is formally obtained for t > 0 by successiveintegration. Let X0 WD C0 .Œ1; 0/; R/ be a set of continuous initial functions such that thelimit limt!0 .t/ exists. X0 can be viewed as a restriction of the standard spaceof initial functions C.Œ1; 0 ; R/ when its elements are considered on the reduceddomain, the half open interval Œ1; 0/. Thus
X0 WD f 2 C0 .Œ1; 0/; R/ j lim .s/ D .0/ existsg: s!0
For arbitrary 2 X0 there exists unique solution x .t/ to Eq. (2) defined for allt 1. The solution is formally obtained for t 0 by successive iterations. Thesolution is typically discontinuous at the integer values tk D k 2 N0 with a jumpdiscontinuity. In the case when the contiguity condition
lim .s/ D f . .1// .cc/ s!0
holds, and f is continuous, the solution x .t/ is also continuous for all t 1. We consider two distinct cases of the nonlinearity f in DDEs (1):(H1) f is continuous and satisfies the negative feedback assumption
xf .x/ < 0 8x 2 Rnf0g: .nf /
Also, f is one-sided bounded on R: f .x/ M or f .x/ M for all x 2 R and some M > 0;(H2) f is a Farey-type type nonlinearity defined by: f .x/ D mx C A if x 0; and f .x/ D mx B if x > 0, for some 0 < m < 1 and A; B > 0. The first case is motivated by a standard set of assumptions normally imposedfor the retarded type DDEs (1) when c D 0; see e.g. [3] and further referencestherein. The second case is an exact reduction of a specific nonlinear boundary valueproblem for a scalar wave equation [5]. Though many aspects of the dynamics ofequations (1) are studied in the case c D 0, virtually no comprehensive research isdone and very little is known in the neutral case, when c ¤ 0. 632 A.F. Ivanov and Z.A. Dzalilov
3 Main Results
3.1 Continuous Dependence on Singular Parameter "
In this subsection we establish certain continuous dependence results on the singularparameter " for the differential delay equation (1). We show that its solutions andsolutions of the difference equation with continuous argument (2) are close on finiteintervals in a properly chosen metric for all sufficiently small ". Since the solutionsto Eq. (1) are continuous for all t 0, while the solutions to Eq. (2) are typicallydiscontinuous at integer values tk D k 2 N0 , we need a proper means of comparisonof those solutions. We shall use the uniform metric to measure the closeness between two functions˛.t/ and ˇ.t/ defined on a set S R:
jj˛ ˇjjS WD supfj˛.t/ ˇ.t/j; t 2 Sg:
For the differential delay equation (1) initial functions '1 ; '2 2 C1 .Œ1; 0 ; R/ DX1 are compared in the standard uniform metric:
jj'1 '2 jjŒ1;0 D supfj'1 .t/ '2 .t/j C j'10 .t/ '20 .t/j; t 2 Œ1; 0 g:
For the difference equation (2) initial functions 1; 2 2 C0 .Œ1; 0/; R/ D X0are compared in the following uniform metric
jj 1 2 jjŒ1;0/ D supfj 1 .t/ 2 .t/j; t 2 Œ1; 0/g:
Given ' 2 X1 and 2 X0 we shall compare them on the initial interval Œ1; 0 asfollows:
jj' jjŒ1;0 D supfj'.t/ .t/j; t 2 Œ1; 0 ; where .0/ WD lim .t/g: t!0
Given two initial functions, ' 2 X1 and 2 X0 , which are close on the initialinterval Œ1; 0 , we would like to estimate the closeness between their respectivesolutions to Eqs. (1) and (2) on any finite interval Œ0; T . Since x"' .t/ is continuousfor all t 0, while x .t/ is typically discontinuous at tk D k 2 N0 , the closenessin the uniform metric on the interval Œ0; T cannot happen. However, the closenesstakes place everywhere else, if the integer values tk D k are excluded on the intervalŒ0; T . For the purpose of such comparison we introduce the following notation
ŒT [ JT WD Œ0; T n U .k/; kD0 Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation 633
where U .k/ D .k ; k C / is the -neighborhood of the point tk D k, and Œ isthe integer value function.Theorem 1 Suppose f 2 C.R; R/ satisfies the Lipschitz condition, j f .u/ f .v/j Kju vjj, and ' 2 X1 is fixed. For arbitrary T > 0; > 0, and > 0 thereexist ı > 0 and "0 > 0 such that if 2 X0 is such that jj 'jjŒ1;0 < ı thenjjx x"' jjJT < for all 0 < " < "0 .The proof of the theorem follows from the following three lemmas.Lemma 1 Suppose that 1 ; 2 2 X0 are given, and let x 1 .t/; x 2 .t/ be thecorresponding solutions to difference equation (2). Then
jjx 1 x 2 jjŒ0;1 Kjj 1 2 jjŒ1;0 :
The proof follows from the Lipschitz continuity of function f .Lemma 2 Suppose that '1 ; '2 2 X1 are given, and let x"'1 .t/; x"'2 .t/ be thecorresponding solutions to differential delay equation (1). Then there exists "0 > 0such that for all 0 < " < "0 one has
jjx"'1 x"'2 jjŒ0;1 M2 jj'1 '2 jjŒ1;0
for some constant M2 > 0.Proof The proof is similar to that of Lemma 2 of paper [3], page 185. Solutions toEq. (1) also solve the following integral equation Z t 1 t st x.t/ D x.0/ expf g C f .x.s 1// expf g ds (4) " " 0 " Z t st c x 0 .s 1/ expf g ds: 0 "
The reasoning and major steps of the proof of Lemma 2 should be repeated andapplied now to the integral equation (4). We leave details to the reader. t uLemma 3 Suppose that ' 2 X1 is given, and let x"' .t/ and x' .t/ be the correspond-ing solutions to equations (1) and (2), respectively. Then for arbitrary > 0 and > 0 there exists "0 > 0 such that for all 0 < " < "0 one has
jjx"' x' jjŒ;1 :
Proof The proof follows ideas and repeats major steps of the proof of Lemma 3 ofpaper [3] when one estimates the difference jx"' .t/ f .x.t 1//j for t 2 Œ0; 1 . Theintegral equation (4) is used in the new calculations. 634 A.F. Ivanov and Z.A. Dzalilov
By integration by parts the last term of Eq. (4) one arrives at the following integralequation
t x.t/ D Œx.0/ C cx.1/ expf g (5) " Z 1 t st C Œ f .x.s 1// C cx.s 1/ expf g ds cx.t 1/: " 0 "
Using the identity Z 1 t st t F.t/ D F.t/ expf g ds C expf gF.t/ " 0 " "
the difference x"' .t/ x' .t/ D x"' .t/ f .'.t 1//; t 2 Œ0; 1 , can be represented as
t x"' .t/ f .'.t 1// D Œ'.0/ C c'.1/ f .'.t 1// c'.t 1/ expf g " Z 1 t st C Œ f .'.s 1// f .'.t 1// expf g ds " 0 " Z c t st C Œ'.s 1/ '.t 1/ expf g ds: " 0 "
The rest of the estimates can be done very similar to those in Lemma 3 of [3]. Weleave details to the reader. t u The proof of Theorem 1 now follows by induction and the triangle inequality
jjx x"' jjJT jjx x' jjJT C jjx' x"' jjJT :
Corollary 1 When the contiguity condition (cc) is satisfied Theorem 1 is valid inthe uniform metric on the whole interval Œ0; T .This essentially follows from the last chain of estimates in Lemma 3 for thedifference x"' .t/ f .'.t 1//; t 2 Œ0; 1 , where one can choose D 0.
3.2 Periodicity
This subsection deals with the existence of periodic solutions to differential delayequation (1). We consider the two cases of the nonlinearity f as described above bythe hypotheses (H1) and (H2). We first introduce a notion of periodic solutions to difference equation (2)associated with cycles of the corresponding map f . Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation 635
Definition 1 (Square wave periodic solutions to DE (2)) Let fa1 ; a2 ; : : : ; aN g bean N-cycle of map f , f .ak / D akC1 ; k D 1; : : : ; N 1; f .aN / D a1 : Define thepiece-wise constant function by
Sqwv.t; a1 ; : : : ; aN / WD ak ; for t 2 Œk 1; k/; k D 1; 2; : : : ; N;
and extend it periodically to all t 2 R. The function Sqwv.t; a1 ; : : : ; aN /; t 2 R; iscalled the square wave periodic solution of difference equation (2) correspondingto the cycle fa1 ; a2 ; : : : ; aN g. We define next the convergence of a parameter dependent family of functions tothe discontinuous function Sqwv.t; a1 ; : : : ; aN / on the interval Œ0; N . Let a familyp" .t/ 2 C.Œ0; N ; R/ with the parameter " be given.Definition 2 We say that p" .t/ converges to Sqwv.t; a1 ; : : : ; aN / on the intervalŒ0; N as " ! 0C if for arbitrary > 0 and > 0 there exists "0 > 0 suchthat for all 0 < " < "0 one has jjSqwv.; a1 ; : : : ; aN / p" ./jjJN < .
3.2.1 Negative Feedback Nonlinearity
Theorem 2 Assume that (H1) is satisfied and f 0 .0/ < 1. There exists "0 > 0such that for all 0 < " < "0 DDE (1) has a periodic solution x"p .t/ with the periodT D 2 C O."/ where O."/ ! 0C as " ! 0C . The periodic solution x"p .t/ convergesas " ! 0 to one of the square wave periodic solutions Sqwv.t; a1 ; a2 / of DE (2),where fa1 ; a2 g is a cycle of period two of the map f .Proof We outline main steps of the proof for the case c D 0, omitting theintermediate details due to the space limitation. Consider the set of initial functions
X D f' 2 C0 .Œ1; 0 ; R/ j .i/; .ii/; .iii/ are satisfied g;
where the assumptions (i), (ii), and (iii) are specified as follows: (i) ' is non-negative with '.1/ D 0; '.t/ > 0 8t 2 .1; 0 , and ' is bounded from above, '.t/ M 8t 2 Œ0; 1 and some M > 0;(ii) ' has an exponential growth in a right neighborhood of t D 1 given by: .tC1/˛ '.t/ l .t/ WD K 1 e " 8t 2 Œ1; 1 C ˇ ;
for some K > 1; 0 < ˛ < 1, and ˇ D O."/;(iii) ' is bounded away from zero, '.t/ ı 8t 2 Œ1 C ˇ; 0 ; for some sufficiently small and fixed ı > 0.Loosely speaking, the set X consists of non-negative bounded above initialfunctions such that all of them start at 0 when t D 1 (assumption (i)), they quickly 636 A.F. Ivanov and Z.A. Dzalilov
grow with the exponential rate in a small right ˇ-neighborhood of t D 1 beyondthe value of ı > 0 (assumption (ii)), and they remain bounded away from 0 by thevalue ı for the remainder Œ1 C ˇ; 0 of the initial interval Œ1; 0 (assumption (iii)). Consider also a negatively symmetric to X set defined by
X WD f 2 C.Œ1; 0 ; R/j 2 X g:
When c D 0 solutions to Eq. (1) satisfy the following integral equation Z t 1 t st x.t/ D x.0/ expf g C f .x.s 1// expf g ds: (6) " " 0 "
It is obtained from the integral equation (4) by setting c D 0. By using (6) one can make precise forward estimations on the solution x"' .t/ for' 2 X . One can show that there exists "10 > 0 such that for every ' 2 X and all0 < " < "10 there exists time t1 D t1 .'/ > 0 such that the corresponding solutionx' .t/ to DDE (1) has the properties:(a) the segment x' .t1 C 1 C s/; s 2 Œ1; 0 ; of the solution belongs to X ;(b) 0 < t1 P1 " for some P1 > 0.Define now a mapping F 1 on X with the values in X by the formula
F 1 .'/ WD x' .t1 C 1 C s/; s 2 Œ1; 0 :
Likewise, there exists "20 > 0 such that for every 2 X and all 0 < " < "20 thereexists time t2 D t2 . / > 0 such that the corresponding solution x .t/ to DDE (1)has the properties:(c) the segment x .t2 C 1 C s/; s 2 Œ1; 0 ; of the solution belongs to X ;(d) 0 < t2 P2 " for some P2 > 0.Define a mapping F 2 on X with the values in X by the formula
F 2 . / WD x .t2 C 1 C s/; s 2 Œ1; 0 :
For arbitrary ' 2 X and all 0 < " < minf"1 ; "2 g WD "0 define now the mappingF as the composition of the above mappings F 1 and F 2 , F WD F 2 ı F 1 . Fmaps the convex set X into itself. It is a compact map, for when c D 0 the DDE (1)is a retarded type equation. Also the set F .X / is bounded due to the one-sidedboundedness of the function f . Therefore, by the Schauder fixed point theorem thereexists '0 2 X such that .F /.'0 / D '0 . Clearly, that the corresponding solutionx'0 .t/ is periodic. Its period is T D 2 C t1 C t2 2 C .P1 C P2 /". The convergencex'0 .t/ ! Sqwv.t; a1 ; a2 /, where fa1 ; a2 g is a 2-cycle of the map f , can be provedby using the continuous dependence on " (Theorem 1) and the fact that its periodT D 2 C O."/ is close to 2 for small " > 0. Details are omitted due to their length. t u Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation 637
3.2.2 Farey-Type Nonlinearity
Theorem 3 Assume that (H2) is satisfied, and let fa1 ; : : : ; aN g be the uniqueglobally attracting cycle of the map f . There exists "0 > 0 such that for all0 < " < "0 DDE (1) has a periodic solution x"p .t/ with the period T D N C O."/,where O."/ ! 0 as " ! 0. The periodic solution x"p .t/ converges as " ! 0 to thesquare wave periodic solution Sqwv.t; a1 ; : : : ; aN / of DE (2).Proof The idea of the proof is similar to that in the proof of Theorem 2. Associatedwith the unique cycle of the map f we construct a sequence of convex bounded setsXk C.Œ1; 0 ; R/; k D 1; : : : ; N; and a sequence of compact maps F k such thatF k W Xk ! XkC1 .N C 1 WD 1/. The composition map F WD F N ı ı F 1 thenmaps X1 into itself. By the Schauder fixed point theorem it has a fixed point '0 2 X1
there. The corresponding solution x"'0 .t/ to DDE (1) is periodic. Its properties andthe convergence to Sqwv.t; a1 ; : : : ; aN / as " ! 0 follow from the construction ofthe sets Xk outlined below. Let fa1 ; : : : ; ak ; akC1 ; : : : ; aN g be the globally attracting cycle of the map f withf .ak / D akC1 ; k D 1; : : : ; N 1 and f .aN / D a1 . In order to be specific in thedefinition of Xk we assume that akC1 > ak . Define now Xk as follows:
Xk D f' 2 C0 .Œ1; 0 ; R/ j .i/; .ii/ are satisfied g;
where the assumptions (i) and (ii) are specified as follows:(i) ' starts in a neighborhood of ak with a fast exponential transition to akC1 defined as: '.1/ D ak C ı, and l .t/ '.t/ u .t/ 8t 2 Œ1; 1 C ˇ ; where .tC1/˛ .tC1/ l .t/ D akC1 C .ak C ı akC1 / e " ; u .t/ D M C .ak C ı M/ e "
with ˇ D O."/; 0 < ˛ < 1 and ı > 0 is small and fixed;(ii) ' stays in the ı-neighborhood of akC1 for the rest of the time on the initial interval Œ1; 0 :
akC1 ı '.t/ akC1 C ı 8t 2 Œ1 C ˇ; 0 :
In the opposite case of akC1 < ak the inequalities in part (i) of the definition of Xkwill have to be reversed, as well as the starting point for ' to be moved to ak ı,'.1/ D ak ı. By using the integral equation (6) one can show that there exists "k0 > 0 such thatfor all 0 < " < "k0 and every ' 2 Xk there exists time tk > 0 for its solution x"' .t/such that the following holds: (a) the segment x' .tk C 1 C s/; s 2 Œ1; 0 ; of the solution belongs to XkC1 ;(b) 0 < tk Pk " for some Pk > 0. 638 A.F. Ivanov and Z.A. Dzalilov
Therefore, the map F k W Xk ! XkC1 can be defined by
.F k '/.t/ WD x' .tk C 1 C t/; t 2 Œ1; 0 :
The composite map F D F N ı ı F 1 has then a fixed point '0 2 X1 whichgenerates the periodic solution x"'0 .t/ to DDE (1) with the stated properties. t u
4 Open Problems and Conjectures
The existence of periodic solutions in Theorem 2 is a well known fact due to Hadelerand Tomiuk [1, 2, 6]. The existence itself only requires the instability of the trivialsolution x 0 and the one-sided boundedness of f . In the case of globally attractingcycle of period two for the map f , fa1 ; a2 g, the square wave limiting shape of theperiodic solutions was derived by Mallet-Paret and Nussbaum [2, 3]. We prove theasymptotic shape in the general case of the negative feedback assumption on f . Thequestions of uniqueness and asymptotic stability of the periodic solutions remainlargely open problems. In fact, multiple periodic solutions and chaotic behaviorsare shown to exist in the case of globally attracting two-cycle [3]. Therefore, toestablish at least some sufficient conditions for the existence and stability of periodicsolutions for the neutral DDE (1) appears to be quite challenging mathematicalproblem. The theoretical results on periodic solutions of this paper can be proved forthe retarded DDEs (1) at this time. The principal reasons being that the SchauderFixed Point Theorem is used, with the essential requirements of the convexity andboundedness of the sets X ; X ; Xk and of the compactness of the shift operatorsF along the solutions of DDE (1) when c D 0. Though the estimates in the proofsof the Theorems 2 and 3 seem to remain valid also in the case c ¤ 0 we are notaware of the existence of appropriate fixed point theorems that can be applied in theneutral case.Conjecture 1 Theorems 2 and 3 remain valid for the case of neutral DDE (1), whenc ¤ 0.
Acknowledgements The first author would like to express his gratitude and appreciation for thesupport and hospitality extended to him during his visit and stay at the CIAO of the FederationUniversity Australia, Ballarat, in December 2014–January 2015. This paper is a result of thecollaborative research work initiated during the visit. Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation 639
References
1. Erneux, T.: Applied Delay Differential Equations. Ser.: Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3, 204 pp. Springer, New York (2009)2. Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer Applied Mathematical Sciences, vol. 99, 447 pp. Springer-Verlag, New York (1993)3. Ivanov, A.F., Sharkovsky, A.N.: Oscillations in singularly perturbed delay equations. In: Jones, C.K.R.T., Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported (New Series), vol. 1, pp. 165–224. Springer, New York (1991)4. Nagumo, J., Shimura, M.: Self-oscillation in a transmission line with a tunnel diode. Proc. IRE 49(8), 1281–1291 (1961)5. Sharkovsy, A.N., Maistrenko, Yu.L., Romanenko, E.Yu.: Difference Equations and Their Perturbations, vol. 250, 358 pp. Kluwer Academic Publishers, Dordrecht/Boston/London (1993)6. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Ser.: Texts in Applied Mathematics, vol. 57, 172 pp. Springer, New York/Dor- drecht/Heidelberg/London (2011)7. Witt, A.A.: On the theory of the violin string. Zhurn. Tech. Fiz. 6, 1459–1470 (1936, in Russian) Localized Spot Patterns on the Sphere forReaction-Diffusion Systems: Theory and OpenProblems
Alastair Jamieson-Lane, Philippe H. Trinh, and Michael J. Ward
Abstract A new class of point-interaction problem characterizing the time evolu-tion of spatially localized spots for reaction-diffusion (RD) systems on the surface ofthe sphere is introduced and studied. This problem consists of a differential algebraicsystem (DAE) of ODEs for the locations of a collection of spots on the sphere,and is derived from an asymptotic analysis in the large diffusivity ratio limit ofcertain singularly perturbed two-component RD systems. In Trinh and Ward (Thedynamics of localized spot patterns for reaction-diffusion systems on the sphere.Nonlinearity Nonlinearity 29(3), 766–806 (2016)), this DAE system was derived forthe Brusselator and Schnakenberg RD systems, and herein we extend this previousanalysis to the Gray-Scott RD model. Results and open problems pertaining to thedetermination of equilibria of this DAE system, and its relation to elliptic Feketepoint sets, are highlighted. The potential of deriving similar DAE systems for morecomplicated modeling scenarios is discussed.
1 Introduction
Spatially localized patterns can occur for two-component reaction-diffusion (RD)systems in the singularly perturbed limit corresponding to a large diffusivity ratiobetween the two components in the system. In particular, on the surface of the unitsphere, the stability and dynamics of localized spot patterns, whereby the solutionconcentrates at discrete points on the sphere, have been analyzed recently for thesingularly perturbed Brusselator and Schnakenberg RD systems (cf. [24, 27]). It wasalso shown that, in certain parameter regimes, these localized patterns can exhibitvarious instabilities, including either spot-shape instabilities, leading to spot self-replication events, or competition instabilities, leading to the annihilation of spots
A. Jamieson-Lane • M.J. Ward ()Department of Mathematics, University of British Columbia, Vancouver, BC, Canadae-mail: [emailprotected]; [emailprotected]. TrinhOCIAM, Mathematical Institute, University of Oxford, Oxford, UKe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 641J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_58 642 A. Jamieson-Lane et al.
in the pattern. This analysis of [24] and [27] extends the previous studies of localizedspot patterns on 2-D planar domains for related RD systems (cf. [6, 11, 29]). By using formal asymptotic methods on the RD system in the singularlyperturbed limit, it is possible to derive a differential algebraic ODE system char-acterizing the dynamics of spot patterns. This new class of dynamically interactingparticle system has some common features with the well-known ODE systemcharacterizing the dynamics of Eulerian point vortices in fluid mechanics. In thislatter context, there has been an intense study of the dynamics and equilibria ofpoint vortices on the sphere over the past three decades (cf. [2, 3, 10, 18, 19, 23]).Similar to the original asymptotic derivation of the limiting point vortex problemin [3] starting with the Euler equations of fluid mechanics, the main result for spotdynamics in [27] for the Brusselator and Schnakenberg model, and herein for theGray-Scott RD model, provides a reduced dynamical system for the time evolutionof the centres of the localized spots on the sphere. Motivated by specific questions related to the development of biological patternson both stationary and time-evolving surfaces (cf. [5, 12, 17, 20, 21]), there havebeen many numerical studies of RD patterns on the sphere and other compactmanifolds (cf. [1, 13, 14, 28] see also the references therein). Prior analyticalstudies of RD pattern formation on the surface of the sphere, have focused onusing weakly nonlinear and equivariant bifurcation theory to derive normal formequations characterizing the development of small amplitude spatial patterns thatbifurcate from a spatially uniform steady-state (cf. [4, 7, 16]). However, as a resultof the typical high degree of degeneracy of the eigenspace associated with sphericalharmonics of large mode number, these amplitude equations typically consist of arather large coupled set of nonlinear ODEs. The latter are known to have an intricatesubcritical bifurcation structure (cf. [4, 7, 16]). As a result, the preferred spatialpattern that emerges from an interaction of these weakly nonlinear modes is difficultto predict theoretically. This intrinsic difficulty is accentuated for RD systems wherethere is a large diffusivity ratio, which effectively yields a large aspect ratio systemwhere center manifold analysis is of more limited use [25]. For such large aspectratio RD systems, there is typically a rather wide band of unstable modes [24, 27],and so the prediction of pattern development based on the conventional paradigm ofusing both a Turing and a weakly nonlinear analysis is not generally possible. However, it is in this singular limit of a large diffusivity ratio that localizedspot patterns robustly appear from a transient process starting from small randomperturbations of a spatially uniform state [24]. A discussion of results and openproblems relating to the study of such “far-from equilibrium patterns” is the topicof this short article. In particular, in certain cases the equilibria of this DAEsystem for spot dynamics have a close relationship to the classical problem inapproximation theory of determining a set of elliptic Fekete points, which are theglobally minimizer of the discrete logarithmic energy for N points on the sphere. Dynamics and Equilibria of Localized Spots on the Sphere 643
The outline of this brief article is as follows. In Sect. 2, we briefly present theDAE system for the dynamics of spots for the Brusselator models as derived in[27]. In Sect. 3 we discuss some results and open questions related to determiningequilibria for these DAE systems. New results for the equilibria of patterns witheither 9 or 10 spots are presented. In Sect. 4 we give a new result for the DAEdynamics for spot patterns for the well-known Gray-Scott model. Finally, in Sect. 5we list a few open problems related to deriving similar DAE dynamics for spotinteractions for more complicated models.
2 Dynamics of Spots on the Sphere
Under the assumption of a large diffusivity ratio and a small “fuel” supply, theBrusselator model of [22] posed on the surface of the sphere, can be scaled intothe following system for u D u.x; t/ and the inhibitor v D v.x; t/ (cf. [24, 27]):
@u @v D "2 S u C "2 E u C fu2 v ; D S v C "2 u u2 v ; (1) @t @t
for some O.1/ constants E > 0, > 0, and 0 < f < 1. We refer to E as the “fuel”parameter. Here S is the Laplace-Beltrami operator on the sphere. Spatial patterns for which u concentrates as " ! 0 at a discrete set of pointsx1 ; : : : ; xN on the sphere are called spot patterns. For " ! 0, we have u D O.1/in the core of the spot, where jx xj j D O."/, and u "2 E away from the spotcenters where jx xj j D O.1/. Then for " ! 0, the effect of the localized spots onthe inhibitor field v in (1) is to introduce a sum of Dirac-delta “forces” where thestrength of the “force” induced by the spot at xj is proportional to Sj (see [24] fordetails). As such, v can be represented as a superposition of the well-known sourceneutral Green’s function for the sphere. In this way, in [24] a quasi-equilibriumspot pattern with frozen locations x1 ; : : : ; xN was constructed using the method ofmatched asymptotic expansions by formulating a nonlinear algebraic system for thespot locations x1 ; : : : ; xN and the spot strengths S1 ; : : : ; SN . The linear stability ofthis quasi-equilibrium spot pattern to O.1/ time-scale perturbations was analyzedin [24]. Provided that the quasi-equilibrium spot pattern is linearly stable, the slowdynamics of the spot pattern on the long time scale D "2 t was derived in [27].The collective coordinates characterizing this slow dynamics are the spot locationsx1 ; : : : ; xN and their corresponding spot source strengths S1 ; : : : ; SN , that both evolveslowly on the long time-scale D "2 t. The slow dynamics derived in [27] is adifferential algebraic system of ODEs as given by the following result:Principal Result 1 (Slow spot dynamics (cf. [27])) Let " ! 0. Provided thatthere are no O.1/ time-scale instabilities of the quasi-equilibrium spot pattern, thetime-dependent spot locations, xj for j D 1; : : : ; N, on the surface of the sphere vary 644 A. Jamieson-Lane et al.
on the slow time-scale D "2 t, and satisfy the dynamics
dxj 2 X Si xiN D I Qj ; Q j xj xTj ; j D 1; : : : ; N ; d A j .Sj / iD1 jxi xj j2 i¤j (2a)coupled to the constraints for S1 ; : : : ; SN in terms of x1 ; : : : ; xN given by the roots ofa nonlinear algebraic system involving the Green’s matrix G h i 2E N .S/ I .I E 0 /G S C .I E 0 /.S/ e D 0: (2b) N
Here I is N N identity matrix, .E 0 /ij D N1 , .S/i D Si , ..S//i D .Si /, .G /ij Dlog jxi xj j for i ¤ j and .G /ii D 0, .e/i D 1, and D 1= log " 1. From (2) the spot locations are coupled to the spot strengths by (2b), yielding aDAE system for the slow spot evolution. Since the spot strengths can be calculatedin terms of the locations from (2b), the DAE system has index 1. One readilyestablished feature of the DAE system (2) is that it is invariant under an orthogonaltransformation, corresponding to a rotation of the spots on the sphere. In this DAE system (2), there are two functions .Sj / and A j .Sj / < 0, whichdepend only on Sj and the Brusselator parameter f , that must be determinednumerically in terms of the local profile of the spot xj (cf. [27]). These are shownin Fig. 1 for a few values of f . In the limit " ! 0, and for E D O.1/, the onlypossible O.1/ time-scale instability of the quasi-equilibrium spot pattern is a linearinstability of the local spot profile to a peanut-shape if Sj > ˙2 . f /. This linearinstability is found in [24] to lead to a nonlinear spot self-replication event. Thethreshold values ˙2 . f / for spot self-replication for a few values of f are given in thecaption of Fig. 1.
0 25 20 −5
15 −10 10 Aj −15χ(Sj) 5 0 −20
−5 −25 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Sj Sj
Fig. 1 Left panel: the function .Sj I f / in (2) for f D 0:4 (heavy solid), f D 0:5 (solid), f D 0:6(dotted), and f D 0:7 (widely spaced dots). The spot self-replication threshold ˙2 . f / for Sj isshown by the thin vertical lines in this figure. If Sj > ˙2 . f / the local spot profile is linearly unstableto a peanut-splitting instability (cf. [24]). The threshold values are ˙2 .0:4/ 8:21, ˙2 .0:5/ 5:96, ˙2 .0:6/ 4:41, and ˙2 .0:7/ 3:23. Right panel: the function A j .Sj / < 0 in (2) with thesame labels as in the left panel Dynamics and Equilibria of Localized Spots on the Sphere 645
3 Equilibria of the DAE Dynamics and Open Questions
In this section we discuss some previous results obtained in [27] as well as somenew results for the equilibria of (2) that have large basins of attraction to initialconditions. We consider patterns for small values of N which Sj D O.1/ as ! 0.A few open problems are mentioned. To determine the possible equilibrium spot configurations of (2) with large basinsof attraction for N 3 when E D O.1/, f , and are given, we performed numericalsimulations of (2) for both pre-specified and for randomly generated uniformlydistributed initial conditions for the spot locations on the surface of the sphere. Togenerate N initial points that are uniformly distributed with respect to the surfacearea we let h and h be uniformly distributed random variables in .0; 1/ and definespherical coordinates D 2h and D cos1 .2h 1/. Newton’s method wasused to solve (2b) for the initial set of N points. If the Newton iterates failed toconverge, indicating that no quasi-equilibrium exists for the initial configurationof spots, a new randomly generated initial configuration was generated. The DAEdynamics (2a) was then implemented by using an adaptive time-step ODE solvercoupled to a Newton iteration scheme to compute the spot strengths. In the simulations below we took f D 0:5 and " D 0:02. We remark thatwhenever e D .1; : : : ; 1/T is an eigenvector of the Green’s matrix G , then theconstraint (2b) admits a solution where S D Sc e. For such an equal spot–sourcestrength pattern, the equilibrium spatial configuration of spots for (2) is independentof E, f , and . In our discussion below, we refer to a ring pattern as a collection of N equally-spaced spots lying on an equator of the sphere. We refer to an .N 2/ C 2 patternas a spot pattern consisting of two antipodal spots with the remaining N 2 spotsequally-spaced on the equatorial mid-plane between the two polar spots. The simulations of [27] of 50 randomly generated initial spot for N D 3; : : : ; 8yielded the following results for equilibria of (2) with large basin of attractions:• N D 3: three equally-spaced spots that lie on a plane through the center of the sphere. (Common spot strength pattern.)• N D 4: four spots centered at the vertices of a regular tetrahedron. (Common spot strength pattern.)• N D 5; 6; 7: an .N 2/ C 2 pattern consisting of a pair of antipodal spots, with the remaining N 2 spots equally-spaced on the equatorial mid-plane between the two polar spots. (Two different spot strengths for polar and mid-plane spots.)• N D 8: a “twisted cuboidal” shape, consisting of two parallel rings of four equally-spaced spots, with the rings symmetrically placed above and below an equator. The spots are phase shifted by 45ı between each ring. The perpendicular distance between the two planes is 1.12924 as compared to a minimum distance of 1.1672 between neighboring spots on the same ring, so that the pattern does not form a true cube. (Common spot strength pattern.) 646 A. Jamieson-Lane et al.
We remark that for the case N D 2 it was shown in Lemma 5 of [27] that any twoinitial spots on the sphere will become antipodal in the long-time limit ! 1. Thiswas done by deriving a simple ODE for the angle ./ between the spot centers x1and x2 at time , as measured from the center of the sphere, i.e. xT2 x1 D cos , andestablishing from this ODE that ! as ! 1 for any .0/. In [27] the linear stability of ring configuration of spots was studied numerically.The following conjecture was formulated in [27] based on numerical experiments.• A ring pattern of N D 3 is orbitally stable, but is unstable if N 4.• For N D 4; 5; 6; 7, an .N 2/ C 2 pattern is orbitally stable, but such a pattern is unstable if N 8. More specifically for N D 3, the numerical computations of [27] suggest that aring solution is orbitally stable to small random perturbations in the spot locationsin the sense that as time increases the perturbed spot locations become colinear on anearby (tilted) ring. For N 4, a similar small, but otherwise arbitrary, perturbationof the spot locations on the ring leads to a breakup of the ring pattern. Similar an.N 2/ C 2 pattern for N D 8 breaks up and forms a twisted cuboidal shape.Open Problem: Establish analytically these results for the equilibria of spotpatterns for N D 3; : : : ; 8 using group theory methods for ODEs. Analyze the linearstability of ring patterns by using an approach similar to that done in [2] for thecorresponding problem of the linear stability of Eulerian point vortices on a ring. For larger values of N it becomes increasingly difficult to visualize the symme-tries of the final equilibrium pattern that emerges under the DAE system (2) frominitial data. More specifically, it becomes increasingly challenging to find a rotationmatrix to put the pattern in a standard reference configuration. We now discuss twonew results for N D 9 and N D 10 not obtained in [27]. For even larger values of Npoint-matching algorithms from computer science may be useful for classifying thesymmetries of the final pattern. For N D 9 the equilibrium state of (2) with a large basin of attraction for initialconditions is the pattern shown in the lower right subfigure of Fig. 2. Our simulationswith 50 random initial configurations have shown that the limiting pattern consistsof 3 parallel planes of 3 spots each. The spots on the equatorial plane and the othertwo planes are phase-shifted 60ı (see the caption of Fig. 2 for details.) For N D 10 the equilibrium state of (2) with a large basin of attraction forinitial conditions is the pattern shown in the lower right subfigure of Fig. 3. Theequilibrium pattern consists of two polar spots together with two parallel planeswith four equally-spaced spots on each plane. The relative phase-shift of the spotson the two planes is 45ı (see the caption of Fig. 3 for details). A classical problem of point configurations on the sphere is theP problem P of find-ing the global minimizer of the discrete logarithmic energy V i¤j log jxi xj j on the sphere where jxj j D 1, which also has applications to minimizingthe mean first passage time for Brownian motion on the sphere (cf. [8]). Suchoptimizing configurations are called elliptic Fekete point sets. By comparing ourresults for equilibrium spot configurations with the optimal energies V of elliptic Dynamics and Equilibria of Localized Spots on the Sphere 647
Fig. 2 The evolution of a 9-spot pattern at different time for the Brusselator (2) when f D 0:5,E D 18, and D 0:02. (a) The initial state D 0. (b) D 1. (c) D 3. (d) The computed steadystate after a suitable rotation. The steady-state consists of 3 planes of 3 spots each. The spots onthe equatorial plane and the other two planes are phase-shifted 60ı . The distance d between theequatorial plane and each of the other two planes is d 0:7014. The common value of the spotstrengths on the equatorial plane differs from that of the other 6 spots
Fekete point sets, as given in Table 1 of [26], we conclude that our equilibriumspot configurations for N D 3; : : : ; 10 having a large basin of attraction are indeedelliptic Fekete point sets. As a remark, if we were to set Sj D 1 in (2) and ignorethe constraint (2b), then it is readily seen upon introducing Lagrange multipliersthat local and global minima of the discrete energy V are stable equilibria of thesimplified DAE dynamics.Open Problem: Explore computationally for N 10 whether there is a relation-ship between elliptic Fekete point sets and equilibria of the full DAE dynamics (2)that have large basins of attraction of initial conditions. 648 A. Jamieson-Lane et al.
Fig. 3 The evolution of a 10-spot pattern at different times for the Brusselator (2) when f D 0:5,E D 22, and D 0:02. (a) The initial state D 0. (b) D 1. (c) D 3. (d) The computed steadystate after a suitable rotation. The steady-state consists of two polar spots together with two parallelplanes of 4 equidistantly spaced spots. The relative phase-shift of the spots on the two planes is45ı . The distance d between the equator and either of the two planes is d 0:4234
4 Dynamics of Spots on the Sphere: The Gray-Scott Model
In this section we give a new result for the slow dynamics of spots on the unit spherefor the well-known Gray-Scott RD model (cf. [6]).
vt D "2 S v v C Buv 2 ; ut D D s u C .1 u/ uv 2 : (3)
Although the analysis of spot dynamics for this problem follows the methodologydone in [27] for the Brusselator model, this new analysis requires the reduced-wave Dynamics and Equilibria of Localized Spots on the Sphere 649
Green’s function G.xI / on the sphere satisfying
1 S G G D ı.x / ; (4a) D 1 G.xI / log jx j C R C o.1/ ; as x ! ; (4b) 2for some R independent of . In terms of this Green’s function we can derive thefollowing result for the slow dynamics of a collection of spots for the GS model.Principal Result 2 (Slow spot dynamics for the GS model) Let " ! 0, andassume that B D O."=/, where D 1= log ". Then, provided that there areno O.1/ time-scale instabilities of the quasi-equilibrium spot pattern, the time-dependent spot locations, xj for j D 1; : : : ; N, on the surface of the sphere vary onthe slow time-scale D "2 t, and satisfy the dynamics for j D 1; : : : ; N,
dxj X N D 2"2 j .Sj / I Q j Si rx G.xj I xi / ; Q j xj xTj ; (5a) d iD1 i¤j
coupled to the nonlinear constraints for S1 ; : : : ; SN in terms of x1 ; : : : ; xN given by
X N
B D Sj .1 C 2R/ C .Sj / C 2 Si G.xj I xi / ; j D 1; : : : ; k : (5b) iD1 i¤j
pwhere D 1= log " and B B"1 D D O.1/. In this system .Sj / and .Sj / depend on the core problem near the spot and arespecific to the GS model. Since the reduced-wave Green’s function and its regularpart R is not available in simple explicit form, and can only be written in terms ofthe Legendre function (see [24]), it is more challenging to investigate the dynamicsand equilibria of spot patterns on the sphere for the GS model. This topic requiresfurther investigation.
5 DAE Spot Dynamics in Complex Models: Open Problems
There are several possible extensions of the methodology for deriving and analyzinglocalized spot patterns for other scenarios. We remark that explicit DAE dynamics for spot patterns is only possible whenthe source-neutral Green’s function , satisfying s G D j˝j1 ı.xx0 / is explicitlyavailable. Such a Green’s function is well-known for the sphere, and this fact is keyto deriving (2). However, recently in [9], this Green’s function has been provided 650 A. Jamieson-Lane et al.
analytically for a particular class of surfaces of revolution. For this class, DAEdynamics on a manifold of varying curvature, and the possibility of pinning oflocalized spot, can be analyzed. The second possible extension of the modeling framework is to allow for spatialheterogeneity in the fuel supply, so that E D E.x/. The corresponding outer solutionv will involve a sum ofR Dirac distributions, one near each spot, together with a termvp of the form vp D ˝ E./ EN G.xI / d, where EN denotes the spatial averageof E over the sphere. In this way, it can be shown that the corresponding DAEdynamics will yield a nonlocal system for the evolution of the spots. Finally, a more intricate method to introduce spatial heterogeneity in (1) isto consider spot patterns on the sphere for a model that couples passive bulk-diffusion in the interior of the sphere to the Brusselator PDE on the surface. Inthis context, the fuel supply E represents an exchange, or flux, between the surface-bound concentrations and their bulk counterparts. This coupling should lead to richbehavior in the DAE dynamics for spots on the sphere. This paradigm of studyingcoupled surface-bulk models is becoming more prominent in scientific computationand in applications (cf. [14, 15]).
Acknowledgements PHT thanks Lincoln College, Oxford and the Zilkha Trust for generousfunding. MJW gratefully acknowledges grant support from NSERC.
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Matthew Fury, Beth Campbell Hetrick, and Walter Huddell
Abstract We prove regularization for certain ill-posed problems in Banach spaceby considering small changes in the model. The problems are of the form du=dt DAu; 0 t < T; u.0/ D where A is assumed to generate a uniformly boundedholomorphic semigroup fezA W <z 0g. Continuous dependence on modeling isestablished by considering an approximate problem where the operator A is replacedby an operator fˇ .A/, ˇ > 0 which approximates A in some sense as the parameter ˇtends to zero. The particular logarithmic approximation we apply originates from amodified quasi-reversibility method recently introduced by Boussetila and Rebbani.
1 Introduction
Ill-posed problems have been the focus of considerable attention over the last fewdecades. In this paper we focus on the abstract Cauchy problem
du D Au ; 0t<T ; (1) dt u.0/ D
where A is the infinitesimal generator of a uniformly bounded holomorphicsemigroup fS.z/ D ezA W <z 0g in a Banach space .X; k k/, 2 X, and uis a function u W Œ0; T ! X. This problem is generally ill-posed in that a solutionmay not exist, or if it does, it may not depend continuously on the data. To estimate
M. Fury ()Penn State Abington, 1600 Woodland Road, Abington, PA 19001, USAe-mail: [emailprotected]. Campbell HetrickGettysburg College, 300 North Washington Street, Gettysburg, PA 17325, USAe-mail: [emailprotected]. HuddellEastern University, 1300 Eagle Road, St. Davids, PA 19087, USAe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 653J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_59 654 M. Fury et al.
the solution, we introduce an approximate well-posed problem
dv D fˇ .A/v ; 0t<T ; (2) dt v.0/ D
where ˇ > 0 and fˇ .A/ approximates A. We prove continuous dependence onmodeling; i.e., that if fˇ .A/ is a suitable approximation of A, then a solution ofthe ill-posed problem (if it exists) is appropriately close to the solution of thewell-posed model problem. This approach, introduced by Lattes and Lions in[11], has been used with several different approximations fˇ . Examples includefˇ .A/ D A ˇA2 , used in [2, 3, 5, 11, 12, 14], and fˇ .A/ D A.I C ˇA/1 , asused in [3, 5, 10, 13]. However, both of these cases induce an error of order eC=ˇ ,which is problematic in extending results to the nonlinear case. In this paper we usea different approximation introduced by Boussetila and Rebbani [4]:
1 fˇ .A/ D ln.ˇ C epTA / ; ˇ>0; p1: (3) pT
Boussetila and Rebbani use this approximation in Hilbert space (see also [15] and[8]); Huang [9] extends the results to Banach space. In both cases, the authors usethe quasi-reversibility method: they use the solution to a well-posed model problemto approximate the initial value for the original problem, then solve the originalproblem backward. They show that the final value of the solution obtained in thismanner depends continuously on the initial data. Our work differs in that we provedirectly continuous dependence on the model. Specifically, we show that p ku.t/ vˇ .t/k C. ˇ/1 T M T ; t t 0t<T (4)
where u.t/ is an assumed solution of (1), vˇ .t/ is the solution of (2) using thelogarithmic function fˇ in (3), and C and M are constants independent of ˇ. The problems above share the same initial data, but we note that in practice theobserved data may differ. Thus we are interested in the approximate problem (2)with replaced by ı where ı > 0 yields a change in the initial data satisfyingk ı k ı. In this context, we directly obtain regularization results by applyingthe estimate (4). In Sect. 2, we introduce the functional calculus used for fˇ .A/ and prove severalauxiliary results. In Sect. 3, we use these results to prove continuous dependenceon modeling (Theorem 1). In Sect. 4, we prove regularization for the problem (1).Finally, Sect. 5 demonstrates the theory of this paper applied to certain partialdifferential equations. Below, B.X/ denotes the space of bounded linear operators onX, and .A/ the resolvent set of A. Also, a classical solution u.t/ of (1) is a functionu W Œ0; T ! X such that u.t/ 2 Dom.A/ for 0 < t < T, u 2 CŒ0; T \ C1 .0; T/, andu satisfies (1) in X. Continuous Dependence Using a Logarithmic Approximation 655
2 Approximation
In order to make sense of the operator fˇ .A/, we employ a functional calculus bydeLaubenfels [6]. By the fact that A generates fS.z/ D ezA W <z 0g, define thefunction Z 1 1 1 cos.sr/ irA G.s; A/ D e dr ; s 0 : 1 r2
It may be shown that G.s; A/ is a continuous function in s mapping into B.X/. Alsowhile kG.s; A/k s sup<z0 kezA k , by switching within equivalent norms, wemay assume without loss of generality that kG.s; A/k s for all s 0 (see [9]).From this emerges deLaubenfels’s functional calculus h i Z 1 f .A/ D lim f .t/ I C f 00 .s/G.s; A/ds (5) t!1 0
for f 2 ACr1 Œ0; 1/ WD fh ı g W h 2 AC1 Œ0; 1 g where g.t/ D .1 C t/1 andAC1 Œ0; 1 WD f f W f 0 exists and is absolutely continuous on Œ0; 1 g. In [9], Huangprovides a concise layout of applying (5) in order to find a formula for fˇ .A/ defined 1by (3). Inserting f .s/ D pT ln.ˇ C epTs / in (5) yields Z 1 1 ˇpTepTs fˇ .A/ D ln ˇ G.s; A/ds : pT 0 .ˇ C epTs /2
Also, Huang shows that 3 p fˇ .A/ is a bounded operator on X satisfying kfˇ .A/k pT ln ˇ for 0 < ˇ < . 5 1/=2.Lemma 1 Let A be the infinitesimal generator of a uniformly bounded p holomor-phic semigroup fS.z/ D ezA W <z 0g on X and let 0 < ˇ < . 5 1/=2.Then
8ˇ 1 k Ax C fˇ .A/xk kS .2pT/xk pT
for all x 2 Dom.S1 .2pT// D Ran.S.2pT//.Proof Note that in the statement of the lemma we have used implicitly the fact thatS.t/ is a bounded, injective operator for each t 0 (cf. [7, Lemma 3.1]). Now, definefor s 0,
1 1 h.s/ D .s ln.ˇ C epTs //e2pTs D ln.ˇepTs C 1/e2pTs : pT pT 656 M. Fury et al.
Using the fact that ln.x C 1/ x for x 0, we have ˇ 2 ˇ ˇ 4ˇ pTe2pTs C 3ˇpTepTs ˇ 2pTs jh00 .s/j D ˇˇ 4pT ln.ˇe pTs C 1/ ˇe ˇ .ˇepTs C 1/2 2 4ˇ pTe2pTs C 3ˇpTepTs C 4ˇpTe pTs e2pTs 8ˇpTepTs : .ˇepTs C 1/2
Then for x 2 Dom.S1 .2pT// D Ran.S.2pT// Dom.A/,
k Ax C fˇ .A/xk D k.A C fˇ .A//e2pTA S1 .2pT/xk Z 1 Dk h00 .s/G.s; A/S1 .2pT/x dsk 0 Z 1 8ˇ 1 8ˇpTs epTs kS1 .2pT/xk ds D kS .2pT/xk : 0 pT
t u In light of Lemma 1, for x 2 Dom.A/, define the operator gˇ .A/ in X by
gˇ .A/x D Ax C fˇ .A/x :
Proposition 1 Let A be the infinitesimal generator of a uniformly p boundedholomorphic semigroup fS.z/ D ezA W <z 0g on X and let 0 < ˇ < . 5 1/=2.Then gˇ .A/ generates a C0 semigroup fetgˇ .A/ gt0 on X satisfying tgˇ .A/ t pT ke k2 ln C1 for 0<tT: pT ˇt
Proof For s 0 and 0 < t T, define the function h.t; s/ D .ˇepTs C 1/t=pT . 1Also, let sˇ D pT ln. pT ˇt /. Then for 0 < t T and x 2 X,
Z 1 tgˇ .A/ @2 ke xk D k h.t; s/G.s; A/x dsk 0 @s2 Z 1 tpTˇe pTs . pTt ˇepTs 1/ Dk G.s; A/x dsk .ˇepTs C 1/ pT C2 t 0 Z sˇ tpTˇs epTs . pTt ˇepTs C 1/ Z 1 tpTˇs epTs . pTt ˇepTs 1/ t kxk ds C t kxk ds 0 .ˇepTs C 1/ pT C2 sˇ .ˇepTs C 1/ pT C2 Continuous Dependence Using a Logarithmic Approximation 657
! 2ˇepTsˇ .tsˇ C 1/ C 2 1 D t t kxk .ˇepTsˇ C 1/ pT C1 .ˇ C 1/ pT ! ln. pT ˇt / 1 t pT < 2kxk C 2kxk ln C1 : ˇt t t . pTt C 1/ pT C1 . pTt C 1/ pT pT
Finally, for any y 2 X, it may be shown by a dominated convergence argument that R1 2limt!0C 0 @s@ 2 f.h.t; s/ 1/epTs gG.s; A/y ds D 0. In other words, ketgˇ .A/ epTA y epTA yk ! 0 as t ! 0C for all y 2 X. The result
ketgˇ .A/ x xk ! 0 as t ! 0C for all x 2 X (6)
then follows from the density of Ran.epTA / in X together with the fact that pT CpT ln. ˇt / ! 0 as t ! 0 . t t uLemma 2 For sufficiently small ˇ > 0, r 2 t ketgˇ .A/ k p C1 for 0tT : ˇ pT pProof By Proposition 1 and the fact that x ln. 1x / x for x > 0, we have s ! r tgˇ .A/ 1 ˇt 2 t p 2 r t ke k2 C1 D p C ˇ p C1 ˇ pT ˇ pT ˇ pT
for 0 < t T. This estimate also holds for sufficiently small ˇ when t D 0 by (6).
3 Continuous Dependence on Modeling
The results proved in Sect. 2 will be used to establish continuous dependence onmodeling. For this, we assume u.t/ is a solution of (1) and let vˇ .t/ be the uniquesolution of (2). We extend these solutions into the complex plane and apply theThree Lines Theorem. Following [7], for " > 0 define a family of bounded operators Z 1 2 C" D e"w .w A/1 dw 2i
where is a complex contour contained within .A/, running from 1ei to 1eiwith 0 < < 2 . It may be shown that fC" g">0 is a strongly continuous holomorphicsemigroup on X generated by A2 . 658 M. Fury et al.
Lemma 3 Let " > 0 and let u.t/ be a classical solution of (1). Then
C" etfˇ .A/ D etgˇ .A/ C" u.t/ for all t 2 Œ0; T :
Proof The result follows from uniqueness of solutions to well-posed problems sinceeach item is a classical solution of (2) with initial data replaced by C" . t u We also require the following based on work of Agmon and Nirenberg [1].Lemma 4 ([1, p. 148]) Let .z/ be a complex function which is bounded andcontinuous on S D fz D C i j 2 Œ0; T ; 2 Rg. For ˛ D t C ir 2 S,define Z Z 1 1 1 ˚.˛/ D .z/ C dd : S z˛ zN C 1 C ˛
NThen ˚.˛/ is absolutely convergent, @˚.˛/ D .˛/ where @N denotes the Cauchy-Riemann operator, and there exist constants K > 0 and L > T such that Z ˇ ˇ ˇ 1 1 1 ˇ ˇ C ˇ d K 1 C ln L if ¤t: ˇ zN C 1 C ˛ ˇ 1 z ˛ j tj
Theorem 1 Let A be the infinitesimal generator of a uniformly bounded p holo-morphic semigroup fS.z/ D ezA W <z 0g on X and let 0 < ˇ < . 5 1/=2.Let u.t/ and vˇ .t/ respectively be classical solutions of (1) and (2) and assume thatu.t/ 2 Dom.S1 .2pT// D Ran.S.2pT// and kS1 .2pT/u.t/k M 0 for all t 2 Œ0; T .Then there exist constants C and M, each independent of ˇ, such that p t t ku.t/ vˇ .t/k C. ˇ/1 T M T for 0t<T:
Proof Define S to be the complex strip S D ft C ir j t 2 Œ0; T ; r 2 Rg. Since eirAis a bounded operator on X for every r 2 R, we may define for ˛ D t C ir 2 S,
" .˛/ D eirA C" .u.t/ vˇ .t// D eirA C" .u.t/ etfˇ .A/ / :
Also, following work of Agmon and Nirenberg [1], for ˛ D t C ir 2 S define Z Z 1 1 1 ˚" .˛/ D @" .z/ C dd ; S z˛ zN C 1 C ˛ where z D Ci 2 S and @N denotes the Cauchy-Riemann operator @N D 12 @t@ C i @r@ .In view of Lemma 4, we will show that @ N " .˛/ is bounded and continuous on S. Continuous Dependence Using a Logarithmic Approximation 659
Note, using the fact that Dom.AC" / D X (cf. [7, Propoisition 2.10]), we have @ d d " .˛/ D eirA C" u.t/ etfˇ .A/ D eirA C" .Au.t/ fˇ .A/etfˇ .A/ / ; @t dt dt
@ @ " .˛/ D eirA C" .u.t/ etfˇ .A/ / D eirA .iA/C" .u.t/ etfˇ .A/ / : @r @rTherefore,
N " .˛/ D eirA C" .Au.t/ fˇ .A/etfˇ .A/ / eirA AC" .u.t/ etfˇ .A/ / 2 @ D eirA .A fˇ .A//C" etfˇ .A/ :
Recall, fC" g">0 is a holomorphic semigroup satisfying C" x ! x as " ! 0 forevery x 2 X. Hence for small ", we may set C D sup0<"<1 kC" k. Also, set J Dsup<z0 kezA k. Then by Lemma 3, Lemma 2, Lemma 1, and stabilizing condition,
keirA .A fˇ .A//C" etfˇ .A/ k D keirA etgˇ .A/ .A fˇ .A//C" u.t/k r r 2 t 8ˇ 1 2 t 8ˇ Jp C1 kS .2pT/C" u.t/k J p C1 CM 0 ˇ pT pT ˇ pT pT
since C" commutes with A. Hence we have shown p N " .˛/k C0 ˇ k@ (7)
where C0 is a constant independent of ˇ, ", and ˛. Thus, @ N " .˛/ is bounded on S. N " .˛/ is continuous on S. Hence, by Lemma 4, ˚" .˛/It is also readily shown that @is absolutely convergent, @˚N " .˛/ D @ N " .˛/ and there exist constants K > 0 andL > T such that Z 1ˇ ˇ ˇ 1 1 ˇ ˇ C ˇ d K 1 C ln L if ¤ t : ˇ zN C 1 C ˛ ˇ 1 z ˛ j tj
Next, define w" W S ! C by
w" .˛/ D x " .˛/ x ˚" .˛/ ;
where x is in the dual space of X. We will show that the Three Lines Theorem maybe applied to w" . First for ˛ D t C ir 2 S, using standard properties of semigroups 660 M. Fury et al.
and our stabilizing assumptions, we have
k" .˛/k JkC" u.t/ C" etfˇ .A/ k D JkC" u.t/ etgˇ .A/ C" u.t/k Z t tgˇ .A/ d sgˇ .A/ D Jk.I e /C" u.t/k D Jk .e C" u.t//dsk 0 ds Z t Z t r 2 s 8ˇ D Jk esgˇ .A/ gˇ .A/C" u.t/dsk J p C1 CM 0 ds (8) 0 0 ˇ pT pT
showing that " .˛/ is bounded on S. Next, for z D C i 2 S, by Lemma 4 and (7), Z Z ˇ ˇ 1 1 T 0 p ˇˇ 1 1 ˇ ˇ dd k˚" .˛/k C ˇˇ C 1 0 z˛ zN C 1 C ˛ ˇ Z p Z T 1 T 0p L e L C ˇ K 1 C ln d D K ˇ 1 C ln d (9) 0 j tj 0 j tj
where eK is a constant. It follows that w" is bounded on S. Furthermore, w" iscontinuous on S as well, and by Lemma 4, for ˛ in the interior of S, @w N " .˛/ D N Nx @" .˛/ x @˚" .˛/ D 0 showing that w" is analytic on the interior of S. Therefore, by the Three Lines Theorem,
jw" .t/j M.0/1 T M.T/ T t t
for 0 t T, where M.t/ D supr2R jw" .t C ir/j. Looking at the sides of the stripS, we have from (9),
M.0/ D sup jw" .ir/j kx kk" .ir/k C kx kk˚" .ir/k r2R
D kx kkeirA .u.0/ vˇ .0//k C kx kk˚" .ir/k D kx kk˚" .ir/k p Z T e L K ˇkx k 1 C ln d : 0
Next by (8), we have k" .T C ir/k J 0 where J 0 is a constant independent of ˇ, ",and r since ˇ < 1. Hence, together with (9) we have that
M.T/ D sup jw" .T C ir/j kx kk" .T C ir/k C kx kk˚" .T C ir/k r2R Z T L kx k J 0 C e K 1 C ln d 0 j tj Continuous Dependence Using a Logarithmic Approximation 661
where again we have used the fact that ˇ < 1. Therefore p Z T 1 Tt Z T Tt L Ljw" .t/j kx k e K ˇ 1C ln d J Ce 0 K 1C ln d : 0 0 j tj
Taking the supremum over all x 2 X with kx k 1, we have h p i1 Tt t p k" .t/ ˚" .t/k e J0 C e t t K ˇM1 KM2 T D C. ˇ/1 T M T
where C and M are constants independent of both ˇ and ". Again using (9),
kC" .u.t/ vˇ .t//k D k" .t/k k" .t/ ˚" .t/k C k˚" .t/k p p Z T L C. ˇ/1 T M T C e t t K ˇ 1 C ln d 0 j tj p C. ˇ/1 T M T t t
for a possibly different value of C still independent of ˇ. Here the bound on the rightis independent of ", so we let " ! 0 to obtain p t t ku.t/ vˇ .t/k C. ˇ/1 T M T :
4 Regularization for Problem (1)
We prove regularization for problem (1) using our estimate from Theorem 1.Definition 1 [10, Definition 3.1] A family fRˇ .t/ j ˇ > 0; t 2 Œ0; T g B.X/ iscalled a family of regularizing operators for the problem (1) if for each solution u.t/of (1) with initial data and for any ı > 0, there exists ˇ.ı/ > 0 such that1. ˇ.ı/ ! 0 as ı ! 0 ,2. ku.t/ Rˇ .t/ı k ! 0 as ı ! 0 for each t 2 Œ0; T whenever k ı k ı .Theorem 2 Let A be the infinitesimal generator of a uniformly bounded holomor-phic semigroup fS.z/ D ezA W p <z 0g on X and let fˇ .A/ be defined by (3). Then tfˇ .A/fRˇ .t/ WD e j 0 < ˇ < . 5 1/=2; t 2 Œ0; T g is a family of regularizingoperators for problem (1).Proof Let u.t/ be a classical solution of (1) satisfying kS1 .2pT/u.t/k M 0 for allt 2 Œ0; T and let k ı k ı. Note that since fˇ .A/ 2 B.X/, etfˇ .A/ satisfies 3t 3t ketfˇ .A/ k etkfˇ .A/k e pT ln ˇ D ˇ pT pfor all t 2 Œ0; T , for 0 < ˇ < . 5 1/=2. 662 M. Fury et al.
p First let t 2 Œ0; T/ and choose ˇ D ı 6 . Then ˇ ! 0 as ı ! 0 and fromTheorem 1,
ku.t/ Rˇ .t/ı k ku.t/ etfˇ .A/ k C ketfˇ .A/ etfˇ .A/ ı k D ku.t/ vˇ .t/k C ketfˇ .A/ . ı /k p 3t C. ˇ/1 T M T C ˇ pT ı t t
p t t t D C.ı 12 /1 T M T C ı 1 2T ! 0 as ı ! 0 : p In the case t D T, from (8) and (9) the estimate ku.T/ vˇ .T/k N ˇ caneasily be p obtained where N is a constant independent of ˇ. Then as above, still withˇ D ı6, p 3 p p ku.T/ Rˇ .T/ı k N ˇ C ˇ p ı D Nı 12 C ı ! 0 as ı ! 0 :
5 Example
For an example, we use that of deLaubenfels in [6]. Let T be the unit circle inthe complex plane and for 1 p 1, let H p .T/ be the set of all functions inLp .T/ that can be extended to a holomorphic function on the unit disc. Define A DH 1 .T/ \ C.T/ and let A D i d d d . Then iA D d is the generator of the familiartranslation group on A
.eirA /.ei / D .ei. Cr/ / ; r2R
for 2 H 1 .T/. This group extends to a uniformly bounded holomorphic semigroupfezA W <z 0g as shown in [6]. In this setting, (1) becomes the ill-posed partial differential equation
ut D iu ; 0t<T ; u.e ; 0/ D .e / : i i
By Theorem 2, Rˇ .t/ D etfˇ .A/ D .ˇ C epTA / pT . If 2 A , deLaubenfels in [6] t
shows G.s; A/ in this case to be given by
Œs X G.s; A/ D .s k/.k/g O k kD0 Continuous Dependence Using a Logarithmic Approximation 663
1 Rwhere gk .ei / D eik and .k/ O D 2 .ei /gk .ei /d. Defining the function t pTs pTh.s/ D .ˇCe / and employing the functional calculus (5), we find for 2 A Z 1 pT tRˇ .t/ D h.A/ D ˇ C h00 .s/G.s; A/ds 0 Z Œs t 1 tpTepTs epTs .t C pT/ X D ˇ pT C 1 .s k/.k/g O k ds : pT.ˇ C epTs / kD0 t 0 .ˇ C epTs / PT C1
Acknowledgements The authors would like to thank Rhonda J. Hughes for her guidance and alsothe 2015 AMMCS-CAIMS Congress.
References
1. Agmon, S., Nirenberg, L.: Properties of solutions of ordinary differential equations in Banach space. Commun. Pure Appl. Math. 16, 121–151 (1963) 2. Ames, K.A.: On the comparison of solutions of related properly and improperly posed Cauchy problems for first order operator equations. SIAM J. Math. Anal. 13, 594–606 (1982) 3. Ames, K.A., Hughes, R.J.: Structural stability for ill-posed problems in Banach space. Semigroup Forum 70, 127–145 (2005) 4. Boussetila, N., Rebbani, F.: A modified quasi-reversibility method for a class of ill-posed Cauchy problems. Georgian Math. J. 14, 627–642 (2007) 5. Campbell Hetrick, B.M., Hughes, R.J.: Continuous dependence results for inhom*ogeneous ill- posed problems in Banach space. J. Math. Anal. Appl. 331, 342–357 (2007) 6. deLaubenfels, R.: Functional calculus for generators of uniformly bounded holomorphic semigroups. Semigroup Forum 38, 91–103 (1989) 7. deLaubenfels, R.: Entire solutions of the abstract Cauchy problem. Semigroup Forum 42, 83– 105 (1991) 8. Fury, M.: Modified quasi-reversibility method for nonautonomous semilinear problems, Ninth Mississippi state conference on differential equations and computational simulations. Electron. J. Differ. Equ. Conf. 20, 99–121 (2013) 9. Huang, Y.: Modified quasi-reversibility method for final value problems in Banach spaces. J. Math. Anal. Appl. 340, 757–769 (2008)10. Huang, Y., Zheng, Q.: Regularization for a class of ill-posed Cauchy problems. Proc. Am. Math. Soc. 133, 3005–3012 (2005)11. Lattes, R., Lions J.L.: The Method of Quasi-Reversibility, Applications to Partial Differential Equations. American Elsevier, New York (1969)12. Miller, K.: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well- posed problems. In: Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Lecture Notes in Mathematics, vol. 316, pp. 161–176. Springer, Berlin (1973)13. Showalter, R.E.: The final value problem for evolution equations. J. Math. Anal. Appl. 47, 563–572 (1974)14. Trong, D.D., Tuan, N.H.: Regularization and error estimates for non hom*ogeneous backward heat problems. Electron. J. Differ. Equ. 4, 1–10 (2006)15. Trong, D.D., Tuan, N.H.: Stabilized quasi-reversibility method for a class of nonlinear ill- posed problems. Electron. J. Differ. Equ. 84, 1–12 (2008) Solving Differential-Algebraic Equationsby Selecting Universal Dummy Derivatives
Ross McKenzie and John D. Pryce
Abstract A common way of making a high index DAE amenable to numericalsolution is that of index reduction. A classical way of reducing a DAE’s index is thedummy derivative method of Mattsson and Söderlind, however for many problemsthis method only provides a local index 1 DAE. Using the Signature Matrix basedstructural analysis of Pryce to inform the dummy derivative method we present away to make this reduction global, where instead of picking new dummy derivativesat run time and thus changing the overall structure of the problem you instead haveto update a list of parameters.
1 Introduction
We consider DAEs of the form:
fi .t; the xj and derivatives of them/; i D 1; : : : ; n (1)
where xj .t/, j D 1; : : : ; n are state variables and functions of some independent vari-able t, usually considered to be time. Usually to solve such a problem differentialsof some of the equations are added to the system, these equations are called thehidden constraints. A DAE has an associated index, mainly we are concerned withthe differential index, which is the minimum number of differentiations requiredto obtain an ODE. Generally speaking the higher the index the more difficultthe numerical solution—there exist good solvers for index 1 systems and as suchwe would like to reduce any DAE to an equivalent (same solution set) index 1formulation. We are interested in reducing the index of our DAE via the DummyDerivative (henceforth DD method) method introduced in [4] by using the structuralanalysis of the Signature Matrix method introduced in [2] (henceforth SA) to
R. McKenzie () • J.D. PryceSchool of Mathematics, Cardiff University, Cardiff, CF24 4AG, Wales, UKe-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 665J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_60 666 R. McKenzie and J.D. Pryce
identify the hidden constraints and otherwise inform the DD method more than ispossible than in the classical approach of DDs via the Pantelides algorithm [1]. TheDD method introduces a choice of derivatives of variables to be considered algebraicso that our enlarged system containing the hidden constraints is made square. Thisprocess is inherently local because it relies on the non-singularity of potentiallydynamic matrices. We show in Sect. 2 how to pick a global index 1 system for asimple example, in Sect. 3 we demonstrate how this approach extends to any DAEon which Pryce’s SA works and in Sect. 4 we present numerical results.
2 Avoiding Dummy Pivoting for a Simple Example
Example 1 Consider the infamous simple pendulum:
s x- 9 @@ f1 .t/ D x00 .t/ C .t/x.t/ D 0> > @ y = 00 @ ? f2 .t/ D y .t/ C .t/y.t/ g D0 @~ (2) > > 2 2 2 ; f3 .t/ D x .t/ C y .t/ L D0 Pendulum bob, mass=1
where g and L are gravitational acceleration and length of the rod respectively. Theproblem’s signature matrix as specified by the Signature Matrix Method [2] is:
x y ci f1 2 1 0ı ! 0 f 1 2ı 0 0 ˙D 2 f3 0ı 0 1 2 dj 2 2 0
Note: there are two highest value transversals (henceforth HVTs) for the simplependulum, marked by and ı in the signature matrix above. The offsets c and d tellus how many times to differentiate each equation and the associated derivative orderfor the variables that will be found. We now proceed to carry out the DD algorithmin [4] but note whilst the matrices we use are the same the numbering is changedso as to line up with the SA notation. The DD algorithm proceeds in stages, usingmatrices GŒ and H Œ for D 0; 1; : : : where GŒ0 is the n n system Jacobian .c / .d /J D @fi i =@xj j . Deleting appropriate rows of GŒ gives H Œ . Deleting appropriatecolumns of H Œ to form a nonsingular matrix gives GŒC1 . This example illustrates Solving DAEs by Selecting Universal Dummy Derivatives 667
the process; for space reasons we refer to [4] for details. We have an initial Jacobian(with highest order equations and variables) and a secondary non-square Jacobian:
x00 y00 ci f1 1 0 x! 0 00 00 Œ0 f2 0 1 y 0 Œ0 x y G D 00 and H D f300 2x 2y 0 : f3 2x 2y 0 2 dj 2 2 0
We have to select 1 column from H Œ0 to get a square non-singular matrix GŒ1 .We cannot choose column 3 because it is structurally 0, as is always the casewith columns corresponding to non-differentiated variables. We thus either choosecolumn 1 or column 2. Consider instead that we want to eliminate the need to choosebetween any variables and instead pick both, since any choice may become invalid(due to a subsequent G matrix becoming locally ill conditioned, i.e. as x or y ! 0)and it will be expensive to pivot between potential systems as this happens for ageneral DAE. If we want to eliminate the choice of candidate dummy derivatives atthis stage we will need to add equations to the DAE so that we can choose a squaresubmatrix of H Œ0 that contains the columns corresponding to all candidate dummyderivatives, i.e. x and y. This is achieved by adding an equation of form:
Z1 WD ˛x0 C ˇy0 z1 D 0 (3)
to the original DAE, where ˛ and ˇ are some parameters that will be chosen atrun time so the row using new equations (newly introduced equations of the formZi D 0) in GŒ1 is roughly orthogonal to the row using old equations (equationspart of the original DAE including hidden constraints) in GŒ1 and z1 is some newvariable to solve for. We would like the H Œ0 for our new DAE to be of the form: 00 00 0 x y z1 f3002x 2y 0 0 H Œ0 D Z10˛ ˇ 0 1
so that it is now possible to pick x00 and y00 as DDs for all time and update ˛ and ˇalong the solution to keep the resulting GŒ1 matrix well-conditioned. Note: checkingthe condition number of the corresponding G matrix and using a Gram-Schmidt orQR type procedure for ‘new’ rows if the matrix becomes ill conditioned would be areasonable way of updating ˛ and ˇ dynamically. Consider however the new DAEs 668 R. McKenzie and J.D. Pryce
signature matrix and canonical (element-wise minimum) offsets:
x y z1 ci f1 0 2 1 0ı 11 0 f2 B1 2ı 0 1C 0 ˙ D f3 @ 0ı 0 1 1A 2 Z1 1 1 1 0 0 dj 2 2 0 0
where the block structure is highlighted, see [3]. Unfortunately we see that entriesin positions .4; 1/ and .4; 2/ are both structurally zero (meaning dj ci ¤ i;j ), sowon’t appear in H Œ0 , see [5]. There is a key inequality that comes from the SA forfinding the offsets c and d:
dj ci i;j with equality on a HVT. (4)
Noting this one can see that changing d4 D c4 (because entry .4; 4/ must be ona HVT) to 1 will not affect the other offsets. Due to our choice of Z1 we see thatd4 D c4 D 1 makes the entries in positions .4; 1/ and .4; 2/ structurally non zero. Ifwe write down H Œ1 for this new system (i.e. the DAE using equations f1 ; f2 ; f3 andZ1 ) we get: 0 0 Œ1 x y H D f30 2x 2y Iagain we add an equation to the system so that we can always choose the variablesthat are DD candidates at this stage:
Z2 WD x C ıy z2 D 0: (5)
So that our new DAE’s signature matrix with canonical offsets is:
x y z1 z2 ci f1 0 2 1 0ı 1 11 0 f2 B1 2ı 0 1 1C 0 B ı C 0 0 ˙ D 3B f 1 1 1C 2 @ A Z1 1 1 1 0 1 0 Z2 0 0 1 1 0 0 dj 2 2 0 0 0
Now to have all necessary entries of H Œ0 and H Œ1 structurally non zero we need toset c4 D d4 D 1 and c5 D d5 D 2. Note that adding these equations to the systemdoes not change the original solution of the DAE, because the equations we addform a new block dependent on the original DAE’s (block 1 above) solution. If we Solving DAEs by Selecting Universal Dummy Derivatives 669
now consider the DD stages of our new system we get:
x00 y00 z01 z002 ci f300 2x 2y 0 0 0 !2 Z10 ˛ ˇ 0 1 0 1 H Œ0 D Z200 ı 0 0 1 2 dj 2 2 0 1 2
We see that a valid choice is x00 , y00 and z002 —what we’re doing is adding DDs for allvariables in an old block that doesn’t contain an equation introduced at this stagein the original system and ‘borrowing’ DDs from the new blocks that do use anequation introduced at this stage in the original system. We’re forcing ourselves into a scheme that is unobtainable by performing a block decomposition of the newDAE and doing DDs on each block. When proceeding to find H Œ1 we see that theequation introduced for stage 1 is removed and we select x0 and y0 as DDs. Now note,that since the second equation we introduced is almost the antiderivative of the firstwe can condense both in to only one additional equation and finish with a DAE thathas the following signature matrix and valid offsets and now only selects x00 , x0 , y00and y0 as DDs. The reason this is possible to do is perhaps not clear at first glance(since whilst such a reduction is clearly structurally valid it may fail numerically).Note that due to Griewank’s Lemma [2] our new DAE’s DD Jacobians GŒ1 and GŒ2 are equal, so if a set of parameters works for stage 1 it will also work for stage 2.
x y z2 ci f1 0 2 1 0ı 11 0 f2 B1 2ı 0 1C 0 ˙ D f3 @ 0ı 0 1 1A 2 Z2 0 0 1 0 2 dj 2 2 0 2
3 Finding ‘Universal’ Dummy Derivatives in General
Before giving a general algorithm for adding such equations to the system we needthe following definition:Definition 1 Let the m and n be the number of rows and columns in the DDmatrix H Œ1 respectively.We have the following algorithm to find a static selection of states (Note: we onlysuggest to carry out the algorithm below in practice if a static selection cannotalready be found.): 670 R. McKenzie and J.D. Pryce
Algorithm 1 The ‘Universal’ Dummy Derivative Algorithm 1: When finding the matrix GŒ : 2: if m D n 3: Proceed as normal 4: else 5: Create a vector say, candidatelist, of variables occurring in H Œ1 ; : : : 6: with structurally non zero columns, to order dj 7: S Dsize(candidatelist)m 8: for j D 1; : : : ; S (using a new vector, parameterlist, in each equation) 9: Add an equation to the DAE of the form: Psize.candidatelist/10: iD1 candidatelist.i/ parameterlist.i/ vj D 011: Set each new equation’s offsets equal to 12: Add DDs to the system for the differentials of entries in candidatelist13: if > 114: for j D 1; : : : ; S .2/ .max ci /15: Add DDs to the system for vj ; : : : ; vj16: Form the now non-square GŒ comprising of all rows for variables in candidatelist17: Proceed with DD algorithm, i.e. form H Œ and proceed to the next stage18: Tidy up final system:19: Check if any equation introduced is the antiderivative (barring new variables and parameters) of one introduced at a later stage.20: Remove all such equations and any corresponding new variables and DDs from the system.
Note: The final check at the end is not needed to keep a static selection of dummyderivatives, but can reduce the size of the resulting index 1 problem, as seen inSect. 2. Note also line 15 where we introduce extra dummy derivatives for the newvariables—we need the new equations at some stage but we do not need them atprevious stages and will instead solve their block based DD system at those stages,which by construction is always square. Let us consider the following example:Example 2 Consider a problem with signature and initial H matrix:
x1 x2 x3 x4 x5 c i f1 0 2 111 x000 1 x002 x003 x004 x005 f2 B 2 2 C0 f10 0 0 0 0 1 B C f3 B 1 0 C2 f300 B 0 0 0C ˙D @ A and H Œ0 D @ A f4 0 0 2 f400 0 0 0 f5 1 1 1 f50 0 0 0 dj 3 2 2 2 2
In H Œ0 is being used to denote a structurally non-zero entry. We have
candidatelist D .x001 ; x02 ; x03 ; x04 ; x05 / (6) Solving DAEs by Selecting Universal Dummy Derivatives 671
and S D 1. This makes sense: there are only two possible choices of GŒ1 . Allvariables that appear in a GŒ1 are also in candidatelist, either we have
x001 x02 x03 x04 x001 x03 x04 x05 0 f1 0 0 01 f1 0 0 0 1 f0 0 0C f30 B 0 0C D 30 B GŒ1 @ A or GŒ1 D 0@ A: f4 0 0 f4 0 0 f5 0 0 0 f5 0 0 0
We add an equation of the form:
Z1 WD ˛x001 C ˇx02 C x03 C ıx04 C x05 v1 D 0 (7)
to the DAE and choose x000 00 00 00 00 1 , x2 , x3 , x4 and x5 to be ‘Universal’ DDs, setting d6 Dc6 D 1. We form the following non square Jacobian:
x001 x02 x03 x04 x05 0 f1 0 0 0 1 00 0 0 0 0 x1 x2 x3 x4 x5 f30 B 0 0 0C f30 0 0 0 GŒ1 D 0@ H Œ1 D 0A and get f4 0 0 f40 0 0 0 f5 0 0 0
again, this system is not square, we have
candidatelist D .x01 ; x3 ; x4 / (8)
and S D 1 again. We add an equation of the form:
Z2 WD x01 C x3 C x4 vv1 D 0 (9)
to the DAE and choose x001 , x03 , x04 and vv100 to be ‘Universal’ DDs, setting d7 Dc7 D 2. Let us just confirm our choice of DDs is valid (given the potential need to updateour parameters to keep any structurally non singular G matrix numerically nonsingular). Our enlarged system has signature matrix and initial H matrix:
x1 x2 x3 x4 x5 v1 vv1 ci f1 0 2 1 11 x000 1 x002 x003 x004 x005 v10 vv100 f2 B 2 2 C0 f10 0 0 0 0 0 0 1 B C f3 B 1 0 C2 f300 B 0 0 0 0 0 C B C B C 0 0 f400 B 0 0 0 0 0 C˙ D 4B C2 f B C and H Œ0 D 0 B C: f5 B 1 1 C1 f5 B 0 0 0 0 0 C @ A @ A Z1 2 1 1 1 1 0 1 Z10 ˛ ˇ ı 1 0 00 Z2 1 0 0 0 2 Z2 0 0 0 1 dj 3 2 2 2 2 1 2 672 R. McKenzie and J.D. Pryce
Where as expected a valid GŒ1 is:
x001 x02 x03 x04 x05 vv10 f1 0 0 0 0 0 1 x001 x02 x03 x04 x05 vv10 f30 B 0 0 0 0 C 0 ! B C f30 0 0 0 f0 0 0 0 0 C D 4B GŒ1 B C yielding H Œ1 D f40 0 0 0 0 f5 B 0 0 0 0 C Z20 0 0 Z1 @ ˛ 0 A 1 ˇ ı Z20 0 0 1
again, as expected a valid choice for GŒ2 is:
x01 x3 x4 f3 0! GŒ2 D f4 0 Z2
where the algorithm terminates. One may think a way to tidy up the above system would be to increase c6 Dd6 D 2 and remove the final equation and variable. Structurally this will still work,but numerically we may fall in to trouble as GŒ2 may be singular.We now go about proving the above algorithm provides a valid DD scheme withsame solution as the original problem. First we need the following definition:Definition 2 (Transversals) A transversal for an m n matrix, with m n is a setof m tuples .i; j/ so that i 2 f1; : : : ; mg and j 2 f1; : : : ; ng where no i or j is repeated.Theorem 1 If a variable’s derivative does not appear in any potential DD schemethe corresponding entries in the H matrix are all 0.This result is easily derived from the following:Theorem 2 Given any rectangular matrix with more columns than rows, and ofstructural full row rank every non empty column contains an element of sometransversal.Proof Assume for the sake of contradiction there exists a non empty column thatdoes not contain any elements that belong to a transversal. Without loss of generalityrearrange so that a non 0 entry is in the bottom right corner, say position .m ; n /.Since we have a valid DD scheme at the stage (i.e. the system is of full row rank) weknow there exists some transversal, T say. This transversal contains some element.m ; J/, for some J 2 f1; : : : ; n 1g. We can therefore form a new transversal ofthe form fT .m ; J/g [ .m ; n /. Solving DAEs by Selecting Universal Dummy Derivatives 673
Theorem 1 is why we consider all non empty columns in our candidatelist. We nowseek to justify our removal of equations in the last part of the algorithm above.Theorem 3 If a scheme produced by the above algorithm yields equations that canbe removed the solution is the same as the scheme without removing those equations.Proof Structurally it is clear that such removal still leaves us with the same potentialchoice of DDs, since if such a removal is possible we will have (at least) structurallyidentical equations at some stage , (at least one) of which is being treated asits own block system until its new variable is needed to make a square G matrix.Numerically if such a reduction is possible then the size of the candidatelist vectormust remain unchanged. Since we only remove equations in our DD stages we havetwo possibilities, either the difference between the size of candidatelist and m hasstayed the same, or it has increased. If it has stayed the same then the proposedremoval of equations gives us the same structural matrix, and by Griewank’s Lemmathe same numerical matrix. If it has reduced we will be adding new equationsthat are not the antiderivatives of equations previously introduced. In which casewe can still make our corresponding G matrix non-singular by varying only thenew parameters introduced at this stage, since we know the rows are linearlyindependent.Note: The above proof yields a necessary condition for introducing equations thatcan be removed that could shorten our algorithm. If at some stage ci > 1 foreach i considered at that stage then at the next stage we must introduce equationswhich are the ‘antiderivatives’ of the ones introduced at this stage. We could shortenAlgorithm 1 by introducing such a condition. Finally we have the following:Theorem 4 The above algorithm provides an always static selection of DDs thatis always valid (provided one chooses suitable parameters throughout integration)and has same solution as the original DAE.Proof Theorems 1 and 3 give us that we have a valid DD scheme, hence all thatis left to prove is that such equation additions do not change the solution set. Asillustrated in above example the inclusion of additional equations in the describedmanner is equivalent to adding more dependent blocks to the system, which will notchange the original block’s solution.Note: ‘same solution’ in the above Theorem may be confusing on a first read sincethe reformulated DAE is of a larger size. We mean that the value of any xi .t/ in theoriginal DAE is also the value of that same xi .t/ in the reformulated DAE. 674 R. McKenzie and J.D. Pryce
4 Preliminary Numerical Results for Universal Dummy Derivatives
We solve the simple pendulum index 1 universal DD reformulation given below(where xd denotes the DD corresponding to selecting x0 as a DD in the algorithmabove, xdd denotes x00 and similarly for y): 9 f1 .t/ D xdd .t/ C .t/x.t/ D 0> > > > f2 .t/ D ydd .t/ C .t/y.t/ g D 0> > > > > > > f3 .t/ D x2 .t/ C y2 .t/ L2 D 0> > > > > > Z2 .t/ D x.t/ C ıy.t/ z2 .t/ D 0= (10) f30 .t/ D 2x.t/xd .t/ C 2y.t/yd .t/ D 0> > > > > > > Z20 .t/ D xd .t/ C ıyd .t/ z02 .t/ D 0> > > > > f300 .t/ D 2x.t/xdd .t/ C 2x2d .t/ C 2y.t/ydd .t/ C 2y2d .t/ D 0> > > > > ; Z200 .t/ D xdd .t/ C ıydd .t/ z002 .t/ D0
in MATLAB using ode45 (using variable step size with initial conditions x D 6,y D 8, x0 D y0 D 0 and parameters L D 10 and G D 9:81) by reformulatingthe problem as an ODE in z2 and z02 and switching parameters whenever theangle between .x; y/ and .; ı/ becomes small. We compare the solution with oneproduced by DAETS [6], an accurate order 30 Taylor series solution, see Fig. 1. We briefly present some information on switching the system (changing andˇ), a full discussion is left to a future work. If we choose our switching condition theangle between .; ı/ and .x; y/ being less that =4, i.e. trying to keep the G matriceswell conditioned then the change in energy from t D 0 to t D 100 grows to around105 . If we instead switch after every time to step to a new .; ı/ orthogonal to .x; y/this change is greatly reduced to around 1012 . One could also attempt to solve thenew ‘Universal’ DD system by coming up with a continuous choice of parametersso that the G matrices are globally non-singular, however in general this will likelybe difficult to achieve so we do not present results for doing this with the simplependulum. Solving DAEs by Selecting Universal Dummy Derivatives 675
−6 10
−8 10
−10 10
−12 10
−14 10 x y
−16 10 0 50 100 150 200 250 300 350 400 450 500 Time
Fig. 1 Difference between the solution of the ‘Universal’ Dummy Derivative index 1 formulationof the simple pendulum solved in MATLAB by reformulating to an ODE and using ode45 andthe solution to the original index 3 formulation solved via an order 30 Taylor Series method usingDAETS
5 Conclusion
We have given and demonstrated by examples a way of finding dummy derivativessuch that one does not have to change the structure of the reduced index 1 problemwhen solving high index DAEs. Instead we have changed the problem from oneof structural pivoting of variables to numerical switching of parameters, which webelieve to be much easier for a general DAE code to handle, since one can exploitunderlying structures of the index 1 formulation throughout a simulation. In a futurework we aim to more formally develop a good switching condition for an arbitraryDAE or discover a way of choosing continuous parameters for an arbitrary DAE.
References
1. Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9, 213–231 (1988)2. Pryce, J.D.: A simple structural analysis method for DAEs. BIT Numer. Math. 41(2), 364–394 (2001)3. Pryce, J.D., Nedialkov, N.S., Tan, G.: DAESA—a matlab tool for structural analysis of differential-algebraic equations: theory. ACM TOMS 41(2), 1–20 (2015) 676 R. McKenzie and J.D. Pryce
4. Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), 677–692 (1993)5. McKenzie, R., Pryce, J.D.: Structural analysis and dummy derivatives: some relations. In: Cojo- caru, M.G. (ed.) Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, pp. 293–299. Springer, Cham (2015)6. Nedialkov, N.S., Pryce, J.D.: DAETS User Guide. 2008–2009 On a Topological Obstruction in the ReachControl Problem
Melkior Ornik and Mireille E. Broucke
Abstract This paper explores aspects of the Reach Control Problem (RCP) to drivethe states of an affine control system to a facet of a simplex without first exiting fromother facets. In analogy with the problem of nonlinear feedback stabilization, weinvestigate a topological obstruction that arises in solving the RCP by continuousstate feedback. The problem is fully solved in this paper for the case of two andthree dimensions.
1 Problem Statement
This paper studies a topological obstruction that arises in solving the Reach ControlProblem (RCP) using continuous state feedback. We consider a simplex S WDcofv0 ; : : : ; vn g with vertices fv0 ; : : : ; vn g and facets fF0 ; : : : ; Fn g. Each facet isindexed according to the vertex it does not contain. Facet F0 is called the exit facet.Let hj be the normal vector to facet Fj pointing outside S. Define I D f1; : : : ; ng. LetI.x/ f0; 1; : : : ; ng be the minimal set of indices such that x 2 cofvi ; j i 2 I.x/g.That is, x is in the interior of cofvi j i 2 I.x/g. We consider the affine control system on S
xP D Ax C Bu C a ; (1)
where x 2 Rn and u 2 Rm where 1 m < n. Let B WD Im.B/ and O WD fx 2Rn j Ax C a 2 Bg. Let u .t; x0 / denote the trajectory of (1) starting at x0 under afeedback u.x/. The problem is to find a state feedback u.x/ such that all trajectoriesu .; x0 / starting in S exit S through F0 in finite time without first leaving S. TheRCP has been extensively studied [4–6]. The purpose of this paper is to announcea topological obstruction in the RCP, paralleling the analogous problem arising inthe problem of continuous state feedback stabilization [3], and to give preliminaryresults on the problem for low dimensional systems.
M. Ornik () • M.E. BrouckeDepartment of Electrical & Computer Engineering, University of Toronto, Toronto,ON M5S 3G4, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 677J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_61 678 M. Ornik and M.E. Broucke
A parallel study of the same problem was made in [10] (some preliminary workappeared in [8]). The contributions of this paper significantly differ from [10]. Thispaper primarily uses retraction theory to study the case of dim.OS / D n 1.This leads to a simple solution in low dimensions of n. Studies of low-dimensionalsystems are prevalent [1]. On the other hand, [10] uses hom*otopy theory to study thecase of dim.B/ D 2. While the conclusions of [10] could also potentially lead to asolution in low dimensions of n, this situation is not explored in [10]. Furthermore,the elegant cone condition for the topological obstruction developed in this paper,B \ cone.OS / D 0, does not make an appearance at all in [10]; its place is taken bya more involved result based on null-hom*otopic maps on a circle. A supplement to this paper is found in [11] where we present supporting resultsand a study of the case of an obstruction using affine feedback. The results of thispaper for the case of n D 2; 3, Theorems 3 and 4, could be obtained from [11].However, while it is tempting to forgo more advanced topological methods in favourof the brute force linear algebra arguments in [11], we insist on the importance ofthe topological approach in order to have a hope of generalizing the results to higherdimensions. For each x 2 S, we define the cone
C.x/ D fy 2 Rn j hj y 0; j 2 InI.x/g : (2)
In other words, C.x/ is the set of all vectors y which, when attached at x, point intoS or through the exit facet F0 . (We note that if x 2 Int.S/, C.x/ D Rn .) In orderfor the trajectory u .t; x0 / to not leave S through any facet except the exit facet, werequire [6]:
du D Ax C Bu.x/ C a 2 C.x/ ; x2S: (3) dt
In addition to this necessary condition, if u.x/ solves the RCP then there are noclosed-loop equilibria in S. The equilibria of an affine system can only lie in theaffine space O, and for all x 2 S \ O and u 2 Rm , Ax C Bu C a 2 B. Defining theclosed-loop vector field f .x/ D Ax C Bu.x/ C a, the previous statements suggest thata necessary condition to solve the RCP by continuous state feedback is: there existsa non-vanishing continuous map f .x/ on the set S \ O such that f .x/ 2 B \ C.x/.Motivated by Brockett’s work [3], we will say that if such a function does not exist,the system contains a topological obstruction. Define OS D S \ O. We define cone.OS / D \x2OS C.x/. For the remainderof the paper we assume that OS ¤ ;. We will also assume v0 … OS , as well as1 dim.OS / n 1. The cases of dim.OS / D 0; n and v0 2 OS are trivial toanalyze. For the sake of completeness, this analysis was formally done in Lemma 9,Lemma 10 and Corollary 11 of [11]. On a Topological Obstruction in the Reach Control Problem 679
We study the following problem.Problem 1 Let S, B, O, and OS be as above. Does there exist a continuous mapf W OS ! Bnf0g such that for every x 2 OS , f .x/ 2 C.x/?
2 Main Results
This section presents the main results on solving Problem 1. We show that ifdim.OS / D n 1, then it is possible to characterize the solution of Problem 1in terms of a smaller polytope OS0 , and OS0 will be amenable to a complete analysisof the problem in low dimensions. The main result is presented in Theorem 1. Theconsequences of Theorem 1 to low dimensional systems are presented in Theorem 3. Let us assume OS is .n 1/-dimensional. According to [7, 9], this means S is cutby O into two parts: one part containing v0 and p 0 other vertices, and the othercontaining the other n p 1 vertices of S. W.l.o.g. we assume fv0 ; v1 ; : : : ; vp gare on one side of OS and fvpC1 ; : : : ; vn g are on the other side, where we assumevertices of S on OS are in the set fvpC1 ; : : : ; vn g. The vertices of OS lie on thoseedges of S connecting vi ’s which are on different sides of OS . Thus, we employ thenotation oij to denote a vertex of OS with I.oij / D fi; jg. If there are no vertices of Son OS , then OS has .p C 1/.n p/ vertices [7], but if OS contains r vertices of S,then OS has .p C 1/.n p/ pr vertices. At this point we introduce a mild abuseof notation with the convention that if vj 2 OS , then oij D vj for all i D 0; : : : ; p. Let us introduce the following notation. Let
fi1 ; i2 ; : : : ; ik jj1 ; j2 ; : : : ; jl g D cofoi˛ jˇ W 1 ˛ k; 1 ˇ lg :
We observe that since I.oij / D fi; jg, if x 2 fi1 ; i2 ; : : : ; ik jj1 ; j2 ; : : : ; jl g then I.x/ fi1 ; i2 ; : : : ; ik g [ fj1 ; j2 ; : : : ; jl g. Also observe that OS D f0; : : : ; pjp C 1; : : : ; ng.Lemma 1 Let OS D f0; : : : ; pjpC1; : : : ; ng, A D fi1 ; : : : ; ik jj1 ; : : : ; jl g OS , andA0 D fi01 ; : : : ; i0k0 jj01 ; : : : ; j0l0 g OS . Let L D fi1 ; : : : ; ik g \ fi01 ; : : : ; i0k0 g. Analogously,let R D fj1 ; : : : ; jl g \ fj01 ; : : : ; j0l0 g. Then, A \ A0 D fLjRg.Proof By definition every vertex of fLjRg is a vertex of A and of A0 , so fLjRg A \ A0 . Conversely, suppose x 2 A \ A0 . Since x 2 A, I.x/ fi1 ; : : : ; ik ; j1 ; : : : ; jl gand since x 2 A0 , I.x/ fi01 ; : : : ; i0k0 ; j01 ; : : : ; j0l0 g. Hence, I.x/ L [ R, where weuse the fact that fi1 ; : : : ; ik g \ fj01 ; : : : ; j0l0 g D ; and fi01 ; : : : ; i0k0 g \ fj1 ; : : : ; jl g D ;. Itfollows x 2 fLjRg. t u Before getting to the crux of the problem, let us introduce the notions of ahomeomorphism and a retraction. Let X and XQ be topological spaces. X and XQare homeomorphic if there exists a continuous bijection h W X ! XQ which hasa continuous inverse. Furthermore, if A is a subspace of X , a continuous mapr W X ! A is a retraction if rjA id. 680 M. Ornik and M.E. Broucke
Example 1 Let us consider the case of n D 3, with OS a quadrilateral with verticeso02 2 cofv0 ; v2 g, o03 2 cofv0 ; v3 g, o12 2 cofv1 ; v2 g, and o13 2 cofv1 ; v3 g. Bydefinition, C.o02 / C.o12 / and C.o03 / C.o13 /. In fact, one can easily show thatthe cone of any point on the edge o12 o02 will be larger than C.o02 /, and analogouslyfor the edge o13 o03 . Thus, we reach the idea that if a continuous function satisfyingProblem 1 exists on the convex set cofo02 ; o03 g containing the most restrictive cones,then that function can easily be extended to the entire OS . Theorem 1 will serve to prove the above claim. The procedure outlined in theproof of Theorem 1, adapted to this example, is as follows. If a function f satisfyingProblem 1 can be defined on the edge o02 o03 , we can also define it on the edge o12 o02by f .x/ D f .o02 / and on the edge o13 o03 by f .x/ D f .o03 /. We note that such f isnon-zero and satisfies the cone condition f .x/ 2 C.x/ because C.o02 / C.o12 / andC.o03 / C.o13 /. So far f has been defined on three edges of OS . Then f can bedefined on the remainder of OS , which consists of its interior as well as the relativeinterior of the edge o12 o13 , by using a retraction r — a continuous map from OS tothe three edges of OS on which f is already defined, such that r is identity on thosethree edges. More formally, it can be shown, as in Theorem 1, that the functionf .x/ D f .r.x// exists and solves Problem 1.Theorem 1 (Dimension Reduction) Let dim OS D n 1, v0 62 OS , and p > 0. 0Define VO S D fo 2 VOS j .9j 2 f1; : : : ; ng/o 2 cofv0 ; vj gg, and let OS0 D co.VO 0 S /.Then the answer to Problem 1 is affirmative if and only if it is affirmative for OS0 . 0Proof Since VO S VOS it follows that OS0 OS . Thus, if there exists f WOS ! Bnf0g solving Problem 1 then f jOS0 W OS0 ! Bnf0g also solves Problem 1.Conversely, suppose there exists f 0 W OS0 ! Bnf0g solving Problem 1. From our 0notational convention, OS D f0; 1; : : : ; p j p C 1; : : : ; ng, VO S D fo0.pC1/; : : : ; o0n g, 0and OS D f0 j p C 1; : : : ; ng. We now proceed with the main topological argument. Informally, we build askeleton of OS , starting with OS0 , and adding in each step additional edges and facesof OS until in the last step all of OS is added. We then use topological methods toshow that Problem 1 for the set obtained in each step can be reduced to the sameproblem applied to the set from the previous step, thus going back from OS to OS0 . We build a skeleton of OS as follows. Let OS1 D OS0 , and for all 2 k n, let [ OSk D OSk1 [ f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g. 0<i1 <:::<i˛ p; p<j1 <:::<jˇ n; ˛CˇDk,˛;ˇ1
Observe that each H D f0; i1 ; : : : ; i˛ j j1 ; : : : ; jˇ g is a closed, convex polytope ofsome dimension d, so it is homeomorphic to the closed ball Bd , and its boundary ishomeomorphic to the .d 1/-dimensional sphere Sd1 [2]. We claim that
@f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g n OSk1 D Int.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/ : (4) On a Topological Obstruction in the Reach Control Problem 681
There are three points to the proof of the claim. (i) fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g 2 @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g. To show that, we note that if x 2 fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g, then
I.x/ fi1 ; : : : ; i˛ ; j1 ; : : : ; jˇ g f0; i1 ; : : : ; i˛ ; j1 ; : : : ; jˇ g,
so x 2 f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g. Moreover, @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g consists of points with jI.x/j ˛ C ˇ. (ii) Int.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/\OSk1 D ;. Assume x 2 Int.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/. Then 0 62 I.x/ and I.x/ D ˛ C ˇ D k. However, if x 2 OSk1 , then either 0 2 I.x/ or jI.x/j k 1.(iii) @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ gn Int.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/ OSk1 . This follows because if x 2 @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ gnInt.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/, then either 0 2 I.x/ and jI.x/j k so x 2 OSk1 ; or 0 62 I.x/ and jI.x/j k 1, so again x 2 OSk1 . What we have shown so far is that f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g is homeomorphic to aclosed ball, and @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g \ OSk1 is homeomorphic to its boundarysphere Sd1 with an open connected set Int.fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g/ of dimension d1cut out of it. Now we show there exists a retraction from Bd to the punctured sphereSd1 n P, where P is homeomorphic to an open ball of dimension d 1. Since Phas dimension d 1, Sd1 is split into two connected parts: P and Sd1 nP. Theretraction argument is standard in topology. We provide the proof since it is integralto our results. First, let us note that Bd is homeomorphic to the upper half-ball BCd D fx 2 Bd Wx1 0g. The precise homeomorphism is not difficult to find, but one can simplyimagine taking the ball and flattening its lower half. Now, our sphere Sd1 wasmapped by this to the boundary of BCd . Furthermore, without loss of generality,1we can assume that the closed part Sd1 nP makes up the bottom of the half-ball:fx 2 Bd W x1 D 0g, while the open part fx 2 Bd W x1 > 0g corresponds to P. 0 Let us define the function rH W BCd ! fx 2 Bd W x1 D 0g by rH 0 .x1 ; x2 ; : : : ; xn / D.0; x1 ; x2 ; : : : ; xn /. Clearly, this is a valid retraction and thus, we have obtained aretraction from BCd to fx 2 Bd W x1 D 0g. Now, using the fact that BCd ishomeomorphic to Bd , while the same homeomorphism takes fx 2 Bd W x1 D 0g 00to Sd1 nP, we know there thus exists a retraction rH W Bd ! Sd1 nP. Finally,reminding ourselves that there exists a homeomorphism between H and Bd which 00takes Sd1 nP to @OS \ OSk1 , by “pushing” rH through that homeomorphism, weobtain a retraction rH W f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g ! @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g\OSk1 . Now we glue these retractions to each other. In order to do that, we needto know that for H’s with constant ˛ C ˇ D k, all the different retractionsrH W f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g ! @f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g \ OSk1 agree on the
1 Really, this is done through another homeomorphism: this time, imagine, before flattening theball, choosing the part that needs to be flattened to be Sd1 nP . 682 M. Ornik and M.E. Broucke
intersections of their domains. That is, if H ¤ H0 , then rH jH\H0 rH0 jH\H0 .(The claim is obvious if H D H0 .) Let H D f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g. Analogously,let H0 D f0; i01 ; : : : ; i0˛0 jj01 ; : : : ; j0ˇ0 g. We noted in Lemma 1 that H \ H0 Df0; fi1 ; : : : ; i˛ g \ fi01 ; : : : ; i0˛0 gjfj1 ; : : : ; jˇ g \ fj01 ; : : : ; j0ˇ0 gg. Since we assumed thatH ¤ H0 , there needs to be an element in fi1 ; : : : ; i˛ ; j1 ; : : : ; jˇ g which is not anelement of the set fi01 ; : : : ; i0˛0 ; j01 ; : : : ; j0ˇ0 g and vice versa (note that both of those setshave k elements, so one cannot be a subset of the other). Thus, H \ H0 will not contain more than k 1 non-zero vertices of S in itsnotation, and hence it will be in both OSk1 (by the definition of OSk1 ), and in @H(as none of its elements can be in the interior of H: the expansion as a convex sum ofevery element in the interior needs to contain every vertex mentioned in the notationof H). Analogously, H \ H0 2 @H0 \ OSk1 , which is the image of the retractionrH0 . Hence, we know that rH0 jH\H0 is an identity map, and so is rH jH\H0 . Thus,these two retractions can indeed be glued together. By iterating this procedure forall H, we obtain a glued retraction rk W OSk ! OSk1 which takes each k-dimensionaledge in OSk to its boundary. Let us note what this retraction does. For every pointx 2 OSk , if x is also in OSk1 , it will not do anything. Thus, C.rk .x// D C.x/. If x 62 OSk1 , then either x is in the interior of some H D f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ gthat is being added to OSk , or it is in the interior of fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g. Now, if x isin the interior of f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g, rk maps it to a point in the boundary of H.In that case, it is easy to verify that C.rk .x// C.x/. This has formally been done inLemma 6 of [11]. If x is in the interior of fi1 ; : : : ; i˛ jj1 ; : : : ; jˇ g, then I.x/ D fi1 ; : : : ; i˛ ; j1 ; : : : ; jˇ g.On the other hand, H D f0; i1 ; : : : ; i˛ jj1 ; : : : ; jˇ g, so for any point y 2 H, I.y/ fi1 ; : : : ; i˛ ; j1 ; : : : ; jˇ g. Thus, C.y/ C.x/. Thus, specifically C.rk .x// C.x/. In all three cases, we deduce that C.rk .x// C.x/. By composing r D r2 ır ı ı rn , we obtain a retraction r W OS D OSn ! OS1 D OS0 . Define 3
f .x/ D f 0 .r.x//. We obtained a nowhere vanishing function f on OS such that f .x/ Df 0 .r2 .r3 .: : : .rn .x// : : :///. Thus, f .x/ is contained in C.r2 .r3 .: : : .rn .x// : : :/// C.r2 .r3 .: : : .rn1 .x// : : :/// : : : C.x/. This function satisfies the conditionsof Problem 1. t u We now proceed to resolving Problem 1 for the case of n D 2 and n D 3. Wehave previously assumed that dim OS ¤ 0 and dim OS ¤ n, as these cases aresimple to analyze. The case of n D 2 is thus reduced to dim OS D 1. As we havealso required 1 dim B < n, we conclude that dim B D 1. However, the case ofdim B D 1 is resolved by Theorem 1 in [12]. This resolves the case of n D 2, as well as dim B D 1. The only cases that remainare when n D 3, dim B D 2, and dim OS is either 1 or 2. We will see that, whendim OS D 1, an argument based on linear algebra applies. On the other hand, apurely topological argument applies when dim OS D 2. First we examine why a sufficiently high dimension for B resolves Problem 1. On a Topological Obstruction in the Reach Control Problem 683
Lemma 2 Suppose OS D cofo1 ; : : : ; oC1 g where the oi ’s are the vertices of OS .If there exists a linearly independent set fbi 2 B \ C.oi / j i D 1; : : : ; C 1g, thenthe answer to Problem 1 is affirmative. P PProof Let f W OS ! B be defined by f . C1 iD1 ˛i oi / D ˛i bi , where ˛i D 1 and˛i 0. Necessarily f .x/ ¤ 0 for x 2 OS for otherwise the bi ’s would be linearlydependent. Also, by a standard convexity argument f .x/ 2 C.x/, x 2 OS . t u The following is the key result in the case of dim OS D 1.Lemma 3 Let n D 3, dim B D 2, and let o1 and o2 be vertices of OS . Then thereexist linearly independent vectors fb1 ; b2 j bi 2 B \ C.oi /g. Moreover, if OS Dcofo1 ; o2 g, the answer to Problem 1 is affirmative.Proof First we assume o1 2 Int.Fi / for some i 2 f0; 1; 2; 3g. By the definition ofC.o1 /, it is a closed half space or R3 , so there exist linearly independent vectorsb11 ; b12 2 B \ C.o1 /. We claim B \ C.o2 / ¤ 0. If o2 2 Int.Fi / for some i 2f0; 1; 2; 3g then the argument above proves the claim. Instead, assume w.l.o.g. thato2 2 F1 \ F2 . Then C.o2 / D fy 2 R3 jh1 y 0; h2 y 0g. Let B D Ker.M T / forsome M 2 R31 . Finding 0 ¤ y 2 B \ C.o2 / is equivalent to solving 2 3 2 3 h1 T s1 4 h2 T 5 y D 4 s2 5 (5) MT 0
for some s1 ; s2 2 R 0 and y ¤ 0. Because fh1 ; h2 g are linearly independent,rank.H/ 2, where H is the matrix appearing on the left hand side of equation 5.If rank.H/ D 3, then let 2 3 1 0 Œy1 y2 D H 1 4 0 1 5 : 0 0
Since .1; 0; 0/ and .0; 1; 0/ are linearly independent, y1 and y2 are linearlyindependent as well. Next, assume rank.H/ D 2. In other words, M D c1 h1 C c2 h2 for some c1 ; c2 2R. Then, by taking s1 D s2 D 0, equation (5) reduces to Œh1 h2 T y D 0. By the rank-nullity theorem, there exists y ¤ 0 satisfying this equation. Moreover, if w.l.o.g.v0 D 0, then y 2 F1 \ F2 D cofv0 ; v3 g, and we can take y D v3 . We have shownthere exist linearly independent b11 ; b12 2 C.o1 / and there exists 0 ¤ b2 2 C.o2 /.We claim at least one of the pairs fb11 ; b2 g and fb12 ; b2 g is linearly independent. Forotherwise there exist c1 ; c2 2 R such that b2 D c1 b11 D c2 b12 , implying b11 andb12 are linearly independent, a contradiction. We conclude there exists a linearlyindependent set fb1 ; b2 j bi 2 C.oi /g. Next we assume neither o1 nor o2 lies in the interior of a facet. W.l.o.g. supposeo1 2 F1 \ F2 and o2 2 F1 \ F3 . If either C.o1 / or C.o2 / contains two linearly 684 M. Ornik and M.E. Broucke
v0 o03 v3
o02 o13 v1 v2 o12
0Fig. 1 The set OS for Example 1. The edge from o02 to o03 forms OS
v0 v0 v0
v3 v3 v3
v1 v2 v1 v2 v1 v2
Fig. 2 Three of the four possible configurations of set OS for n D 3 and dim OS D 2, with thefourth one given in Fig. 1. The leftmost configuration is addressed by Theorem 2, while the othertwo can be reduced using Theorem 1
independent vectors, then by the previous argument, we are done. Otherwise, bythe previous argument again v3 2 C.o1 / and v2 2 C.o2 /. Since fv2 ; v3 g are linearlyindependent, we are done. Finally, if OS D cofo1 ; o2 g, then by Lemma 2 the answerto Problem 1 is affirmative. t u The remaining case to study is when n D 3, dim OS D 2, and dim B D 2.Assuming v0 62 OS (which is a trivial case discussed in Lemma 10 of [11]), thereare four topologically distinct cases for OS , depending on the way O cuts S. Theseare given in Figs. 1 and 2. In Fig. 1, OS is a quadrangle. In that case, p D 1; thatis, there are two vertices of S on each side of O. Then we can apply Theorem 1 toreduce OS to OS0 , and according to the construction in the proof, OS0 has dimension1, and we can apply Lemma 3. Similarly, in the cases given in middle and therightmost configuration of Fig. 2, we can apply Theorem 1 to reduce OS to OS0with dim OS0 being either 0 or 1, respectively. Finally, in the situation given by theleftmost configuration of Fig. 2, we draw upon a proof method already utilized in[4], which is based on Sperner’s lemma. Here we employ a variant found in [13].Lemma 4 Let P D cofw1 ; : : : ; wnC1 g be an n-dimensional simplex. Furthermore,let fQ1 ; : : : ; QnC1 g be a collection of sets covering P such that(P1) Vertex wi 2 Qi and wi 62 Qj for j ¤ i.(P2) If w.l.o.g. x 2 cofw1 ; : : : ; wl g for some 1 l n C 1, then x 2 Q1 [ [ Ql . TThen nC1 1D1 Qi ¤ ;. On a Topological Obstruction in the Reach Control Problem 685
Theorem 2 Let n D 3 and suppose OS D cofo1 ; o2 ; o3 g with v0 62 OS and oi 2.v0 ; vi , i D 1; 2; 3. The answer to Problem 1 is affirmative if and only if
B \ cone.OS / ¤ 0.
Proof Sufficiency is clear: if 0 ¤ b 2 B \ cone.OS /, a constant function f .x/ D bsatisfies Problem 1. For necessity, suppose there exists f W OS ! B n f0g such thatf .x/ 2 C.x/, x 2 OS . By way of contradiction suppose B \ cone.OS / D 0. Sinceoi 2 .v0 ; vi , i D 1; 2; 3, we have
cone.OS / D fy 2 Rn j hj y 0 ; j D 1; 2; 3g :
Define the sets
Qi WD fx 2 OS j hi f .x/ > 0g ; i D 1; 2; 3 : (6)
Now we verify the conditions of Lemma 4. Firstly, we claim that fQi g cover OS . For suppose not. Then there exists x 2OS such that hj f .x/ 0, j D 1; 2; 3. Hence f .x/ 2 B \ cone.OS /, so f .x/ D0, a contradiction to f being non-vanishing on OS . Secondly, we verify property(P1). We claim that oi 2 Qi for i D 1; 2; 3. For suppose not. Then hi f .x/ 0.Additionally, because f .oi / 2 C.oi /, hj f .x/ 0, j 2 f1; 2; 3g n fig. We concludef .oi / 2 B \ cone.OS /, so f .oi / D 0, a contradiction. Next we claim oi 62 Qj ,j ¤ i. This is immediate since f .oi / 2 C.oi / implies hj f .oi / 0, j ¤ i. Thirdly,we verify property (P2). Suppose w.l.o.g. (by reordering the indices f1; 2; 3g) x 2cofo1 ; : : : ; or g for some 1 r 3. We claim x 2 Q1 [ [ Qr . For supposenot. Then hj f .x/ 0, j D 1; : : : ; r. Also, it is easily verified that C.x/ D fy 2Rn j hj y 0 ; j D r C 1; : : : ; 3g. Thus, hj f .x/ 0, j D r C 1; : : : ; 3. Hence,f .x/ 2 B \ cone.OS /, so f .x/ D 0, a contradiction to f being non-vanishing on OS .T3We have verified (P1)-(P2) of Lemma 4. Applying the lemma, there exists x 2 iD1 Qi ; that is, hj f .x/ 0, j D 1; 2; 3. We conclude that f .x/ 2 B \ cone.OS /,so f .x/ D 0, a contradiction. t u The following result finally resolves Problem 1 in all cases of interest.Theorem 3 Let S, B and OS be as above, and let n 2 f2; 3g. If n D 3, dim B D 2and OS does not satisfy the conditions of Theorem 2, then the answer to Problem 1is affirmative. Otherwise, the answer to Problem 1 is affirmative if and only if B \cone.OS / ¤ 0.Proof The discussion prior to Lemma 2, as well as Lemma 3 and Theorem 2,covered all the cases except for the one where dim B D dim OS D 2 and OSdoes not satisfy the conditions of Theorem 2. However, in that case, as describedprior to Lemma 4, Theorem 1 reduces OS to OS0 of dimension 0 or 1. ApplyingLemma 3, we now obtain that the answer to Problem 1 is affirmative. t u 686 M. Ornik and M.E. Broucke
An interesting modification of Problem 1 requires that f be not only continuous,but also affine. This corresponds to a classic problem of designing an affine statefeedback, applied to RCP. We note that in most of the configurations consideredabove, the same claims that work for Problem 1 can also be used in the affine case.The only significantly different case is dim OS D dim B D 2, i.e., the situationcovered by Theorem 1. In this case, the full analysis of the problem of affineobstruction can be done using linear algebra. Such an analysis is not difficult, butis computationally long. It is presented in full in Section 3.2.2 of [11]. With theresults contained in [11] (in fact, Theorem 16 in [11] is essentially the same as ourTheorem 3), we reach the following theorem:Theorem 4 Let n 2 f2; 3g. The problem of affine obstruction is solvable if and onlyif Problem 1 is solvable. It is not known if such a result holds in general. Hence, we end this paper withthe following conjecture.Conjecture 1 Let n 4. The problem of affine obstruction is solvable if and onlyif Problem 1 is solvable.
3 Conclusion
This paper introduces a topological obstruction to solving the RCP via continuousstate feedback. The results show an interplay between linear algebra-based argu-ments regarding the number of control inputs and purely topological argumentsregarding a cone condition on B. We show that for n D 2 and n D 3 these twoproperties together fully characterize when a topological obstruction arises.
References
1. Bemporad, A., Morari, M., Dua, V., Pistikopulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002) 2. Bredon, G.E.: Topology and Geometry. Springer, New York (1997) 3. Brockett, R.W.: Asymptotic stability and feedback stabilization. In: Brockett, R.W., Millman, R.S., Sussmann, H.J. (eds.) Differential Geometric Control Theory, pp. 181–191. Birkhauser, Boston (1983) 4. Broucke, M.E.: Reach control on simplices by continuous state feedback. SIAM J. Control Optim. 48(5), 3482–3500 (2010) 5. Broucke, M.E., Ganness, M.: Reach control on simplices by piecewise affine feedback. SIAM J. Control and Optim. 52(5), 3261–3286 (2014) 6. Habets, L.C.G.J.M., van Schuppen, J.H.:A control problem for affine dynamical systems on a full-dimensional polytope. Automatica 40(1), 21–35 (2004) 7. Kettler, P.C.: Vertex and Path Attributes of the Generalized Hypercube with Simplex Cuts. University of Oslo, Oslo (2008) On a Topological Obstruction in the Reach Control Problem 687
8. Mehtaa, K.: A topological obstruction in a control problem. Master’s thesis, University of Toronto (2012) 9. Min, C.: Simplicial isosurfacing in arbitrary dimension and codimension. J. Comput. Phys. 190(1), 295–310 (2003)10. Ornik, M., Broucke, M.E.: A topological obstruction to reach control by continuous state feedback. In: This work has now been published in 54th IEEE Conference on Decision and Control (CDC), 2015, pp. 2258-2263, doi: 10.1109/CDC.2015.7402543. For full citation, see here: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7402543&tag=111. Ornik, M., Broucke, M.E.: Some results on an affine obstruction to reach control. arXiv:1507.02957 [math.OC] (2015)12. Semsar Kazerooni, E., Broucke, M.E.: Reach controllability of single input affine systems. IEEE Trans. Autom. Control 59(3), 738–744 (2014)13. Talman, D.: Intersection theorems on the unit simplex and the simplotope. In: Imperfections and Behavior in Economic Organizations, pp. 257–278. Springer, New York (1994) Continuous Approaches to the UnconstrainedBinary Quadratic Problems
Oksana Pichugina and Sergey Yakovlev
Abstract Two continuous approaches to discrete problems over sets inscribedinto a sphere are presented. They use an analytic description of the sets and aconvex extension of objective functions from the sets onto Euclidean space. Thefirst approach is the Branch and Bound Polyhedral-Spherical Method (B&BPSM)for “two-layer sets”, which utilizes the sets representation as an intersection of apolyhedron with a sphere and the global minimizers on a sphere. The concept of afunctional representation of a discrete set as an intersection of surfaces is introducedand used, for constructing penalty functions, in the second approach – the PenaltyMethod based on Functional Representations (FRPM). The methods are applied tothe Unconstrained Binary Quadratic Problem. For the purpose, the representation ofthe binary set as a “touching set” of smooth surfaces is derived.
1 Introduction
The Unconstrained Binary Quadratic Problem (UBQP):
z D f .x/ ! min; (1) f .x/ D x Ax C c x; T T (2) x 2 Bn D f0n ; 1n g (3)
is a famous NP-hard problem known by numerous applications and a varietyof exact and heuristic methods (the solution methods alongside with differentapplications on graphs, facility and resources allocation problems, clustering andpartitioning problems, variations of assignment problems, etc. see in details in [4]).The methods are roughly divisible into combinatorial and continuous. The division
O. Pichugina ()Department of Mathematics & Statistics, Brock University, St. Catharines, ON, Canadae-mail: [emailprotected]. YakovlevDepartment of IT & Protection of Information, National University of Internal Affairs, Kharkov,Ukrainee-mail: [emailprotected]© Springer International Publishing Switzerland 2016 689J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_62 690 O. Pichugina and S. Yakovlev
is not very precise since the continuous techniques are often used for solution ofsubproblems in Branch & Bound, Branch & Cut and other combinatorial algorithms.In [3] connection between continuous and combinatorial approaches to discreteproblems are discussed, and an overview of the continuous ones is given. The lastapproaches are typically classified into continuous relaxation and reformulationbased. Among the relaxation techniques there are polyhedral and semi-definiterelaxations. For instance, the semi-definite programming is applicable to the UBQPafter reformulation of (1), (2) and (3) as a linear problem in Rnn over the cutpolytope. Since the analytic description of the polytope is unknown, its semi-definiterelaxation is used [2] and efforts of researchers are focused on strengthening thisbasic relaxation [5]. The UBQP equivalent reformulations depend on ways how the discrete con-straint (3) is rewritable as continuous one, e.g.:– Way 1 [3]: xi C yi D 1; xi yi D 0; xi ; yi 0; i 2 Jn D f1; : : :; ng, transform the UBQP into a constrained quadratic problem in R2n ;– Way 2 [3, 4, 6]: replacement of (3) by (4) allows to consider the unconstrained problem (5) instead of the UBQP. The next two approaches utilizing (4) reduce the UBQP to concave or convexproblems and are based on the following property: if f .x/ satisfies (7), then9min ; max ; min max such that ˚ 1;2 .x; / are concave 8 max and convex8 min .
xi .1 xi / D 0; i 2 Jn : (4) X ˚ l .x; / D f .x/ C .xi .1 xi //l ! min : (5) i
0 x 1; (6) f .x/ 2 C2 .Rn /; @2 f .x/=.@xi @xj / < 1; i; j: (7)
For example, concave methods [3] are applied to (1), (2) and (3) equivalentreformulation (2), (5), (6) (l D 1) where is chosen to provide concavity of (5).The smoothing methods [6] utilize (5), (6) (l D 2) in a penalty function; use theboxing constraints (6) in forming barriers and providing convexity of the functionfor a range of . We continue study of the continuous approaches to binary optimization basedon nonlinear reformulations at the same space. For the purpose, we introduce theconcepts of an extension of a function from sets onto larger sets (see Sect. 2)and a functional representation of discrete sets (see Sect. 2.3). The presented inSects. 2.2 and 2.3 methods operate with sets inscribed into a sphere and convexextensions from the sets. Ways of constructing the extensions, in particular for (2),are described in Sect. 2.1. The Branch and Bound Polyhedral-Spherical Method(B&BPSM) for “two-layer sets”, combining branch and bound techniques withpolyhedral and spherical relaxations, is described and specified for the UBQP Conltinuous Approaches to the Unconstrained Binary Quadratic Problems 691
in Sect. 2.2. The Penalty Method based on Functional Representations (FRPM),presented in Sect. 2.3, is applicable to any sets representable as intersection ofsurfaces. In particular, for the binary set (3) a new representation as a touchingset of a sphere and an order four surface is given and recommended to use in theFRPM.
2 Main Part
In the section we consider the following generalization of the UBQP – theproblem (1) over a finite discrete set inscribed into a hypersphere:
x 2 E Rn ; (8) jEj D N < 1; (9) E Sr .a/; (10) 2 2 Sr .a/ D fx 2 R W .x a/ D r g: n (11)
There is a number of combinatorial sets inscribed into a sphere, such as the binaryset Bn [6, 12], the permutation and polypermutation sets, some classes of the partialpermutations and combinations [8]. So, the class (1), (8), (9), and (10) includes allunconstrained problems over these sets, in particular, the UBQP. The set (9), (10) is vertex located, i.e. E coincides with the vertex set of thecorresponding polyhedron:
E D vert P where P D conv E is the polyhedron: (12)
Definition 1 ([11]) Function F.x/ is an extension of f .x/ from M onto M 0 M ifit is defined over M 0 and coincides with f .x/ over M: F.x/ D f .x/; x 2 M.Depending on a type of the function F.x/ the f .x/-extension can be continuous,differentiable, smooth, convex or concave, quadratic, etc.Theorem 1 ([11]) If f .x/ is defined on a vertex located set E, then there exists F.x/,which is a convex differentiable extension (CDE) of f .x/ from E onto Rn .By (12), Theorem 1 is applicable to the sets (9)–(10). Therefore, we can assumethat, instead of the original function (1), its CDE from E onto Rn is considered. A CDE can be constructed in different way, e.g., by using the next result:Theorem 2 ([10]) If f .x/ satisfies (7) then 9K W 8k > K F.x/ D f .x/ C k .x a/2 r2 (13)
is a CDE of f .x/ from Sr .a/ onto Rn . 692 O. Pichugina and S. Yakovlev
Remark 1 In addition, in [10] the lower estimated bound of K is given.This property of reducing the original problem (1), (8), (9), and (10) to optimizationof a convex function, F.x/ ! min, seems useful but the problem is still discrete. EThe convexity can be really helpful if continuous optimization is applicable to it. The methods presented in Sects. 2.2 and 2.3 are especially effective for thefunctions and the combinatorial sets with the known solutions of the followingsubproblems:1. an analytic description of the polyhedron P: P D fx W Ax bg;2. a linear problem over E: xLP D argmin cT x; x2E3. the global solution of (1) on a sphere:
zS D min f .x/; xS D argmin f .x/I (14) x2Sr .a/ x2Sr .a/
4. the minimal circ*mscribed sphere around E, Smin .Alongside with (10), these four cases can simplify the solution of the next problemsconsidered typically complicated:– minimization of f .x/ over a polyhedron:
zP D minf .x/; xP D argmin f .x/: (15) x2P x2P
Knowledge of the exact solution of the subproblem 2 allows to utilize the modification of conditional gradient methods such as the Frank-Wolfe algorithm, where linear direction-finding subproblems are solved explicitly [12];– search of a projection of an arbitrary point x 2 Rn onto a discrete set E, y D PrE x. This problem for sets satisfying (10) is reduced to the linear subproblem 2. Note that for the UBQP the optimization approaches are applicable effectively.Indeed, the explicit solution of the subproblems: #1 is the unit hypercube (6); #2 – i D 0; if ci 0; otherwise 0 .i 2 Jn /; #3 – is presented in [1]; #4 – is:is xLP
Smin D S pn .0.5/ : (16) 2
2.1 Convex Extensions of Quadratic Functions
We present two ways of constructing convex extensions of (2) from Bn onto Rn :Way 1: Notice that ˚ 1 .x; / in (5) is rewritable as ˚ 1 .x; / D f .x/ .x 0.5/2 n=4 : (17)
By (16), (17) is an extension of any f .x/ from Smin and 8E Smin onto Rn . Conltinuous Approaches to the Unconstrained Binary Quadratic Problems 693
Using Theorem 2 and taking D k in (13), we obtain that (17) is the CDEof (7) 8 < min D K, where K can be estimated according to Remark 1. In particular, (17) is a CDE of (2) from Bn onto Rn if < min , where the boundmin can be found exactly: min is the minimal eigenvalue of the matrix A.Way 2 : The convex extensions can be formed specifically for a discrete set. Forinstance, a convex extension of (2) from Bn onto Rn can be constructed as follows:– the function (2) is represented as a sum of a convex linear form f1 .x/ and a quadratic form f2 .x/:
X n X n X n X n X n f1 .x/ D ci xi ; f2 .x/ D aij xi xj D jaij j .˙xi xj /I iD1 iD1 i j iD1 i j
– the convex extensions are formed for each nonconvex term jaii j x2i and ˙jaij j xi xj using the property (4): 8i 2 Jn ; x2i D xi . For the both types of nonconvex Bn terms the convex extensions are formed as follows: (a) if i D j and aii < 0, then the CDE of the component fii D x2i is Fii D xi ; (b) if i ¤ j and aij ¤ 0, then the CDEFij ofthe component fij D ˙xi xj is: 1 2 2 2 1 2 fij D ˙xi xj D 2 xi ˙ xj xi xj D 2 xi ˙ xj xi xj D Fij ; Bn
– all these convex extensions are incorporated to the final CDE F.x/ of (2):
X n X n X X F.x/ D f .x/ C D x2i D xi , where D D jaii j C jaij j: iD1 iD1 aii <0 aij ¤0;i<j
2.2 The Polyhedral Spherical Method
The discrete problem (1), (8)–(10) is equivalent to the f .x/-optimization problem (1)over intersection of a sphere (11) and a polyhedron (12), f .x/ ! min. It yields P\Sr .a/two types of continuous relaxations: Relaxation 1 – optimization over a polyhedron,which is convex problem (15); Relaxation 2 – optimization over a sphere, which isnonconvex problem (14). The Polyhedral-Spherical Method (PSM) is the cutting-plane method wherecuttings of infeasible parts of a hypersphere are conducted by polyhedron facets. In [9, 12] the approximating version of this method was presented and appliedto quadratic optimization over the permutation set and the binary set, respectively.This version essentially uses: (a) the convex extensions of a quadratic function;(b) simplicity of search of all stationary points of a convex quadratic functionon a sphere; (c) easiness of solving (15) over the hypercube (6). The search of 694 O. Pichugina and S. Yakovlev
x D argminf .x/ is restricted to facets cutting the stationary points. Then the Eproblem is reduced to search of these points and solving the subproblems oftype (14), (15) on these facets, etc. As a result, a series of optimization problems ofdifferent dimensions with convex objective functions over sets inscribed into spheresis solved. In [7] the method is generalized for any sets inscribed into a sphere and is namedthe Polyhedral-Spherical Method. In [12] the exact version for (1), (2), and (3) is also presented, which utilizes thedecomposition of Bn into two subsets located on parallel facets of (6). In this paper, we modify the exact method and generalize it, in theBranch&Bound PSM (B&BPSM), to any “two-layer” combinatorial set, i.e. itallows the decomposition: 8a; b W ax D b is a P-facet dim conv.fx 2 E W ax ¤bg/ < dim P.The B&BPSM Algorithm for two-layer setsStep 1. The unconstrained convex problem (1) is solved, z D min n f .x/ D f .x /, R and x is projected onto E, y D PrE x . The initial upper zu and lower zl estimated bounds are: zl D z ; zu D z.y / D zy .Step 2. Depending on location of x we choose: Scheme 1 – solving the Relax- ation 2 problems at each step – if x 2 P (xP D x ); Scheme 2 – solution of Relaxation 1–2 problems at each iteration – if x … P.Step 3. Initialization H D f;g, BH D Bf;g is the root of the search tree. The candidate solution set B D Bf;g .Step 4. Take the branch BH 2 B with the least lower bound and jHj < n. If the 0 choice is not unique, then among BH 2 fBh ; Bh g the branch Bh is examined first, h0 B – next. If for BH the relaxation problems were solved, go to Step 4.2, otherwise to Step 4.1: Step 4.1. For BH solve the relaxation and projection problems on RnjHj -facet depending on the scheme: (a) for Scheme 1 – xS;H ; yS;H , the bounds are zl;H D x ; z D min.z ; zy /; (b) for Scheme 2 – x zS;H ; xP;H ; yS;H ; yP;H , the bounds u u S;H S;H
are z D max.zx ; zx /; z D min.z ; zy ; zy /; l;H S;H P;H u u S;H P;H
Step 4.2. Let: – ˘.xS;H / D fx W ax D bg be a cutting plane for the point xS;H nearest to xP such that fx W ax bg is a right cut of xS;H , EH D E \ ˘.xS;H /; 0 – ˘ 0 .xS;H / D fx W a0 x D b0 g be the second layer, EH D EnEH , facet. 0 The sets EH ; EH form two branches from the node BH – BfH;jHC1jg and 0 fH;jHC1jg fH;jHC1j0 gBfH;jHC1j g - with the same lower bound zu;B H D zu;B D zu;B . They 0are added to B instead of BH , B D BnBH [ fBfH;jHC1jg ; BfH;jHC1j g g. Step 4 is repeated until the candidate solutions with lower bound less than zuremain. Conltinuous Approaches to the Unconstrained Binary Quadratic Problems 695
The B&BPSM for the UBQP. Step 4.2 becomes: ˘.xS;H / D fx W xi D bg. The rulefor cutting plane choice on Step 4.2 becomes: ˘.xS;H / D fx W xi D bg, ˘ 0 .xS;H / Dfx W xi D 1 bg where b 2 f0; 1g, i 2 Jn is chosen depending on which one of therestrictions xi 0 or xi 1 is mostly violated at xS;H . Respectively, new branches 0BfH;jHC1jg ; BfH;jHC1j g correspond to fixing the coordinate xi .Example 1 Solve the following problem: f .x/ D 33:1 x21 C 23:7 x22 C 21:6 x23 11:4 x1 x2 C 38:8 x1 x3 15:6 x2 x3 52 x1 4 x2 39 x3 ! min; x 2 B3 : Solution outline:1. n D 3, f .x/ is convex, x D 12 A1 c D .0:51; 0:40; 0:59/, y D .1; 0; 1/. The initial bounds are zl D z D 25:54; zu D zy D 2:5;2. x is an interior point of the unit hypercube (xP D x , yP D y ). The results of the relaxation Scheme 1: xS D .0:06; 0:51; 1:25/, yS D .0; 1; 1/. The bounds are zl D zS D 21:30, zu D min.zu ; zSy / D min.2:50; 13:30/ D 13:30 (the record occurs at xu D .0; 1; 1/).3. H D f;g, the right cut for xS;H D xS is uniquely defined by x3 1. Respectively, x3 is fixed and exploring of the search tree starts with the branch BfH;jHC1jg D 0 0 B1 W x3 D 1, the next one is BfH;jHC1j g D B1 W x3 D 0.4. The reduction is conducted (H D f1g) and, at first, the Relaxation 2 problem on the plane x3 D 1 is solved yielding xS1 D .0:09; 0:11; 1/, zl1 D zS1 x D 17:7. yS1 D .0; 0; 1/, zS1 y D 17:4 > zu ; new record zu D 17:4 occurs at xu D .0; 0; 1/. The right cut for xS1 is x1 0 yields two branches with H D fH; jH C 1jg D 0 f1; 2g, B12 W x1 D 0; x3 D 1 and B12 W x1 D 1; x3 D 1, with fixed x1 and the 0 lower bound zl12 D zl12 D zl1 D 17:7. 0 0 0 S105. The same is conducted for B1 : xS1 D .1:04; 0:04; 0:00/, zS1 x D 18:92; y D 0 .1; 0; 0/, zS1 y D 18:9. New record is zu D 18:9 at x u D .1; 0; 0/ and the 0 0 0 bound is zl1 D 18:92. The right xS1 -cut is x1 1 yields the branches B1 2 W 10 20 l10 2 l10 20 x1 D 1; x3 D 0 and B W x1 D 0; x3 D 0, with the lower bound z Dz D 0 zl1 D 18:9. 0 0 06. The least lower bound 18:92 corresponds to B1 2 and B1 2 , which are examined 0 consecutively. First, for H D f1 ; 2g Relaxation 2 is conducted. Since, jHj D n 0 0 1, B1 2 corresponds to 1-sphere, where, in addition to yS1 D .1; 0; 0/, it is enough S10 0 to analyse another endpoint t D .1; 1; 0/ of the hypercube edge, zS1 t D 10:6. l10 2 S10 S10 0 0 z D min.zy ; zt / D min.18:9; 10:6/ D 18:9. For H D f1 ; 2 g and B1 2 0 0 0 0 0 0 0 0 0 0 the result of Relaxation 2 is xS1 2 D .0; 0; 0/. Since zl1 2 D zS1 2 D 0 > zu , B1 2 is discarded.7. No bud nodes with less lower bound than zu D 18:9 left. The examination of the search tree is terminated. The solution is x D .1; 0; 0/, z D 18:9.Illustration of the solution is shown in Figs. 1, 2, and 3. 696 O. Pichugina and S. Yakovlev
Fig. 1 The touching the levelsurface at xS . In Figs. 2 and 3the reduced problems solvingat Step 4 are demonstrated,namely, the ellipsoids 0yielding xS1 ; xS1 alongside 0with the projections yS1 ; yS1 ,their cuts and the branches ofthe next level
Fig. 2 B1 exploration 1.2
0.8
0.6 x2 B12’ 0.4 B12
0.2
xS1 –0.2 0 0.2 0.4 0.6 0.8 1 1.2 x1 –0.2 yS1
2.3 The Penalty Method Based on Functional Representations
Definition 2 The following system of functions:
f.x/ D f fi .x/gi2Jm (18)
is called a functional representation of a set M Rn if
x0 2 M , fi .x0 / D 0 8fi .x/ 2 f.x/: (19)
Similar to the extensions of functions, depending on a type of functions in (18), thesets’ representations can be continuous, differentiable, convex, smooth, etc. Conltinuous Approaches to the Unconstrained Binary Quadratic Problems 697
0Fig. 3 B1 exploration 1.2
1 tS1’2
0.8
x2 0.6 B1’2’ 0.4 B1’2
0.2 S1’2’ X
xS1’ –0.2 0 0.2 0.4 0.6 0.8 1 1.2 x1 yS1’=XS1’2’ –0.2
Definition 3 The representation (18) is irreducible if it is impossible to extractanything from (18) without violating (19): 8j 2 Jm M ¤ fx W fj .x/ D 0gfi .x/2f.x/;i¤j .The underlying idea of the Penalty Method based on Functional Representations(FRPM) is constructing and applying irreducible differential representations of theinitial discrete set E in penalty optimization. Suppose, for E its differentiable representation is known. Then, by (19), thediscrete problem (1), (8) is transformed into the constrained continuous problem (1)subject to f.x/ D 0. After placing these constraints in a penalty function, sayquadratic, the problem becomes unconstrained one of minimizing the differentiablefunction:
X m 2 ˚.x; / D f .x/ C f i .x/ ! min: (20) >0 iD1
The problem (20) is solved for an increasing sequence
2 D fk W k < kC1 ; 0 D 0gk2JK0 DJK [f0g : (21)
Its solutions gradually approach a local minimizer x of (1): xk ! x . Notice k!1that ˚.x; / in (20) is an extension of (1) onto Rn and a sum of a convex objectivefunction (the first term) and a nonconvex penalty function (the second term). Hence9k0 W ˚.x; k / is convex for k k0 and the solution of (20) is unique, otherwise isnot and the rest of the search is held in the vicinity N.x / of x . For the problem (1), (8), (9), and (10), by using Theorem 2, the region of ˚.x; /-convexity can be increased and, respectively, the vicinity N.x / can be narrowed by 698 O. Pichugina and S. Yakovlev
switching from (20) to the following extension of (1) from E onto Rn :
X m 2 ˚.x; ; / D f .x/ C ..x a/2 r2 / C f i .x/ .; > 0/: (22) iD1
The problem (22) can be solved for two increasing sequences, (21) and fj W j <jC1 gj , in the way described below:The FRPM Algorithm for (1), (8) based on (22).Step 0. Set j D 0, 0 D 0, " > 0, the upper and lower bounds zl D 1, zu D 1;Step 1. Solve (22) for (21), D j , and obtain fxjk gk ; xj D xjK . zl D max.zl ; zj /, zu D min.zu ; f .PrE .xjk //; kStep 2. For (22), by Remark 1, find the lower bound 0j : 8 > 0j ˚.x; K ; / is convex. Solve (22) for D K , 2 Mj D fj0 > 0 W j0 < 0 j0 C1 ; J 0 > 0j gj0 2JJ0 , the solutions are fxjK gj ; xK D xJ K . zl D max.zl ; zK /, zu D min.zu ; f .PrE .xjK //; jStep 3. Set f .x/ D ˚.x; K ; J 0 /;Step 4. Repeat Steps 1–3. The process terminates if zu zl < ". Assuming that n > 3, we classify functional representations of discrete setsdepending on a number of functions in (18) as follows: (a) intersecting functionalrepresentation if m D n implying that E is formed as an intersection of n surfaces;(b) touching functional representation if m D 2, hence, E is the set of touchingpoints of two surfaces; (c) otherwise mixed functional representation. Deriving functional representations of combinatorial sets is an interesting on itsown problem, because it provides better understanding the sets topological structure.Among this class of problems finding the touching representations is a problem ofparticular interest since it investigates extremal properties of the sets. Indeed, letassume that a touching representation of E is known and the functions in (18) areenumerated such that the surface S2 D f f2 .x/ D 0g is inscribed into S1 D f f1 .x/ D0g. Then E is the set of local minimizers of f2 .x/ on the surface S1 . Touching sets representations, especially differentiable ones, are useful in opti-mization, e.g., they allows usage in the FRPM, instead of (22), the penalty function:
2 X 2 X 2 ˚.x; ; / D f .x/ C i f i .x/ C i f i .x/ ; (23) iD1 iD1
and managing a few penalty parameters 1 ; 2 ; 1 ; 2 for obtaining the solutionscloser to exact one. Conltinuous Approaches to the Unconstrained Binary Quadratic Problems 699
Fig. 4 Illustration of thetouching representations (24).The Fig. 4 shows the unithypercube inscribed into theorder four surface S2 . In turn,S2 is inscribed into the sphereS1 D Smin . The binary set B3is the “touching set” of S1 , S2
The Functional Representations of Bn . We show that the FRPM based on anyof (22), (23) can be used for (1), (3):– Equation (22) is applicable since the irreducible intersecting representation fi .x/ D xi x2i ; i 2 Jn , is derived directly from (4) and Bn is inscribed into the sphere (16).– Equation (23) can be used for the following irreducible touching differentiable representation of Bn , which we derived (the geometric interpretation is shown in Fig. 4):
X n X n f1 .x/ D .xi 0:5/2 D n=4; f2 .x/ D .xi 0:5/4 D n=16: (24) iD1 iD1
We recommend for the UBQP utilizing (24) in the penalty function (23) of (1), (2), and (3).
3 Conclusion
Two approaches, the B&BPSM and FRPM, to discrete optimization over setsinscribed into a sphere are presented. For the UBQP they use analytic descriptionsof Bn as an intersection of a hypercube with a sphere and as a touching set; convexextensions of quadratic functions. The approaches can be applied to other sets withknown analytic description of a convex hull, a circ*mscribed surface, a functionalrepresentation, and to any function with known the global minimizer on the surface. 700 O. Pichugina and S. Yakovlev
References
1. Dahl, J.: Convex optimization in signal processing and communications. Ph.D. dissertation. Department of Communication Technology, Aalborg University (2003) 2. Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82(3), 291–315 (1998) 3. Hillier, F.S., Appa, L., Pitsoulis, L., Williams, H.P., Pardalos, P.M., Prokopyev, O.A., Busygin, S.: Continuous approaches for solving discrete optimization problems. In: Handbook on Modelling for Discrete Optimization, pp. 1–39. Springer, New York (2006) 4. Kochenberger, G., Hao, J.-K., Glover, F., Lewis, M., Lu, Z., Wang, H., Wang, Y.: The unconstrained binary quadratic programming problem: a survey. J. Comb. Optimiz. 1, 58–81 (2014) 5. Krislock, N., Malick, J., Roupin, F.: Improved semidefinite bounding procedure for solving Max-Cut problems to optimality. Math. Program. 143(1–2), 61–86 (2014) 6. Murray, W., Ng, K.-M.: An algorithm for nonlinear optimization problems with binary variables. Comput. Optimiz. Appl. 47(2), 257–288 (2008) 7. Pichugina, O., Yakovlev, S.: Polyhedral-spherical approach to solving some classes of com- binatorial optimization problems. In: Proceedings of the 6th International School-Seminar on Decision Theory, Uzhgorod, pp. 152–153 (2012) 8. Stoyan, Y.G., Yemets, O.: Theory and Methods of Euclidean Combinatorial Optimization. ISSE, Kiev (1993) 9. Stoyan, Y.G., Yakovlev, S.V., Parshin, O.V.: Quadratic optimization on combinatorial sets in Rn . Cybern. Syst. Anal. 27(4), 562–567 (1991)10. Stoyan, Y., Yakovlev, S., Emets, O., Valuiskaya, O.: Construction of convex continuations for functions defined on a hypersphere. Cybern. Syst. Anal. 34(2), 176–184 (1998)11. Yakovlev, S.: The theory of convex continuations of functions at the vertices of convex polygons. Comput. Math. Math. Phys. 34(7), 959–965 (1994)12. Yakovlev, S., Grebennik, I.: Certain classes of optimization problems on set distributions and their properties. Izvestiya Vysshikh Uchebnikh Zavedenii Mathematika 11, 74–86 (1991) Fixed Point Techniques in Analog Systems
Diogo Poças and Jeffery Zucker
Abstract Analog computation is concerned with continuous rather than discretespaces. Most of the physical processes arising in nature are modeled by differentialequations, either ordinary (example: spring/mass/damper system) or partial (exam-ple: heat diffusion). In analog computability, the existence of an effective way toobtain solutions (either exact or approximate) of these systems is essential. We develop a framework in which the solutions can be seen as fixed pointsof certain operators on continuous data streams, using the framework of Fréchetspaces. We apply a fixed point construction to retrieve these solutions and presentsufficient conditions on the operators and initial inputs to ensure existence anduniqueness of these corresponding fixed points.
1 Introduction
Analog computation, as conceived by Kelvin [10], Bush [1], and Hartree [4], isa form of experimental computation with physical systems called analog devicesor analog computers. Historically, data are represented by measurable physicalquantities, including lengths, shaft rotation, voltage, current, resistance, etc., andthe analog devices that process these representations are made from mechanical orelectromechanical or electronic components [5, 7, 9]. The main objects of our study are analog networks or analog systems, [6, 11–13],whose main components are described as follows:
Analog network D data C time C channels C modules:
We will model data as elements from a topological vector space A that isactually a Fréchet space. We will use the nonnegative real numbers as a continuousmodel of time T D Œ0; 1/. Each channel carries a continuous stream, representedas a function u W T ! A (this space is denoted by C.T; A /). Each module M is
D. Poças () • J. ZuckerDepartment of Mathematics and Statistics, McMaster University, Hamilton, ON, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 701J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_63 702 D. Poças and J. Zucker
Fig. 1 Analog network (with gtwo modules and threechannels) for the time t u Lu uevolution problem L 0
specified by a stream function
FM W C.T; A /k A ` ! C.T; A /:
In this way, we can think of analog networks as directed graphs where modulesare nodes and channels are edges (see Fig. 1). We will use analog systems to studythe time evolution problem (also called the Cauchy problem).Definition 1 (Time evolution problem, [3]) For a given initial condition g 2 Aand an operator L W A ! A , the time evolution problem is given by the system du dt D Lu; t 2 TI (1) u.0/ D g:
We look for a solution u 2 C.T; A /. To construct an analog system that represents the time evolution problem, we cansimply integrate the differential equation (1) to obtain Z t u.t/ D g C Lu.s/ds DW Fg .u/; (2) 0
where we use the right hand side to define an operator Fg W C.T; A / ! C.T; A /,which can be computed using an analog network with two modules. Introducing afeedback to implement the equality, we obtain the analog system of Fig. 1. We can informally define a specification of the analog network as a tuple ofstreams describing the data on all channels which satisfy the equations given by themodules. We can then observe the equivalence between the notions of (a) solutionsto the time evolution problem of Definition 1; (b) specifications of the analog systemof Fig. 1; and (c) fixed points of the operator Fg of Equation (2). Henceforth we willfocus on the last notion. Our goal is to provide sufficient conditions on Fg that ensureexistence and uniqueness of fixed points, as well as the existence of a constructivemethod to obtain fixed points when they exist. In Sect. 2 we introduce Fréchet spaces, which form the framework for ourproblem at hand. In Sect. 3 we assume analyticity of g to prove local existence andconvergence of fixed points (Theorem 2). In Sect. 4 we first extend our results toglobal existence and convergence (Theorem 3) and then extend our constructive Fixed Point Techniques in Analog Systems 703
method for different choices of initial input (Theorem 5). Finally, we turn touniqueness of fixed points and prove it for certain operators (Theorem 6).
2 Fréchet Spaces
For the remainder of this paper, we use the following notation:Notation 1 A is the space of infinitely differentiable functions, A D C1 .R/.
Notation 2 The operator L W A ! A is given by Lu D ˛@x u, for some ˛ 2 R. Thus, the operator Fg becomes Z t Fg .u/.t; x/ D g.x/ C ˛ @x u.s; x/ds: (3) 0
As we know, A D C1 .R/ is not complete under the supremum norm; however,for each x0 2 R, X 2 RC and k 2 N, we can define a pseudonorm by ˇ k ˇ ˇ@ f ˇ k f kX;x0 ;k ˇ D sup ˇ k .x/ˇˇ : jxx0 jX @x
It turns out that the notion of interest is that of Fréchet space, which we nowbriefly review (a detailed exposition can be found in [8, Ch. V]).Definition 2 (Fréchet space [8]) A Fréchet space is a topological vector spaceX whose topology is induced by a countable family of pseudonorms fk k˛ g˛2A .Moreover,• the family fk k˛ g˛2A separates points, that is,
u D 0 if and only if kuk˛ D 0 for all ˛I
• X is complete with respect to fk k˛ g˛2A , that is, for every sequence .xn / which is Cauchy with respect to each pseudonorm k k˛ , there exists x 2 X such that .xn / converges to x with respect to each pseudonorm k k˛ .Example 1 The space A D C1 .R/ of infinitely differentiable functions in R is aFréchet space with the countable family of pseudonorms given by ˇ k ˇ ˇ@ f ˇ k f kN;k ˇ D sup ˇ k .x/ˇˇ ; (4) NxN @x
for N; k 2 N. 704 D. Poças and J. Zucker
Example 2 The space C.T; A / of continuous streams is also a Fréchet space, withthe countable family of pseudonorms given by ˇ k ˇ ˇ@ u ˇ kukM;N;k D sup sup ˇˇ k .t; x/ˇˇ ; (5) 0tM NxN @x
for M; N; k 2 N. We can see that the family of pseudonorms in Example 2 is closely relatedto the family in Example 1. In fact, this illustrates a useful property of Fréchetspaces; in general, the space of continuous streams over a Fréchet space is itself aFréchet space. In other words, Fréchet spaces work well with the operation of takingcontinuous streams.Proposition 1 (New Fréchet spaces from old) If A is a Fréchet space with acountable family of pseudonorms fk k˛ g˛2A , then so is C.T; A / with the countablefamily of pseudonorms fk kM;˛ gM2N;˛2A , where
kukM;˛ D sup ku.t/k˛ : 0tM
Even though Fréchet spaces, as they stand, are not necessarily normed spaces, wecan define a metric from the pseudonorms, under which these spaces are complete.Proposition 2 Given a Fréchet space, we can define a complete metric from thefamily of pseudonorms which induces the same topology. For a proof, see [8, Ch. V], in particular Theorem V.5. The usefulness of complete metric spaces is evident due to the following.Theorem 1 (Banach fixed point theorem) Given a complete metric space .X; d/,suppose that T W X ! X is a contracting operator in the sense that there exists0 < 1 with
d.T.x/; T.y// d.x; y/ for all x; y 2 X:
Then T has a unique fixed point x . Moreover, for all x0 2 X the sequence ofiterations xn WD T n .x0 / converges to x .
3 Local Convergence Theorem
Consider the space C.T; A /. Take any (arbitrary but fixed) x0 2 R, X 2 RC , T 2 T.Then, for any k 2 N, we have a pseudonorm ˇ k ˇ ˇ@ u ˇ kukT;X;x0 ;k ˇ D sup ˇ k .t; x/ˇˇ : (6) 0tT @x jxx0 jX Fixed Point Techniques in Analog Systems 705
Fig. 2 Compact rectangles t
· T,X,x0 ,k
x x0 − X x0 x0 + X
Observe that we are taking suprema on compact rectangles of the form Œ0; T Œx0 X; x0 C X (see Fig. 2). The reason for taking suprema on compact rectangleswill be made clear shortly with Theorem 2 (local convergence theorem). We alsoobserve that, for each compact rectangle X0 D Œ0; T Œx0 X; x0 CX , we can definethe space of compact continuous streams C.Œ0; T ; C1 .x0 X; x0 CX//. Clearly, anyfunction in C.T; A / can be mapped to a function in C.Œ0; T ; C1 .x0 X; x0 C X//via the restriction u 7! uX0 . Moreover, C.Œ0; T ; C1 .x0 X; x0 C X// can be seento be a Fréchet space with the family of pseudonorms k kT;X;x0 ;k given by (6). Notethat x0 , X and T are fixed and the indexing is on k 2 N. Finally, observe that the operator Fg W C.T; A / ! C.T; A / has a restrictionFgX0 to the space C.Œ0; T ; C1 .x0 X; x0 C X//. Our next step is to prove contraction inequalities, which play an important rolein fixed point techniques.Lemma 1 (Contraction inequalities) Consider the Fréchet space C.T; A / withpseudonorms kkT;X;x0 ;k given by (6). Let g 2 C1 .R/ and Fg W C.T; A / ! C.T; A /be given by (3). Then, for any u; v 2 C.T; A /, any pseudonorm k kT;X;x0 ;k and anym 2 N, we have the following bound:
.j˛jT/m kFgm .u/ Fgm .v/kT;X;x0 ;k ku vkT;X;x0 ;kCm : (7) mŠProof By induction on m. t u Let us see how we can use these bounds in a proof.Theorem 2 (Local Fréchet space convergence theorem) Consider the Fréchetspace C.T; A / with pseudonorms k kT;X;x0 ;k given by (6). Take an initial inputu0 2 C.T; A / and initial condition g 2 C1 .R/. Assume also that u0 D 0 and g isanalytic at x0 with some radius of convergence1 R. Let Fg W C.T; A / ! C.T; A /be given by (3). Then, for any T; X 2 RC such that j˛jT C X < R, the sequence .um /
1 Or equivalently, that g has a holomorphic extension on a disk of the complex plane with center x0and radius R; see Remark 1. 706 D. Poças and J. Zucker
given by um D Fgm .u0 / converges in the rectangle X0 D Œ0; T Œx0 X; x0 C X toa fixed point of FgX0 .Proof To ease the exposition we introduce the pseudonorms on g given by ˇ k ˇ ˇ@ g ˇ kgkX;x0 ;k ˇ D sup ˇ k .x/ˇˇ ; for x0 2 R; X 2 RC ; k 2 N: (8) jxx0 jX @x
Since g is analytic at x0 with radius of convergence R, there is a sequence of realcoefficients .aj / such that, for all x 2 .x0 R; x0 C R/,
1 X g.x/ D aj .x x0 /j : jD0
p 1 It also follows that lim supn!1 n jan j R (see Footnote 1). Moreover, we havethe following bound, for any X < R: ˇ ˇ ˇ X1 ˇ X 1 ˇ . j C k/Š ˇ . j C k/Š kgkX;x0 ;k D ˇˇ sup ajCk .x x0 /j ˇˇ jajCk jX j : (9) ˇjxx0 j<X jD0 jŠ ˇ jD0 jŠ
Let T; X 2 RC such that j˛jT C X < R. We show that .um / is a Cauchy sequencewith respect to the pseudonorm k kT;X;x0 ;k . First observe that 1 X 1 X kumC1 um kT;X;x0 ;k D kFgm .g/ Fgm .0/kT;X;x0 ;k mD0 mD0 1 X 1 Tm j˛jm kgkX;x0 ;kCm mD0 mŠ 1 X X 1 2 .k C m C j/Š j˛jm T m X j jakCmCj j mD0 jD0 mŠjŠ 1 X X s 3 .k C s/Š D j˛jm T m X sm jakCsj sD0 mD0 mŠ.s m/Š 1 X 4 .k C s/Š D .j˛jT C X/s jakCsj ; sD0 sŠ
where (1) is justified by the Contraction Inequalities (Lemma 1), (2) by equation (9),(3) by rearranging the sum and adding over diagonals s D m C j and (4) by takingthe binomial expansion of .j˛jT C X/s . Fixed Point Techniques in Analog Systems 707
By the root test, the above series is convergent, since r s .k C s/Š p R lim sup .j˛jT C X/s jakCs j D .j˛jT C X/ lim sup n jan j 1 < D 1: s!1 sŠ n!1 R
Since the series is convergent, it follows that, for i < j,
X j1 1 Xkuj ui kT;X;x0 ;k kumC1 um kT;X;x0 ;k kumC1 um kT;X;x0 ;k ! 0: i!1 mDi mDi
Hence .um / is a Cauchy sequence with respect to the pseudonorm k kT;X;x0 ;k .Since this holds for all k 2 N and C.Œ0; T ; C1 .x0 X; x0 C X// is complete, itfollows that .um / has a limit in X0 . Now, using continuity of FgX0 , we conclude thatthis limit must be a fixed point of FgX0 . t uRemark 1 The reader should distinguish between the following two concepts:• the existence of a holomorphic function, defined in a disk of the complex plane C, which coincides with g at the real axis fy D 0g;• the convergence of the construction um D Fgm .0/ to a fixed point u, defined in a rectangle of T R, which coincides with g at initial time ft D 0g. As seen from Theorem 2, the existence of a holomorphic extension impliesconvergence to a fixed point. Both these functions (the holomorphic extensionand the fixed point) can be depicted by planar diagrams, and both can be seen asextensions of g (see Fig. 3). However, these functions, and the domains which theyinhabit, are substantially different.
y t C T×R R T g(x + iy) u(t, x) · · · x · · · x x0 − R x0 x0 + R x0 − X x0 x0 + X
Fig. 3 On the left: a function g.x C iy/ of type C ! C, defined in a disk, that coincides with gat fy D 0; x0 R < x < x0 C Rg. On the right: a fixed point u.t; x/ of type T R ! R, definedin a rectangle, that coincides with g at ft D 0; x0 X < x < x0 C Xg. The rectangle and diskdimensions follow the relation j˛jT C X < R 708 D. Poças and J. Zucker
4 Global Convergence Theorems
As an immediate corollary of Theorem 2, we have:Theorem 3 (First global Fréchet space convergence theorem) Consider theFréchet space C.T; A / with pseudonorms k kT;X;x0 ;k given by (6). Take an initialinput u0 2 C.T; A / and initial condition g 2 C1 .R/. Assume also that u0 D 0and g is entire (i.e. has a holomorphic extension to the complex plane). LetFg W C.T; A / ! C.T; A / be given by (3). Then the sequence .um / given byum D Fgm .u0 / converges to a fixed point of F.Proof Since g is entire, it is analytic at any x0 with any radius of convergence R.Thus, by Theorem 2, the sequence um converges to a fixed point on any compactrectangle Œ0; T Œx0 X; x0 C X . Therefore, we have convergence of um for anypseudonorm k kT;X;x0 ;k , so that we have convergence in C.T; A /. t u The next step is to generalize Theorem 3 to a larger class of initial functions u0(other that u0 D 0). We do that proof in two steps: first assume g D 0 to establishsufficient conditions on u0 ; then consider the more general case g 2 C1 .R/.DefinitionP1 3 We say that a function u 2 C.T; A / is uniformly entire if u.t; x/ D jD0 a j .t/x j for some sequence of functions .aj / 2 C.R/ such that r lim j sup jaj .t/j D 0 for all T 2 T: 0tT
The motivation for this terminology is that, for p such a function u, the sectionx 7! u.t; x/ is entire for all t, and the convergence j jaj .t/j ! 0 is uniform in t.Theorem 4 Consider the Fréchet space C.T; A / with the family of pseudonormsk kT;X;x0 ;k given by (6). Let also u0 2 C.T; A / be an initial input, and g 2 C1 .R/be an initial condition. We assume in addition that u0 is uniformly entire and g D 0.Let F0 W C.T; A / ! C.T; A / be given by Z t F0 .u/.t; x/ D ˛ @x u.s; x/ds: (10) 0
Then the sequence .um / given by um D F0m .u0 / converges to 0.Proof To ease the exposition we introduce the pseudonorms on aj given by
kaj kT D sup jaj .t/j, for T 2 T: (11) 0tT Fixed Point Techniques in Analog Systems 709
X We show that kum kT;X;0;k is a convergent series for any pseudonorm kkT;X;x0 ;k mwith x0 D 0. We have that (see proof of Theorem 2) 1 X 1 X kum kT;X;0;k D kF0m .u0 / F0m .0/kT;X;0;k mD0 mD0
X1 j˛jm T m ku0 kT;X;0;kCm mD0 mŠ ˇ ˇ 1 ˇX ˇ X j˛jm T m ˇ 1 .j C k C m/Š ˇ D sup ˇˇ ajCkCm .t/xj ˇˇ mD0 mŠ 0tT ˇ jD0 jŠ ˇ jxjX 1 X X 1 .j C k C m/Š j˛jm T m X j kajCkCm kT mD0 jD0 mŠjŠ 1 X X s .k C s/Š D j˛jm T m X sm kakCs kT sD0 mD0 mŠ.m s/Š
X1 .k C s/Š D .j˛jT C X/s kakCs kT : sD0 sŠ
By the root test, the above series is convergent, since r s .k C s/Š .j˛jT C X/s ! j˛jT C X sŠ s!1
p Pand s kakCs kT ! 0 by assumption. Therefore kum kT;X;0;k is convergent, so that s!1kum kT;X;0;k ! 0 and thus um converges to 0, as we wanted to prove. t u m!1
We now combine Theorems 3 and 4 to prove our most general result.Theorem 5 (Second global Fréchet space convergence theorem) Consider theFréchet space C.T; A / with pseudonorms k kT;X;x0 ;k given by (6). Take an initialinput u0 2 C.T; A / and initial condition g 2 C1 .R/. Assume also that u isuniformly entire and that g is entire. Let Fg W C.T; A / ! C.T; A / be given by (3).Then the sequence .um / given by um D F m .u0 / converges to a fixed point of F.Proof Let Fg ; F0 W C.T; A / ! C.T; A / be given by (3), (10). We observe that, forany u; v 2 C0;1 .X/ we have Z t Z t Z tFg .u C v/ D g C ˛ .u C v/x ds D g C ˛ ux ds C ˛ vx ds D Fg .u/ C F0 .v/: 0 0 0 710 D. Poças and J. Zucker
We can then infer that u1 D Fg .u0 / D Fg .0 C u0 / D Fg .0/ C F0 .u0 /. Also,u2 D Fg .u1 / D Fg .Fg .0/ C F0 .u0 // D Fg2 .0/ C F02 .u0 /, and, in general,
um D Fgm .0/ C F0m .u0 /:
By Theorem 3, .Fgm .0// converges to a fixed point of Fg . By Theorem 4, .F0m .u0 //converges to 0. Therefore, .um / and .Fgm .0// have the same limit. In particular, .um /converges to a fixed point of Fg . t u A nice consequence of the proof is that it allows us to also establish uniquenessin a certain class of functions.Theorem 6 (Uniqueness of uniformly entire fixed points) Consider the Fréchetspace C.T; A / with pseudonorms k kT;X;x0 ;k given by (6). Take an initial conditiong 2 C1 .R/ and assume also that g is entire. Let Fg W C.T; A / ! C.T; A / begiven by (3). Then there is at most one uniformly entire fixed point of Fg .Proof Let u be any uniformly entire fixed point of Fg . By the proof of Theorem 5,we know that u D Fgm .u/ D Fgm .0/ C F0m .u/. Since .F0m .u0 // converges to 0, weget that .Fgm .0// converges to u. Thus any uniformly entire fixed point of Fg mustcoincide with the limit of .Fgm .0//. t u
5 Conclusion and Further Research
In this paper we have seen how to study solutions to differential equations as outputsof analog networks, and how to obtain them using fixed point techniques. Theexample we have considered (L D ˛@x ) is a well-known problem whose solutioncan be obtained analytically by taking a Fourier transform. However, the methodpresented in this paper provides a different perspective which is suitable for analogcomputability and the study of analog systems as in [6, 11–13], where Fréchetspaces clearly provide a natural framework. We intend to investigate this approach (fixed points in Fréchet spaces) in moregeneral settings. In fact, as a next step one can look at a more general operatorL W A ! A using higher-order derivatives, for example with bounds of the form
kLukT;X;x0 ;k Ck@`x ukT;X;x0 ;k CkukT;X;x0 ;kC` : (12)
We observe that analyticity of the initial condition g is not enough to ensureexistence of solutions. A counterexample is given by the heat equation ut D uxx 1with initial condition g.x/ D 1x . Even though g is analytic near zero, the solutionfails to be analytic at a neighborhood of the origin (see [2]). Thus, the general casemay require different tools such as explicit bounds on the pseudonorms of g. We also plan to investigate properties of the fixed points, such as computability,continuity and stability, as functions of the parameters and initial conditions. Fixed Point Techniques in Analog Systems 711
References
1. Bush, V.: The differential analyzer. A new machine for solving differential equations. J. Frankl. Inst. 212(4), 447–488 (1931) 2. Egorov, Y.V., Shubin, M.A.: Foundations of the Classical Theory of Partial Differential Equations. Springer-Verlag, Berlin (1998) 3. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Dover, New York (1952) 4. Hartree, D.R.: Calculating Instruments and Machines. Cambridge University Press, Cambridge (1950) 5. Holst, P.A.: Svein Rosseland and the Oslo Analyzer. IEEE Ann. Hist. Comput. 18(4), 16–26 (1996) 6. James, N.D., Zucker, J.I.: A class of contracting stream operators. Comput. J. 56, 15–33 (2013) 7. Johansson, M.: Early analog computers in Sweden – with examples from Chalmers University of Technology and the Swedish aerospace industry. IEEE Ann. Hist. Comput. 18(4), 27–33 (1996) 8. Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Functional Analysis. Academic Press Inc., San Diego (1980) 9. Small, J.S.: General-purpose electronic analog computing: 1945–1965. IEEE Ann. Hist. Comput. 15(2), 8–18 (1993)10. Thompson, W., Tait, P.G.: Treatise on Natural Philosophy, vol. 1, (Part I), 2nd edn. Cambridge University Press, Cambridge (1880)11. Tucker, J.V., Zucker, J.I.: Computability of analog networks. Theor. Comput. Sci. 371, 115–146 (2007)12. Tucker, J.V., Zucker, J.I.: Continuity of operators on continuous and discrete time streams. Theor. Comput. Sci. 412, 3378–3403 (2011)13. Tucker, J.V., Zucker, J.I.: Computability of operators on continuous and discrete time streams. Computability 3, 9–44 (2014) A New Look at Dummy Derivativesfor Differential-Algebraic Equations
John D. Pryce and Ross McKenzie
Abstract We show the dummy derivatives index reduction method for DAEs,introduced in 1993 by Mattsson & Söderlind, is a particular case of the Pryce˙-method solution scheme. We give a pictorial display of the underlying blocktriangular form. This approach gives a simple general method to cast the reduced system in semi-explicit index 1 form, combining order reduction and index reduction in one process. It also shows each DD scheme for a given DAE is uniquely described by aninteger “DDspec” vector ı. The method is illustrated by an example. We give various reasons why, contrary to common belief, converting further fromsemi-explicit index 1 form to an explicit ODE, can be a good idea for numericalsolution.
1 Introduction
We give a brief summary, wherein terms in slanted font are defined more fully later. The dummy derivatives (DDs) method for a differential-algebraic equation(DAE), described as a way to reduce it to a (locally) equivalent DAE of index 1,was introduced by Mattsson and Söderlind [2] (here, MS denotes this paper or itsauthors). MS use the Pantelides method [4] for structural analysis (SA) of the DAE; thiscan be replaced by the Pryce signature matrix method (˙-method) [5] which can dothe same analysis more simply, and is more powerful in that Pantelides only appliesto DAEs of first order. Hence the DAEs to which DDs, as described by MS, canbe applied are precisely the SA-friendly DAEs: those for which SA (Pantelides orPryce) succeeds , by giving a nonsingular System Jacobian at some point. The batches of derivatives formed by DDs are exactly those in stages of the ˙-method’s standard solution scheme , but in reverse order, as proved in [3]. This paperfollows up this connection. Adapting notation in [5], we give a concise description
J.D. Pryce () • R. McKenzieSchool of Mathematics, Cardiff University, Cardiff CF24 4AG, Wales, UKe-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 713J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_64 714 J.D. Pryce and R. McKenzie
of the block-triangular form (BTF) of dependencies between batches, plus a pictorialview. Each DD scheme for a given DAE is uniquely described by an integer DD-specification vector ı D .ı1 ; : : : ; ın /. The DAE resulting from DDs is generallycalled “index 1” but has the somewhat stronger property of being an implicit ODE . MS ignores the conceptually trivial but practically important issue of reductionto order 1, needed when giving a DAE to a solver such as DASSL [1]. Our approachunifies index and order reduction into one process.
2 Basic Theory
DAEs and index As in MS (but with a different notation), we consider DAEs ofthe general form
fi . t; the xj and derivatives of them / D 0; i D 1; : : : ; n; (1)
where the xj .t/; j D 1; : : : ; n are state variables that are functions of an independentvariable t, usually regarded as time. The classic differentiation index d is the largest number of times some equationfi of the system must be differentiated w.r.t. t to produce an enlarged system ofequations, solvable as an algebraic system to give an equivalent ordinary differentialequation system (ODE) for the state variables xj .t/.Example 1 To contrast the above with the DDs viewpoint below, consider the DAE
0 D A WD x u.t/; 0 D B WD xP y; (dot denotes d=dt); (2)
where u is a given forcing function. It is an edge-case (in computer science terms) inhaving a unique solution—zero degrees of freedom—but the general d definitionstill applies. One must differentiate B once to give an equation with yP in it; now xRappears, so differentiate A twice to produce xR which can then be eliminated. Thusd D 2, since combining xR yP D 0 and xR uR .t/ D 0 with the original set one obtainsan ODE
xP D y; yP D uR .t/: (3)
Then (3) has the same solution as (2) through any consistent point of (2). But (3) hasDOF D 2 so its general solution x D u.t/C˛ Cˇt; y D u P .t/Cˇ (with ˛; ˇ arbitraryscalars) includes many beside the unique solution x D u.t/; y D uP .t/ of (2). (DOF means the phrase “degrees of freedom”, DOF is the correspondingnumber.) Solving the ODE numerically as a proxy for the DAE results in progressive driftfrom the DAE’s solution manifold, which in the above example, as a subset of theODE’s .t; x; y/ state space, is the set M D f .t; x; y/ 2 R3 j x D u.t/, y D uP .t/ g. A New Look at Dummy Derivatives for Differential-Algebraic Equations 715
By contrast the DDs method gives a system genuinely equivalent to the original,but with no extra DOF that cause numerical drift. It is not concerned with producingan ODE in all the xj but, like the ˙-method, seeks merely the least number ofdifferentiations that allows us to find all the xj as functions of t. In Example 1 theanswer is clearly to differentiate A once and B not at all, and rearrange to get
x D u.t/; y D uP .t/; (4)
which unlike (3) has no degrees of freedom, i.e. shows the unique solution. Strictlyby DD rules one obtains (4) via a DAE with three equations and variables, namely
0 D A D x u.t/; 0 D B D x0 y; 0 D AP D x0 uP .t/;
where the “genuine” derivative xP has been replaced by the dummy derivative x0 .˙ -method summary The ˙-method finds the n n signature matrix ˙ D .ij /,where ij is the order of the highest derivative of xj in fi , if xj occurs in fi , and 1if not. A transversal is a set T of n matrix positions .i; j/, with just one in eachrow and each column, P and we seek a highest-value transversal (HVT), such that thenumber Val.T/ D .i;j/2T ij is maximised. We find corresponding dual variables,the offsets, integer n-vectors c D .c1 ; : : : ; cn / and d D .d1 ; : : : ; dn / satisfying
dj ci ij ; with equality on a HVT: (5)
By default we choose the canonical offsets, the unique elementwise smallest oneshaving mini ci D 0. Then ci says how many times equation fi D 0 is to bedifferentiated, and (5) fulfils the requirement stated in Sect. 2.2 of MS, that “thedifferentiated problem is structurally nonsingular with respect to its highest-order .d / .c /derivatives”—i.e., xj j occurs in fi i for each .i; j/ in some transversal. The ˙-method’s standard solution scheme (SSS) says: let kd D maxj dj andkc D maxi ci , then
for k D kd ; kd C1; : : : do Solve the mk equations: fi.kCci / D 0 for those i such that k C ci 0 .kCdj / (6) for the nk unknowns: xj for those j such that k C dj 0, using already known values from previous (with smaller k) stages.
The sizes mk ; nk satisfy mk nk , i.e., each set of equations is underdetermined orsquare. Clearly, mk ; nk increase with k, and D n for k 0. The iteration starts atk D kd because mk D 0 for k < kc , nk D 0 for k < kd , and kd kc . When solving a DAE by Taylor series expansion, k goes to some positiveexpansion order. In the DDs context we stop at k D 0. Here, a consistent point .q/of the DAE shall mean a solution of (6) up to stage k D 0, i.e. values of xj for . p/q D 0 W dj , j D 1 W n, that satisfy fi D 0 for p D 0 W ci ; i D 1 W n. A fuller definitionis in [5]. 716 J.D. Pryce and R. McKenzie
Henceforth we assume SA succeeds—or the DAE is SA-friendly—meaning then n System Jacobian .c / .d / J D @fi i =@xj j : (7) i;jD1 W n
is nonsingular P at some P consistent point. Then, the value Val.T/ of a HVT, whichalso equals j dj i ci , is the number of degrees of freedom DOF of the DAE; also .kCc / .kCd / the mk nk Jacobian Jk D @fi i =@xj j of (6) is of full row rank; (8)
and Jk is the sub-matrix (rows where k C ci 0, columns where k C dj 0) of J. An implicit ODE is an SA-friendly DAE with c D 0, so one can solve to getan ODE without differentiating any equations. For instance the DAE xP C yP D 1,x y D 0 is index 1 but not an implicit ODE, while xP C y D 1, x y D 0 is animplicit ODE.Example 2 Figure 1 shows the DAE of the simple pendulum in Cartesian coordi-nates, with signature matrix ˙ annotated with the canonical offsets, and its HVTsmarked ; ı . The Jacobian J is annotated in the style of MS, also showing the offsets. For conciseness we use notation introduced in [5]. Write xjl to mean the lth .l/ .l/derivative xj of variable xj D xj .t/, similarly fil to mean the lth derivative fi offunction fi . Thus for an index-pair .i; l/ or . j; l/ unless said otherwise it is assumedthat i; j; l are integers with i; j 2 1 W n and l 0. For a set S of index-pairs let xS be the vector comprising all the xjl for . j; l/ 2 S,listed in some arbitrary but fixed order. It may mean a vector of symbolic variables,or real numbers, or functions of t, depending on context. Similarly, let fS be thevector comprising all the fil for .i; l/ 2 S, in some arbitrary but fixed order. For a given DAE with offsets ci , dj and for any integer k define index-pair sets
Ik D f .i; l/ j l D k C ci 0 g; Jk D f . j; l/ j l D k C dj 0 g: (9)
Write jXj for the number of elements in a finite set X. Then the sizes mk ; nk in (6)are given by jIk j D mk and jJk j D nk . Further define Ik to be the union of Il for all l k, and similarly I<k , Jk , J<k .Then the SSS, equation (6), taken up to k D 0, can be written
Fig. 1 Pendulum DAE with signature matrix and Jacobian A New Look at Dummy Derivatives for Differential-Algebraic Equations 717
Fig. 2 Ik , Jk , fIk and xJk for the pendulum, using offsets c D .0; 0; 2/ and d D .2; 2; 0/. Forexample, J1 is the set f.1; 1/; .2; 1/g so that xJ1 D .x11 ; x21 /, which is the same as .Px; yP /
for k D kd ; kd C1; : : : ; 0 do (10) solve fIk .t; xJk / D 0, or equivalently fIk .t; xJ<k I xJk / D 0, for xJk .
The second form separates the “already known” t and xJ<k from the “to befound” xJk . For example, Fig. 2 tabulates Ik , Jk , fIk and xJk for the pendulum, taking x1 ; x2 ; x3to mean x; y; and f1 ; f2 ; f3 to mean A; B; C, in the order given.
3 DDs as a Particular Case of the SSS
In the “vanilla” SSS, all components in xJk are on an equal footing. For numericalsolution, in those stages k < 0 where an underdetermined system is to be solvedbecause mk < nk , typically a trial xJk vector is projected on the manifold defined byfIk D 0, using a Gauss-Newton method or similar, to obtain the accepted solution.Example 3 For the pendulum, stage k D 2 consists in projecting a trial value ofxJ2 D .x10 ; x20 / D .x; y/ on the zero set of fI2 D . f30 / D .h/, i.e. on the circlex2 C y2 D L2 in the x; y plane. Stage k D 1 consists in projecting a trial valueof xJ1 D .x11 ; x21 / D .Px; yP / on the zero set of fI1 D . f31 / D .h/, P i.e. on the linexPx C yPy D 0 in the xP ; yP plane defined by the previously found values of x and y. In DDs, the components in xJk are not on an equal footing. The method as .d /presented in MS starts with the high-order derivatives xj j and works down, at eachstage selecting some derivatives to be dummy, which essentially means they arefound, as functions of others. For the pendulum, cf. [2, Sect. 4, Example 3], wheny ¤ 0 one can choose yR ; yP to be dummy derivatives y00 ; y0 , giving a reduced DAEwhere everything is a function of x and xP ; when x ¤ 0 one can do the reverse. From the ˙-method angle it is natural to go the opposite way, starting with thelow-order derivatives. DDs then becomes a particular way to perform the SSS, aswe now describe. It is proved in [6] that for a given DAE, the set of DD schemesconstructible by this “forward” method is identical with the set constructible by theclassical “reverse” method. 718 J.D. Pryce and R. McKenzie
For brevity let all the unknowns that are found by the SSS in stages k D kd W 0 be .q/called items. That is, an item is any xj with 0 q dj , j 2 1 W n. Equivalently, theitems are precisely the components of xJ0 .The forward DDs method We assume henceforth that the functions fi are suffi-ciently smooth for all needed uses of the Implicit Function Theorem (IFT). For eachstage k the mk nk system (10) is of full row rank by (8). Hence we can find subsetsFk Jk and Sk D Jk n Fk such that, denoting nk mk by DOFk
jFk j D mk ; jSk j D DOFk ;
and the mk mk sub-matrix of Jk defined by
Gk D .columns of Jk corresponding to Fk /; (11)
is nonsingular at the current consistent point. (Gk is the same as GΠin MS, where D 1 k.) Then by the IFT the mk items forming the vector xFk can locally befound as functions of the remaining DOF k items, which form xSk . DOF k is the number of DOF introduced at solution stage k. E.g, for the pendulum,nk mk D 2 1 D 1 for k D 2 and k D 1. That is, one DOF is introduced ateach of these stages: an arbitrary position on the circle, and an arbitrary velocity. The subvector xFk of xJk comprises the items found at this stage as functions ofthe items in xSk . We call the latter state items. We call the sequence of Fk , whichdefines all the found items, a solving scheme. The state vector is the vector of all state items, that is S xS ; where S D kd k1 Sk : (12)
Note we stop at k D 1 because S0 is empty. The components of xS are the statevariables of the implicit ODE, to which this solving scheme reduces the DAE. Allthe remaining items are “found” and form the found vector S xF ; where F D kd k0 Fk : (13)
Since each vector xJk is the concatenation of xFk and xSk , the SSS in the form (10),combined with the IFT used in a solving scheme, becomes
for k D kd ; kd C1; : : : ; 0 do (14) solve fIk .t; xS<k ; xF<k I xSk ; xFk / D 0 for xFk as a function of xSk and xF<k .
This set of equations has a block-triangular form (BTF) such that, by repeated useof the IFT, and repeated substitution, one hasTheorem 1 Each found item can be expressed as a function of the state vector xS . A New Look at Dummy Derivatives for Differential-Algebraic Equations 719
Proof We show this for the case kd D 2, from which the general proof is clear. For kD2, xS<k and xF<k are empty vectors, so we find xF2 as a function of xS2 .For k D 1, we find xF1 as a function of xS1 , xS2 and xF2 , which by substitutingxF2 becomes a function of xS1 . For k D 0, we find xF0 as a function of xS0 , xS1 ,xS2 , xF1 and xF2 , which by substituting xF1 and xF2 becomes a function ofxS0 . Since xS0 equals xS , of which xS1 and xS2 are subvectors, the result follows. Numerically, this involves one root-finding at each k-stage, taking given inputvalues of xS and using the nonsingular matrices Gkd ; Gkd C1 ; : : : ; G0 D J in turn. Not all solving schemes are useful in practice. As so far described, for thependulum one can choose state items y at stage 2 and xP at stage 1, which doesnot lead to an implicit ODE. The extra rule needed to make a useful scheme isDefinition 1 A dummy derivative scheme (DD scheme) is a solving scheme forwhich, if an item is a derivative of a found item, then it is also a found item. Equival-ently, the projections FO k Df j j . j; l/ 2 Fk g, of Fk on the j component, increasewith k.For instance, in the preceding paragraph, choosing y and xP as state items violatesthis constraint, since x must be a found item while its derivative xP is not. Some derivative of each xj is a found item, since stage k D 0 is an n n system .d /that solves for each leading derivative xj j . Hence there is a unique least integer .ı /ıj 2 0 W dj such that xj j is a found item. The vector ı D .ı1 ; : : : ; ın / is the DD-specification vector, or DDspec, of a DD scheme. Then .l/– An xj with l < ıj is a state item. .l/– An item xj with l > ıj is necessarily a found item and the derivative of a found item. It is a dummy derivative—this accords with the MS definition. .ı /– The remaining n items xj j are loop-closers if ıj > 0, as they create the relations in the reduced DAE that make an ODE, and algebraic items if ıj D 0 (see below).The total number of state items equals DOF, and there are ıj state items P for each j.Hence any DDspec vector satisfies 0 ı d (elementwise), and j ıj D DOF .Not every ı with these properties defines a DD scheme. E.g., for the pendulum,ı D .2; 0; 0/, .0; 2; 0/ and .1; 1; 0/ obey these constraints but only the first two“work”. Figure 3 shows the process pictorially. On the left is the general scheme for thecase kd D 2. Each set of equations fIk D 0 has its output (what is solved for) aboveit, and downward lines from it to the previously computed data it uses as input. Onthe right is the more detailed data for the particular case of Example 4 below. Theseare created by MATLAB functions, in the second case using structural data producedby DAESA’s function daeSA from the MATLAB code for the DAE in Example 4. The diagram uses natural notation for derivatives, e.g. x02 (MATLAB graphics ispoor at dots) instead of the x21 notation used in Example 4. 720 J.D. Pryce and R. McKenzie
Fig. 3 DDs process picture. Left: general scheme, kd D 2. Right: specific picture for Example 4
Scheme (14), whether done conceptually or numerically, splits the xj into“differential variables” having ıj > 0, and “algebraic variables” having ıj D 0.E.g., if there are variables u; v; w whose ı’s are 2; 1; 0, then u; v are differential andw is algebraic. Two loop-closer equations create a 2nd-order ODE uR D U.u; uP ; v/,vP D V.u; uP ; v/. After one solves the ODE to get u; uP ; v as functions of t, the relationwDW.u; uP ; v/ gives w. As another example, the DAE in (2) has empty differentialpart. Depending on DAE structure, one might find higher derivatives along the way,e.g. if doing DDs for the pendulum with state variables x; xP , one cannot find thealgebraic variables y and without also finding yP and yR at the same time.
4 Preparing for Numerical Solution
Converting to First Order Implicit ODE How to reduce the result of DDs tofirst order seems not well explained in the literature. Online Modelica tutorialscommonly order-reduce before SA and DDs: clumsy and giving a larger final systemthan necessary. In fact this task fits almost trivially into DDs, in the following steps. In Step 1, write down the SSS equations (6) up to stage 0—this is independent ofany DD scheme. We use xjl notation to show the itemsP count as unrelated variables. PThere are DOF more variables than equations, ND.nC j dj / versus MD.nC i ci /. In Step 2, make the system “square” by adding DOF new equations
xP jl D xj;.lC1/ ; (15)
one for each state variable xjl in xS , to say what its derivative “really is”. If xj;.lC1/ isa state item, this implements order-reduction; otherwise it is a loop-closer. A New Look at Dummy Derivatives for Differential-Algebraic Equations 721
In Step 3, rearrange the variables with the state vector xS followed by the foundvector xF , see (12) and (13), to get the result of DDs in the form
xP S D E.t; xS ; xF /; 0 D F.t; xS ; xF /: (16)
Function E is very simple: its components are all the xj;.lC1/ ’s of (15); it does notreally depend on t. Function F is actually fI0 . Write this as the concatenation of fIklisted k D kd ; kd C1; : : : ; 0, and xF as the concatenation of xFk in the same order. Then @F=@xF is block lower triangular with the nonsingular matrices Gk on itsblock diagonal, so it is nonsingular. Hence the two equations (16) form a semi-explicit index-1 DAE, see [1, p. 34]. From this easily follows:Theorem 2 System (16) is SA-friendly with offsets ci all 0, so it is an implicit ODE.Example 4 MS use the following linear, constant coefficient DAE [2, Example 1]:
MSEXAMPLE1 x1 x2 x3 x4 ci A 20 32 xR 1 xR 2 xR 3 xP 4 0ı0 D A D x1 Cx2 Cu1 .t/ AR2 1 1 0 030 D B D x1 Cx2 Cx3 Cu2 .t/ B6 0 0 0ı 7 2 R 1 ˙D C 4 0 JD B6 1 1 0 7: ; 1 0ı 5 1 4 50 D CD x1 CPx3 Cx4 Cu3 .t/ CP 0 0 1 1 D 2ı 2 2 1 00 D DD 2Rx1 CRx2 CRx3 CPx4 Cu4 .t/ D 2 1 1 1 dj 2 2 2 1
The ui .t/ are given forcing functions. The two HVTs of ˙ are shown and ı . Wehave c D .2; 2; 1; 0/ and d D .2; 2; 2; 1/, so J D @.A;R B; R C;P D/[emailprotected] ; xR 2 ; xR 3 ; xP 4 /. Itis nonsingular, so SA succeeds and there are 2 DOF. Step 1, using the xil notation, writes down M D 9 equations in N D 11 variables,grouped here by stages k D 2; 1; 0: 0 D A D x10 Cx20 Cu1 .t/ (17) 0 D B D x10 Cx20 Cx30 Cu2 .t/ 9 0 D AP D x11 Cx21 CPu1 .t/ = 0 D BP D x11 Cx21 Cx31 CPu2 .t/ (18) ; 0 D C D x10 Cx31 Cx40 Cu3 .t/ 9 0 D AR D x12 Cx22 CRu1 .t/ >> = 0 D BR D x12 Cx22 Cx32 CRu2 .t/ (19) 0 D CP D x11 Cx32 Cx41 CPu3 .t/ >> ; 0 D D D 2x12 Cx22 Cx32 Cx41 Cu4 .t/
There are two possible DDspecs, ı D .2; 0; 0; 0/ and .0; 2; 0; 0/. Following MS,we choose the latter, so the state variables are x20 and x21 . Step 2 now defines twoequations; the upper one does order reduction and the lower is a loop-closer: xP x xP S D 20 D 21 D E.t; xS ; xF /: (20) xP 21 x22 722 J.D. Pryce and R. McKenzie
Step 3 splits the variables into xS D .x20 I x21 /T and xF D .x10 ; x30 I x11 ; x31 ; x40 Ix12 ; x22 ; x32 ; x41 /T , where the semicolons group into stages k D 2; 1 for xS and2; 1; 0 for xF . Then (17), (18), and (19) jointly define the equations
0 D F.t; xS ; xF /; (21)
and (20), (21) give the desired semi-explicit index 1 form. The overall Jacobian is below, where a blank means h 1 an all-zero i block of the 00 appropriate size, and G0 D J with sub-matrices G1 D 1 1 0 , G2 D 11 01 . 011
xS xF2 xF1 xF0 Œx20 ; x21 Œx10 ; x30 Œx11 ; x31 ; x40 Œx12 ; x22 ; x32 ; x41 " #2 " # " # 3 x21 01 0000 ED 6 7 "x22 #6 6 "0 0# 0100 7 7 A 66 10 7 7 fI2 D 6 G2 7 B 6 1 0 7 2 36 2 3 2 3 7 @.E; F/ AP 6 0 1 00 7 D 6 7 : (22) @.xS ; xF / 6 P 76 6 7 6 7 7 fI1 D 4 B 56 40 15 40 05 G1 7 6 7 C 6 00 10 7 2 36 6 2 3 7 7 AR 6 000 7 6 76 6 7 7 6 BR 76 60 0 07 7 fI0 D 6 76 6 7 7 6 P 76 6 7 G0 7 4C 54 41 0 05 5 D 000
This formulation was confirmed by giving it to the MATLAB implicit solver ode15i,for certain ui .t/ and initial values, and comparing with the analytic solution.Converting to explicit ODE Conceptually it is obvious from the nonsingularity of@F=@xF in the DAE (16) that near a consistent point we can convert it to an explicitODE yP D f.t; y/—solve the second equation for xF and substitute in the first. It mayseem strange to do this numerically, in view of the impressive track record of codessuch as DASSL [1], for solving DAEs of the form G.t; z; zP / D 0. But there are reasons for doing so. Some applications generate models with n inthe thousands and only a few DOF, and for which stiffness is not a problem. Thenit makes sense to convert to a small explicit ODE and solve by, say, a Runge–Kuttacode, using far less working memory than would a DAE code. The ESI-CyDesignModelica system does numerical solution this way, having found it more efficientfor their typical models. The Numerical Algorithms Group (NAG) Ltd have recentlyput a reverse communication RK code into their library, for just such uses. Also, ways to compute the Jacobians Gk for an SA-friendly DAE are well known,while the full Jacobian of the DASSL-style formulation—in the example, the below-diagonal blocks of (21)—is messier to find; but see the discussion in Sect. 5. A New Look at Dummy Derivatives for Differential-Algebraic Equations 723
Further (Andreas Rauh, Rostock, personal communication 2015), some calcula-tions of robust design optimisation in control theory are far easier if the governingDAE (which is SA-friendly) has been converted to an explicit ODE.
5 Discussion
We have shown the dummy derivatives method is a particular case of the ˙-method’s standard solution scheme, and displayed the underlying BTF pictorially.This gives a simple method to cast the DD’ed system as a semi-explicit index 1DAE, hence an implicit ODE, combining order reduction and index reduction inone process. We pointed out that there may be compelling reasons for the furtherstep of converting to an explicit ODE before numerical solution. Step 3 of the method in Sect. 4 is optional—being an implicit ODE is independentof how the equations and variables are arranged. Thus different DD schemes fora DAE are distinguished purely by their sets of added equations (15). This set isdeducible immediately from the DDspec vector ı. All this, except the vector ı, isimplicit in Mattsson and Söderlind’s paper, but the simplicity of the algorithm—hence of programming it—does not seem to be known in the literature. A Jacobian such as (21) is used for Newton iterations inside a DASSL-stylesolver. In this context it seems one can ignore its “messy to compute” below-diagonal blocks, i.e. set them to 0, assuming a sufficiently good initial guess.Namely, as G2 is correct it will make the xF2 component converge quadratically.Once this has happened, as G1 is correct it will now make the xF1 componentconverge quadratically; and so on. In the context of solving Example 4 by ode151we have checked that this (surely well known) fact is true, but we have done noexperiments to validate it on harder, nonlinear, problems.
References
1. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)2. Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), 677–692 (1993)3. McKenzie, R.: Structural analysis based dummy derivative selection for differential-algebraic equations. Technical report, Cardiff University (2015). Submitted to BIT Numerical Analysis4. Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM. J. Sci. Stat. Comput. 9, 213–231 (1988)5. Pryce, J.D.: A simple structural analysis method for DAEs. BIT Numer. Math. 41(2), 364–394 (2001)6. Pryce, J.D.: A simple approach to Dummy Derivatives for DAEs. Technical report, Cardiff University, July 2015. In preparation New Master-Slave SynchronizationCriteria of Chaotic Lur’e Systemswith Time-Varying-Delay Feedback Control
Kaibo Shi, Xinzhi Liu, Hong Zhu, and Shouming Zhong
Abstract This study focuses on the issue of designing a time-delay output feedbackcontroller for master-slave synchronization of chaotic Lur’e systems (CLSs). Thetime delay is assumed to be a time-varying continuous function which is boundedbelow and above by positive constants. By constructing an appropriate Lyapunov-Krasovskii functional (LKF), a novel delay-dependent synchronization condition isobtained. Besides, by employing a new free-matrix-based inequality (FMBI), thedesired controller gain matrix can be achieved by solving a set of linear matrixinequalities (LMIs). Finally, one numerical example of Chua’s circle is given toillustrate the effectiveness and advantages of the proposed results.
1 Introduction
During the past few decades, the chaotic synchronization problem has attractedincreasing attention due to its extensive applications in many fields includingsecure communication, physical, chemical and ecological systems, human heartbeatregulation, and so on [1–4]. As is well-known that many nonlinear systems can bemodeled precisely in the form of Lur’e systems, such as Chua’s circuit, networksystems and hyper chaotic attractors, which include a feedback connection of
K. Shi () • H. ZhuSchool of Automation Engineering, University of Electronic Science and Technology of China,Chengdu, Sichuan 611731, P.R. Chinae-mail: [emailprotected]; [emailprotected]. LiuDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canadae-mail: [emailprotected]. ZhongSchool of Mathematical Sciences, University of Electronic Science and Technology of China,Chengdu, 611731, P.R. Chinae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 725J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_65 726 K. Shi et al.
a linear system and a nonlinear element satisfying the sector condition [5–7].Therefore, master-slave synchronization for CLSs has been a focused research topic. Recently, the synchronization problem of chaotic systems with time delay hasbeen intensively investigated due to the unavoidable signal propagation delayfrequently encountered in remote master-slave synchronization scheme [8–10].Especially, many delay-independent and delay-dependent synchronization criteriahave been derived by constructing an appropriate LKF in [11]. Compared with theresults in [11], less conservative synchronization criteria were obtained and somefairly simple algebraic conditions are derived for easier verification in [12]. Basedon Lyapunov method and LMIs approach, the authors in [13] further generalizedand improved the proposed results in [11, 12]. However, in order to obtain sufficientconditions for master-slave synchronization, the authors in [11, 13] employed modeltransformation, which leads to some conservatism for inducing additional terms.Based on the free weighting matrix approach, a delay-dependent synchronizationcondition is obtained in [14]. By using a delay-partition approach, several delay-dependent synchronization criteria are derived in the form of LMIs in [15, 16]. Byconstructing an appropriate LKF including the information of time-varying delayrange, new delay-range-dependent synchronization criteria for Lur’e systems areestablished in [17]. Besides, it should be noted that only constant delay is consideredin [11–16, 18]. In practice, as everyone knows that the range of time-varying delaynon-zero lower bound is often encountered, and such systems are referred to asinterval delayed systems. Motivated by the issues discussed above, the delay-dependent master-slavesynchronization problem of CLSs with time-varying-delay feedback control isinvestigated in this paper. By taking full advantage of the information of time-varying-delay range and nonlinear term of the error system, a less conservativedelay-dependent synchronization criterion is obtained. In addition, based on anappropriate LKF combined with a new FMBI, an explicit expression of the desiredcontrol law can be obtained in terms of LMIs. Finally, one numerical example ofChua’s circle is presented to demonstrate the effectiveness and advantages of thedesign methods.Notation Notations used in this paper are fairly standard: Rn denotes the n-dimensional Euclidean space, Rnm the set of all n m dimensional matrices;I the identity matrix of appropriate dimensions, AT the matrix transposition ofthe matrix A. By X > 0 (respectively X 0), for X 2 Rnn , we meanthat the matrix X is real symmetric positive definite (respectively, positive semi-definite); diagfr1 ; r2 ; ; rn g denotes block diagonal matrix with diagonal elementsri ; i D 1; ; n, the symbol represents the elements below the main diagonal of asymmetric matrix, SymfMg is defined as SymfMg D 12 .M C MT /. New Master-Slave Synchronization Criteria of Chaotic Lur’e Systems with. . . 727
2 Preliminaries
First, consider the following master-slave synchronization scheme of two identicalchaotic Lur’e systems with time-varying delay feedback control: ( xP .t/ D Ax.t/ C B'.Dx.t//; M W (1) p.t/ D Hx.t/; ( yP .t/ D Ay.t/ C B'.Dy.t// C u.t/; S W (2) q.t/ D Hy.t/;
C W u.t/ D K.p.t d.t// q.t d.t///; (3)
which consists of master system M , slave system S and controller C . M and Swith u.t/ D 0 are identical chaotic time-delay Lur’e systems with state vectors x.t/,y.t/ 2 Rn , outputs of subsystems p.t/, q.t/ 2 Rl , respectively, u.t/ 2 Rn is theslave system control input, and A 2 Rnn , B 2 Rnnd , D 2 Rnd n , H 2 Rln areknown real matrices, K 2 Rnl is the time-varying delay controller gain matrix tobe designed. It is assumed that './ is the nonlinear function in the feedback path.Assumption A Time-varying delay d.t/ is differential function and satisfies thefollowing condition:
0 < dL d.t/ dU ; P < 1: d.t/ (4)
where dL , dU and are three real constants.Assumption B The nonlinear function 's .˛/ satisfies the following sector condi-tion: .s D 1; ; nd /
's .˛/ 2 KŒk ;ksC D f's .˛/ j 's .0/ D 0; ks ˛ 2 ˛'s .˛/ ksC ˛ 2 ; ˛ ¤ 0g (5) s
Next, given the synchronization schemes (1), (2) and (3), the synchronizationerror is defined as r.t/ D x.t/ y.t/, and we can get the following synchronizationerror system:
rP .t/ D Ar.t/ C B.Dr.t/; y.t// KHr.t d.t//; (6)
where .Dr.t/; y.t// D '.Dr.t/ C Dy.t// '.Dy.t//. Let D D Œd1 ; ; dnd T withds 2 Rn .s D 1; 2; ; nd /. Under Assumption B, it is easy to obtain that s .dTs r; y/satisfies the following condition
s .dTs r; y/ 's .dTs .r C y// 's .dTs y/ ks D ksC ; 8r; y 2 Rn ; dTs r ¤ 0: (7) dTs r dTs r 728 K. Shi et al.
Lemma 1 ([19]) For differentiable signal r.t/ in Œa; b ! Rn , for symmetricmatrices R 2 Rnn , and Z1 2 R3n3n , Z3 2 R3n3n , and any matrices Z2 2 R3n3n , 2 3 2 3 2 3 N11 N21 Z1 Z2 N1and N1 D 4 N12 5, N2 D 4 N22 5 2 R3nn satisfying R N D 4 Z3 N2 5 > 0, the N13 N23 Rfollowing inequality holds: Z b rP T .s/RPr.s/ds $ T Œ˝0 C SymfN1 ˙1 C N2 ˙2 g $; a
where ˝0 D .b a/.Z1 C 13 Z3 /, ˙1 D ŒI; I; 0 , ˙2 D ŒI; I; 2I , 1 Rb T $ D ŒrT .a/; rT .b/; ba a r .s/ds . T
Remark 1 It should be noted that the set of slack variables in the above inequalitycan provide great freedom in deriving less conservative stability conditions, which itis possible to obtain a much sharper bound. It is easy to prove that some well-knownintegral inequalities are special cases of this one in [19]. For example, if we let N1 DŒYT ; 0 T , N2 D 0, Z1 D diagfX; 0g, Z2 D 0 and Z3 D 0, this integral inequality 1can reduce to the integral inequality in [20]. And if we let N1 D ba ŒR; R; 0 T , 3 T 1 T 1 T 1N2 D ba ŒR; R; 2R , Z1 D N1 R N1 , Z2 D N1 R N2 , Z3 D N2 R N2 , this T
integral inequality also becomes the celebrated Wirtinger integral inequality [21].
3 Main Results
Theorem 1 Under Assumptions A and B, for given scalars , dL , dU , the errorsystem (6) is globally asymptotically stable if there exist positive matrices P, Ri.i D 1; ; 5/, any positive definite diagonal matrices G D diagfg1; ; gnd g,L D diagfl1 ; ; lnd g, W D diagfw1; ; hnd g, symmetrical matrices X1 , Y1 ,Z1 , X3 , Y3 and Z3 , and any matrices X2 , Y2 , Z2 , Ni .i D 1; 2; ; 6/, N and Uwith the appropriate dimensions, such that the following symmetric linear matrixinequalities hold: 2 3 2 3 2 3 X1 X2 N1 Y1 Y2 N3 Z1 Z2 N5 4 X3 N2 5 > 0; 4 Y3 N4 5 > 0 and 4 Z3 N6 5 > 0; (8) R4 R5 R5
< 0; (9) New Master-Slave Synchronization Criteria of Chaotic Lur’e Systems with. . . 729
where
D e1 .R1 C R3 DK WKC DT /eT1 C e2 .dL R4 C dUL R5 /eT2 e3 .R1 R2 /eT3 e4 R2 eT4 .1 /e5 R3 eT5 2e6 WeT6 C Symfe1 Pe2 C e1 D.K C KC /eT6
C e2 DT .G L/eT6 C e1 DT ŒKC L K G DeT2 C Œe1 x C e2 y NŒeT2 C AeT1 C BeT6 1 KHeT4 g C Œe1 ; e3 ; e7 dL X1 C X3 C SymfM1 ˘1 C M2 ˘2 g Œe1 ; e3 ; e7 T 3 1 C Œe3 ; e4 ; e8 dUL Y1 C Y3 C SymfM3 ˘1 C M4 ˘2 g Œe3 ; e4 ; e8 T 3 1 C Œe4 ; e5 ; e9 dUL Z1 C Z3 C SymfM5 ˘1 C M6 ˘2 g Œe4 ; e5 ; e9 T ; 3
Moreover, the gain matrix of state estimator is given by K D N1 U.Proof Consider the following a newly augmented LKF for the error system (6):
V.rt / D V1 .rt / C V2 .rt / C V3 .rt /; (10)
where Z t V1 .rt / D r .t/Pr.t/ C T rT .s/R1 r.s/ds thL Z tdL Z t C r .s/R2 r.s/ds C T rT .s/R3 r.s/ds; (11) tdU td.t/
Z 0 Z t Z dL Z t V2 .rt / D rP T .s/R4 rP .s/dsd C rP T .s/R5 rP .s/dsd; (12) dL tC dU tC
X nd Z dsT r.t/ X nd Z dsT r.t/ V3 .rt / D 2 gs Œs ./ ks d C2 ls ŒksC s ./ d; sD1 0 sD1 0 (13)
Taking the derivative of V.xt / along the trajectory of the error system (6) yields:
P 1 .rt / D 2rT .t/PPr.t/ C rT .t/.R1 C R3 /r.t/ rT .t dL /.R1 R2 /r.t dL / V rT .t dU /R2 r.t dU / .1 d.t//r P T .t d.t//R3 r.t d.t// 2rT .t/PPr.t/ C rT .t/.R1 C R3 /r.t/ rT .t dL /.R1 R2 /r.t dL / rT .t dU /R2 r.t dU / .1 /rT .t d.t//R3 r.t d.t//; (14) 730 K. Shi et al.
P 2 .rt / DPrT .t/.dL R4 C dUL R5 /Pr.t/ V Z t Z tdL rP .s/R4 rP .s/ds T rP T .s/R5 rP .s/ds tdL tdU Z t DPr .t/.dL R4 C dUL R5 /Pr.t/ T rP T .s/R4 rP .s/ds tdL Z tdL Z td.t/ rP T .s/R5 rP .s/ds rP T .s/R5 rP .s/ds; (15) td.t/ tdU
By using Lemma 1, we can have Z t rP T .s/R4 rP .s/ds tdL 1 $1T .t/ dL X1 C X3 C SymfM1 ˘1 C M2 ˘2 g $1 .t/; 3 (16) Z tdL rP T .s/R5 rP .s/ds td.t/ 1 $2T .t/ dUL Y1 C Y3 C SymfM3 ˘1 C M4 ˘2 g $2 .t/; 3 (17) Z td.t/ rP T .s/R5 rP .s/ds tdU 1 $3T .t/ dUL Z1 C Z3 C SymfM5 ˘1 C M6 ˘2 g $3 .t/; 3 (18) Rtwhere dUL D dU dL , $1T .t/ D ŒrT .t/; rT .t dL /; d1L tdL rT .s/ds , $2T .t/ D 1 RtŒrT .t dL /; rT .t d.t//; d.t/d tdL r .s/ds ,˘1 D ŒI; I; 0 $3 .t/ D Œr .t T T T
1 R t Ld.t//; rT .t dU /; dU d.t/ tdL r .s/ds , ˘2 D ŒI; I; 2I . T
X nd X nd P 3 .rt / D2 V P s .dT r.t// C 2 .gs ls /dsT r.t/ P s kC gs k /dT r.t/ dsT r.t/.l s s s s sD1 sD1
D2Pr .t/D .G L/.Dr.t// C 2r .t/DT ŒKC L K G DPr.t/; T T T (19) New Master-Slave Synchronization Criteria of Chaotic Lur’e Systems with. . . 731
Next, for any scalars x and y, and arbitrary matrix N with appropriate dimensions,we can obtain
0 D2ŒPrT .t/x C rT .t/y NŒPr.t/ C Ar.t/ C B.Dr.t/; y.t// KHr.t d.t// ; (20)
From Eq. (7), for any positive diagonal matrix W D diagfw1 ; ; wnd g, it yieldsthat
X nd 2 Œs .dTs r.t/; y.t// ks dTs r.t/ ws Œs .dTs r.t/; y.t// ksC dTs r.t/ sD1
D 2rT .t/DK WKC DC r.t/ C 2rT .t/D.K C KC /W.DT r.t/; y.t// 2 T .DT r.t/; y.t//W.DT r.t/; y.t//; (21)
Combining Eqs. (14), (15), (16), (17), (18), (19), (20), and (21) yields
V.rt / T .t/ .t/; (22)
where T .t/ D ŒrT .t/; rP T .t/; rT .t dL /; rT .t d.t//; rT .t dU /; T .DT r.t/; y.t//;$1T .t/; $2T .t/; $3T .t/ , From (9), we can have V.t; rt / " k r.t/ k2 holds for any sufficiently small" > 0. Therefore, this implies the error system (6) is globally asymptotically stable. Theproof is completed. t u
4 Numerical Example
In this section, a numerical simulation example is given to show the effectivenessand correctness of the main results derived above. Consider Chua’s circuit example and the system equation in [14, 16–18] is givenby 8 ˆ ˆ xP 1 .t/ D˛.x2 .t/ h.x1 .t///; < xP 2 .t/ Dx1 .t/ x2 .t/ C x3 .t/; (23) ˆ :̂ xP 3 .t/ D ˇx2 .t/; 732 K. Shi et al.
with nonlinear characteristic
h.x1 .t// D m1 x1 .t/ C 0:5.m0 m1 /Œjx1 .t/ C cj jx1 .t/ cj ;
and parameters m0 D 17 , m1 D 27 , ˛ D 9, ˇ D 14:28, and c D 1. The system canbe written in chaotic Lur’e system framework (1) with 23 2 3 2 3 ˛m1 1 ˛ 0 ˛.m0 m1 / 1 AD4 1 2 1 5 ; B D 4 0 5 ; D D H D 4 05: 0 ˇ 1 0 0
Under the above conditions and D 0, the maximum upper bounds on theallowable delays of h obtained from the above references [14, 16–18] and Theorem 1are listed in Table 1. Thus, it is also to see that our result is more effective than therecently reported ones. In addition, set '.˛/ D 0:5.j˛C1jj˛1j/, h D 0:191, x.0/ D Œ0:2; 0:3; 0:2 and y.0/ D Œ0:5; 0:1; 0:6 , the simulation results are shown in Figs. 1, 2, 3, 4, 5,and 6 for the above gain matrix. The trajectories of the master-slave systems with
Table 1 Maximum allowed delays h and the best gain matrices K for D 0Method [18] [14] [17] [16] Theorem 1h 0.141 0.180 0.183 0.185 0.188 2 3 2 3 2 3 2 3 2 3 6:0229 3:9125 4:1455 4:0779 4:6512 6 7 6 7 6 7 6 7 6 7K 4 1:3367 5 4 0:9545 5 4 0:9250 5 4 0:9087 5 4 0:5992 5 2:1264 3:8273 4:2596 4:3430 4:5177
0.4
0.2 x 2 (t)
−0.2
−0.4 4 2 3 2 0 1 0 x (t) −2 −1 3 −2 x (t) −4 −3 1
Fig. 1 State trajectories of master system x.t/ New Master-Slave Synchronization Criteria of Chaotic Lur’e Systems with. . . 733
0.4
0.2 y 2 (t)
−0.2
−0.4 4 2 3 2 0 1 y (t) 0 3 −2 −1 −2 y1(t) −4 −3
Fig. 2 State trajectories of slave system y.t/ without u.t/
4 x (t) x (t) x (t) y (t) y (t) y (t) 1 2 3 1 2 3
1 Amplitude
−1
−2
−3 0 1 2 3 4 5 6 7 8 9 10 t
Fig. 3 State trajectories of master-slave systems x.t/ and y.t/ with u.t/
u.t/ D 0 are shown in Figs. 1 and 2. Under the above gain matrix K, the responsesof the state x.t/ and y.t/, the error signal r.t/, outputs of subsystems p.t/ and q.t/are represented in Figs. 3, 4, 5, and 6, respectively. Therefore, it is clear to show thatthe synchronization error is tending asymptotically to zero. 734 K. Shi et al.
r (t) r (t) r (t) 1.5 1 2 3
0.5 Amplitude
−0.5
−1
−1.5
−2 0 1 2 3 4 5 6 7 8 9 10 t
Fig. 4 State trajectories of synchronization error system r.t/
2 p(t) 1.5
0.5 Amplitude
−0.5
−1
−1.5
−2
−2.5 0 1 2 3 4 5 6 7 8 9 10 t
Fig. 5 State trajectories of output vector p.t/ New Master-Slave Synchronization Criteria of Chaotic Lur’e Systems with. . . 735
2 q(t) 1.5
0.5
0 Amplitude
−0.5
−1
−1.5
−2
−2.5 0 1 2 3 4 5 6 7 8 9 10 t
Fig. 6 State trajectories of output vector q.t/
5 Conclusions
This paper consider the delay-dependent master-slave synchronization problem ofCLSs with time-varying-delay feedback control. First, by choosing an augmentedLKF, a novel delay-dependent synchronization criterion is derived. Second, byemploying a new FMBI, which has been proved to be lesser conservative than theexisting integral inequalities and showed to have a great potential efficient in theactual operation, the desired controller gain matrix can be achieved by solving a setof LMIs. Finally, one numerical example of Chua’s circle is given to illustrate theeffectiveness and advantages of the proposed results. The foregoing methods havethe potential to be useful for the further study of CLSs. Meanwhile, it is expectedthat these approaches can be further used for other time-delay systems.
Acknowledgements This work was supported by National Basic Research Program of China(2010CB732501), National Natural Science Foundation of China (61273015), The NationalDefense Pre-Research Foundation of China (Grant No. 9140A27040213DZ02001), The Programfor New Century Excellent Talents in University (NCET-10-0097).
References
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3. Grzybowski, J.M.V., Rafikov, M., Balthazar, J.M.: Synchronization of the unified chaotic system and application in secure communication. Commun. Nonlinear Sci. Numer. Simul. 14, 2793–2806 (2009) 4. Voss, H.: Anticipating chaotic synchronization. Phys. Rev. E 61, 5115–5119 (2000) 5. Mkaouar, H., Boubaker, O.: Chaos synchronization for master slave piecewise linear systems: application to Chua’s circuit. Commun. Nonlinear Sci. Numer. Simul. 17, 1292–1302 (2012) 6. GKamez-GuzmKan, L., Cruz-HernKa ndez, C., LKopez-GutiKerrez, R.M., GarcKia-Guerrero, E.E.: Synchronization of Chua’s circuits with multi-scroll attractors: application to communication. Commun. Nonlinear Sci. Numer. Simul. 14, 2765–2775 (2009) 7. LRu, J.H., Murali, K., Sinha, S., Leung, H., Aziz-Alaoui, M.A.: Generating multi-scroll chaotic attractors by thresholding. Phys. Lett. A 372, 3234–3239 (2008) 8. Stepp, N.: Anticipation in feedback-delayed manual tracking of a chaotic oscillator. Exp. Brain Res. 198, 521–525 (2009) 9. Pyragas, K., Pyragas, T.: Extending anticipation horizon of chaos synchronization schemes with time-delay coupling. Philos. Trans. R. Soc. A 368, 305–317 (2010)10. Milton, J.G.: The delayed and noisy nervous system: implications for neural control. J. Neural Eng. 8, 065005 (2011)11. Yalcin, M.E., Suykens, J.A.K., Vandewalle, J.: Master-slave synchronization of Lur’e systems with time-delay. Int. J. Bifurc. Chaos 11(6), 1707–1722 (2001)12. Liao, X.X., Chen, G.R.: Chaos synchronization of general Lur’e systems via time-delay feedback control. Int. J. Bifurc. Chaos 13(1), 207–213 (2003)13. Cao, J.D., Li, H.X., Daniel, W.C.H.: Synchronization criteria of Lur’e systems with time-delay feedback contro. Chaos, Solitons Fractals 23, 1285–1298 (2005)14. He, Y., Wen, G.L., Wang, Q.G.: Delay-dependent synchronization criterion for Lur’e systems with delay feedback control. Int. J. Bifurc. Chaos 16(10), 3087–3091 (2006)15. Ding, K., Han, Q.L.: Master-slave synchronization criteria for horizontal platform systems using time delay feedback control. J. Sound Vib. 330(11), 2419–2436 (2011)16. Ge, C., Hua, C.C., Guan, X.P.: Master-slave synchronization criteria of Lur’e systems with time-delay feedback control. Appl. Math. Comput. 244, 895–902 (2014)17. Li, T., Yu, J.J., Wang, Z.: Delay-range-dependent synchronization criterion for Lur’e systems with delay feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(5), 1796–1803 (2009)18. Han, Q.L.: New delay-dependent synchronization criteria for Lur’e systems using time delay feedback control. Phys. Lett. A 360(4), 563–569 (2007)19. Zeng, H.B., He, Y., Wu, M., She, J.H.: Stability of time-delay systems via Wirtinger-based double integral inequality. IEEE Trans. Autom. Control. 60(10), 2768–2772 (2015)20. Zhang, X.M., Wu, M., She, J.H., He, Y.: Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41, 1405–1412 (2005)21. Seuret, A., Gouaisbaut, F.: Wirtinger-based integral inequality: application to time-delay systems. Automatica 49, 2860–2866 (2013) Robust Synchronization of Distributed-DelaySystems via Hybrid Control
Peter Stechlinski and Xinzhi Liu
Abstract Drive and response systems which exhibit time-delays and uncertaintiesare synchronized using hybrid control. Classes of dwell-time satisfying switchingrules are identified under which synchronization can be achieved in a robust manner.The theoretical results are established using multiple Lyapunov functions andHalanay-like inequalities.
1 Introduction
Synchronization problems arise in secure communications [1], multi-agent net-works [2], chaotic systems [3], neural networks [4], and many other importantapplications. Many real-world phenomena exhibit time-delays (e.g., see [5, 6])and uncertainties in parameter measurements and data input [7], which can havesignificant impacts on control strategies for synchronization. The main focus ofthe present article is to investigate the synchronization of uncertain systems withdistributed delays and nonlinear perturbations. The approach here is to use acombination of switching and impulsive control. This type of hybrid control, whichcan have a number of advantages over continuous control (see [8–10] for details),leads to a switched system model for the error between the drive and responsesystems. Switched systems, which are a type of hybrid system, are governed by a mixtureof continuous/discrete dynamics and logic-based switching, and have applicationsin a wide-range of problems in applied mathematics, engineering, and computerscience [10]. In the switched systems literature, finding classes of switching ruleswhich guarantee a performance goal (e.g., stability, synchronization, etc.) is a majorarea of research (e.g., see [8, 10–14]), where concepts such as multiple Lyapunovfunctions and dwell-time satisfying switching rules are used. This line of researchhas been extended to switched systems with time-delays composed of stable andunstable modes by way of Halanay-like inequalities (e.g., see [15–20]). Robustcontrol (e.g., see [21, 22]) is important to take into account since relatively small
P. Stechlinski () • X. LiuUniversity of Waterloo, Waterloo, ON, N2L 3G1, Canadae-mail: [emailprotected]; [emailprotected]
© Springer International Publishing Switzerland 2016 737J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_66 738 P. Stechlinski and X. Liu
uncertainties can lead to instability (or, desynchronization in the present context),and has been studied in the time-delay switched systems literature (e.g., see [7, 23–25]). Motivated by the above discussion, the main objective of this article is toextend the literature by studying the robust hybrid synchronization of systems withdistributed delays. Contributions include establishing a switched and impulsiveHalanay-like inequality using multiple Lyapunov functions to prove synchroniza-tion that is robust to uncertainties, and developing verifiable synchronizationconditions in the face of time-delays. Classes of dwell-time, average dwell-time,and periodic switching rules are identified under which synchronization can beachieved via hybrid control. Methods are developed in the present article to dealwith switching desynchronization (where impulsive control is crucial), nonlinearperturbations, as well as impulsive disturbances. The results are generalizable tostability studies of switched systems composed of a mixture of stable and unstablemodes. The theoretical findings are augmented with numerical simulations.
2 Problem Formulation
Let RC denote the set of non-negative real numbers, let N denote the set of positiveintegers, and let Rn denote the Euclidean space of n-dimensions (equipped with theEuclidean norm k k). For a positive constant , denote PC D PC.Π; 0 ; Rn / to bethe space of piecewise continuous functions mapping Π; 0 to Rn , equipped withthe norm k k WD sup s0 k .s/k. Let max ./ and min ./ denote the maximumand minimum eigenvalues of a symmetric matrix, respectively. In the spirit of Guan et al. [14], consider the following drive system: Z 0 xP .t/ D .A C A.t//x.t/ Q C .C C C.t// Q x.t C s/ds C F.t; xt /; (1)
where > 0 is an upper bound on the distribution of time-delays; xt 2 PC isdefined by xt .s/ WD x.t C s/ for s 0; F W R PC ! Rn satisfiesF.t; 0/ 0 for all t t0 ; A; C 2 Rnn are known; A; Q CQ are uncertainties ofthe form A.t/ WD A %A .t/A , C.t/ WD C %C .t/C , where A , C , A , C are Q Qknown constant matrices of appropriate dimensions and %A , %C are time-dependentuncertainty matrices. The response system is given by Z 0 yP .t/ D .A C A.t//y.t/ Q C .C C C.t// Q y.t C s/ds C F.t; yt / C u; (2) Robust Synchronization of Distributed-Delay Systems via Hybrid Control 739
where u WD u1 C u2 is a controller designed as follows: 1 X Z 0 u1 .t; x; y/ WD .Hik C Hik .t//e.t/ C .Gik C Gik .t// Q Q e.t C s/ds 1I k .t/; kD1
1 X u2 .t; x; y/ WD .Eik C EQ ik .t//e.t/ı.t tk /; kD1
where e WD y x is the error between the drive and response systems; Hi , Gi , Ei ,i D 1; : : : ; m (for some positive integer m) are known constant control matrices,H Q i , EQ i , i D 1; : : : ; m, are time-dependent uncertainties of the form H Q i, G Q i .t/ WD
H;i %H;i .t/H;i , G Q i .t/ WD G;i %G;i .t/G;i , EQ i .t/ WD E;i %E;i .t/E;i , where H;i , G;i ,
E;i , H;i , G;i , E;i are known constant matrices of appropriate dimensions and%H;i , %G;i , %E;i are time-dependent uncertainty matrices; ik 2 P WD f1; : : : ; mg foreach k 2 N, where m is a positive integer; I k WD Œtk1 ; tk / is called the switching orhybrid interval, 1I k ./ is the indicator function, and ı./ is the generalized Diracdelta function. Under this construction, u1 and u2 are switching and impulsivecontrollers containing some uncertainties, respectively (see, e.g., [14] for a similarhybrid control construction without time-delays or uncertainties present). The error system can be written as the following hybrid system: Z 0 eP .t/ D A .t/e.t/ C C .t/ e.t C s/ds C F.t; yt / F.t; xt /; t ¤ tk ;
e.t/ D .I C E .t //e.t /; t D tk ; et0 D 0 ; k D 1; 2; : : : ; (3)where t0 2 R is the initial time; 0 WD yt0 xt0 2 PC is the initial error; Ai .t/ WD Q C Hi C HA C A.t/ Q i .t/, Ci .t/ WD C C C.t/ Q C Gi C G Q i .t/, and Ei .t/ WD Ei C EQ i .t/ foreach i D 1; : : : ; m. The switching rule W Œt0 ; C1/ ! P is a piecewise constantfunction that maps each switching interval Œtk1 ; tk / to an integer in P, i.e., it selectsthe active mode of (3). Denote the set of all such switching rules by S .Definition 1 The uncertainties are said to be admissible if they are continuousmatrix functions satisfying k%A .t/k < 1, k%C .t/k < 1, k%H;i .t/k < 1, k%G;i .t/k <1, and k%E;i .t/k < 1 for all t t0 , i D 1; : : : ; m. The drive system (1) and theresponse system (2) are said to be robustly synchronized (RS) if limt!1 ke.t/k D 0for all 0 2 PC and admissible uncertainties. The systems are said to be robustlyexponentially synchronized (RES) if there exist > 0, C > 0 such that, for t t0 ,ke.t/k Ck0 k expŒ.t t0 / for all 0 2 PC and admissible uncertainties.Remark 1 If 2 S inf-dwell WD f 2 S W 9 > 0 s.t. infk2N ftk tk1 g g,Fi is composite-PC in the sense of Ballinger and Liu [26], continuous in its firstvariable, and locally Lipschitz in its second variable for each i D 1; : : : ; m, then 740 P. Stechlinski and X. Liu
there exists a unique solution of (3) by [26] and using the method of steps over thehybrid moments [27]. Given the hybrid control matrices f.Hi ; HQ i ; Gi ; G Q i ; Ei ; EQ i / W i 2 Pg, the goal ofthis article is to design the hybrid time mode sequences (i.e., the set f.ik ; tk / W k 2Ng) such that the drive and response systems achieve robust synchronization.
2.1 Main Results
Consider the following notions of activation time and total number of switches.Definition 2 Let t1 ; t2 2 R such that t2 t1 and let A P. Denote the total R t2activation time of the modes in A by TA .t1 ; t2 / WD t1 1A ..t//dt. Denote the totalnumber of switches to a mode in A by ˚A .t1 ; t2 / WD jftk W .tk / 2 A; t1 tk < t2 gj. Zhu established the following Halanay-like lemma.Lemma 1 ([18]) Let a; b > 0. Assume that u W Œt0 ; 1/ ! RC satisfies uP .t/ bkut k au.t/ for t t0 . If b a 0 then u.t/ kut0 k expŒ.b a/.t t0 / for t t0 , while if b a < 0 then there exists a positive constant satisfying C b exp. / a < 0 such that u.t/ kut0 k expŒ.t t0 / for t t0 : For use in the main theorem, the following Halanay-like result is proved.Proposition 1 Let ai ; bi ; di ; hi 0 for i D 1; : : : ; m. Assume that u W Œt0 ; C1/ ! RC satisfies ( uP .t/ b kut k a u.t/; t ¤ tk ; t t0 ; (4) u.t/ d u.t / C h kut k ; t D tk ; k D 1; 2; : : : ;
for some 2 S which satisfies tk tk1 for all k 2 N. Then, for t t0 , 0 1 8 9 ˚P .t0 ;t/ Y <X X = u.t/ kut0 k @ ıij A exp i Tfig .t0 ; t/ i TQ fig .t0 ; t/ ; (5) : ; jD1 i2P u i2P s
where TQ fig .t0 ; t/ WD Tfig .t0 ; t/ ˚fig .t0 ; t/ , i WD bi maxi2P f1=ıi ; 1g ai , ıi WDdi Chi e , WD maxi2P s fi g, i > 0 is chosen for i 2 P s so that i Cbi ei ai < 0,P u WD fi 2 P W i 0g, P s WD fi 2 P W i < 0g, and i > 0 is chosen fori 2 P s so that i C bi maxi2P f1=ıi ; 1g exp.i / ai < 0.Proof Lemma 1 implies that, for t 2 Œt0 ; t1 /, ( kut0 k expŒi1 .t t0 / ; if i1 2 P u ; u.t/ kut0 k expŒi1 .t t0 / ; if i1 2 P s ; Robust Synchronization of Distributed-Delay Systems via Hybrid Control 741
where kut0 k D sup s0 u.t0 C s/. Let 0 1 ˚P .t0 ;t/ Y w.t/ WD kut0 k @ ıij A exp Œ .t0 ; t/ jD1
where X X .t0 ; t/ WD i Tfig .t0 ; t/ i TQ fig .t0 ; t/: i2P u i2P s
Suppose the result holds for t 2 Œtk1 ; tk /, that is, u.t/ w.t/. We claim that u.t/ w.t/ for t 2 Œtk ; tkC1 /. If not, then there exists a time t 2 Œtk ; tkC1 / such that u.t / Dw.t /, u.t/ w.t/ for all t 2 Œtk ; t / and for any " > 0 there exists a time t" 2.t ; t C "/ such that u.t" / > w.t" /. Suppose that ikC1 2 P s , then
uP .t / ˇikC1 sup u.t C s/ ˛ikC1 u.t /; s0 0 1 Y k1 ˇikC1 maxfıi ; 1gkut0 k @ ıij A expŒ .t0 ; tk / expŒikC1 .t tk / i2P jD1 0 1 Yk1 ˛ikC1 ıik kut0 k @ ıij A expŒ .t0 ; tk / expŒikC1 .t tk / ; jD1
and so 0 1 Yk 1 uP .t / ˇikC1 max ; 1 exp.ikC1 / ˛ikC1 kut0 k @ ıij A exp. .t0 ; t //: i2P ıi jD1
It follows that uP .t / ikC1 w.t / D w.t P /, a contradiction to the definition of t" .On the other hand, if ikC1 2 P u , then
uP .t / ˇikC1 sup u.t C s/ ˛ikC1 v.t /; s0 0 1 Y k1 ˇikC1 maxfıi ; 1gkut0 k @ ıij A expŒ .t0 ; tk / expŒikC1 .t tk / i2P jD1 0 1 Y k1 ˛ikC1 ık kut0 k @ ıij A expŒ .t0 ; tk / expŒikC1 .t tk / ; jD1 742 P. Stechlinski and X. Liu
and therefore, 0 1 Yk 1 uP .t / ˇikC1 max ; 1 ˛ikC1 kut0 k @ ıij A expŒ .t0 ; t / : i2P ıi jD1
Thus, uP .t / ikC1 w.t / D w.t P /, a contradiction. The result holds by induction. The nonlinear perturbations are assumed to satisfy the following condition.Assumption 1 Assume there exist #1 ; #2 0 such that, for all .t; ; / 2 RC R0PC PC, kF.t; / F.t; /k #1 k .0/ .0/k C #2 k .s/ .s/kds. The following lemmata are required for the main result.Lemma 2 (Matrix Cauchy Inequality) For any symmetric positive definite matrixW 2 Rnn and x; y 2 Rn , 2xT y xT Wx C yT W 1 y.Lemma 3 ([22]) For any positive scalar ", matrices U; V; W with W T W I,UWV C V T W T U T "UU T C "1 V T V.Lemma 4 ([28]) For any positive definite symmetric matrix W 2 Rnn , R v T R v nonnegative Rv T scalar v, and w W Œ0; v ! Rn , 0 w.s/ds W 0 w.s/ds v 0 w .s/Ww.s/ds:Lemma 5 ([23]) Let P be a symmetric positive definite matrix and let M, U, V andW.t/ be real matrices of appropriate dimensions, with W.t/ being a matrix function.Then, for any " > 0 such that P1 "UU T is positive definite and W.t/T W.t/ I,.M C UW.t/V/T P.M C UW.t/V/ M.P1 "UU T /1 M C "1 V T V for all t 2 R. We are now in a position to present the main synchronization result.Theorem 1 Suppose that Assumption 1 holds. For i D 1; : : : ; m, let Pi 2 Rnn bepositive definite symmetric matrices and let "A;i ; "C;i ; "G;i ; "H;i be positive constantssuch that P1 i "E;i E;i E;i is positive definite. Let T T
ˇi WD 2 max .P1 1 i .C C Gi / Pi .C C Gi // C 2 kPi k#2 max .Pi / T
C 2 Œ"G;i max .P1 1 T i G;i G;i / C "C;i max .Pi C C / ; T
˛i WD 1 2kPi k#1 max .P1 1 1 i / max .Pi Œ.A C Hi / Pi C Pi .A C Hi / C "A;i Pi A A Pi T T
C "A;i AT A C "1 T T 1 T 1 H;i Pi H;i H;i Pi C "H;i H;i H;i C "C;i Pi C C Pi C "G;i Pi G;i G;i Pi /; T
max .Pi / 1i WD ; WD max i ; i WD ˇi max ; 1 ˛i ; min .Pi / i2P i2P ıi
ıi WDmax .P1 T 1 i Œ.I C Ei / .Pi "E;i E;i T E;i /1 .I C Ei / C "1 E;i E;i E;i /; T
where i > 0, i 2 P s , satisfy i C ˇi maxi2P f1=ıi; 1g exp.i / ˛i < 0. LetP u WD fi 2 P W i 0g, P s WD fi 2 P W i < 0g, C WD maxi2P u i , Robust Synchronization of Distributed-Delay Systems via Hybrid Control 743
WD mini2P s i , T C .t0 ; t/ WD TP u .t0 ; t/, T .t0 ; t/ WD TP s .t0 ; t/ ˚P s .t0 ; t/,ı WD maxi2P ıi . Then, the following results hold: (i) If there exist > 0, > 0 such that tk tk1 for all k 2 N, ı < 1, T C .t0 ; t/ T .t0 ; t/, and ln.ı/= C C < 0, then RES is achieved. (ii) If there exist > 0, > 0 such that tk tk1 for all k 2 N, ı > 1, T C .t0 ; t/ T .t0 ; t/, and ln.ı/= C C < 0, then RES is achieved.(iii) If there exist > 1, k > 0 for k 2 N such that tk tk1 k , P s ;, and ln. ıik / C ik k 0 for all k 2 N, then RS is achieved.(iv) If there exist > 1, k > 0 for k 2 N such that k tk tk1 , P u ;, and ln. ıik / ik .k / 0 for k 2 N, then RS is achieved. (v) If there exist tavg > 0, N0 > 0 such that k 1 N0 C .t t0 /=tavg for any t 2 Œtk1 ; tk /, ı > 1, and ln.ı/ tavg < 0, then RS is achieved.(vi) If there exist i > 0 for i 2 P such that, for t 2 Œtk1 ; tk /, .t/ D k and .t C / D .t/, where i WD ti ti1 , WD 1 C 2 C : : : C m , then RS is achieved if
X m X X ln ıi C i i i .i / < 0: (6) iD1 i2P u i2P s
R0Proof Let Vi .e/ WD eT Pi e for i D 1; : : : ; m, and let eQ .t/ WD e.t C s/ds. Thetime-derivative of Vi along the ith mode of (3) for t ¤ tk is given by
dVi .e.t// D eT .t/Œ.A C A.t/ Q i .t//T Pi C Pi .A C A.t/ Q C Hi C H Q C Hi C H Q i .t// e.t/ dt C 2eT .t/Pi .C C C.t/ Q C Gi C G Q i .t//Qe.t/C2eT .t/Pi ŒFi .t; yt / Fi .t; xt / :
By Lemma 2, 2eT .t/Pi .C C Gi /Qe.t/ eT .t/Pi e.t/ C eQ T .t/.C C Gi /T Pi .C C Gi /Qe.t/.Furthermore, Lemma 3 gives that
eT .t/.AQ T .t/Pi C Pi A.t//e.t/ Q "1 A;i e .t/Pi A A Pi e.t/ C "A;i e .t/A A e.t/; T T T T
eT .t/.H Q i .t//e.t/ "1 Q iT .t/Pi C Pi H H;i e .t/Pi H;i H;i Pi e.t/ C "H;i e .t/H;i H;i e.t/; T T T T
2eT .t/Pi .C.t/ Q CG Q i .t//Qe.t/ "1 C;i e .t/Pi C C Pi e.t/ C "C;i e T T Q T .t/CT C eQ .t/ C "1 G;i e .t/Pi G;i G;i Pi e.t/ C "G;i e T T Q T .t/G;i T G;i eQ .t/:
Lemma 4 implies that Z 0 eQ T .t/.C C Gi /T Pi .C C Gi /Qe.t/ eT .t C s/.C C Gi /T Pi .C C Gi /e.t C s/ds;
and can be applied similarly to the terms eQ T .t/C;i T C;i eQ .t/ and eQ T .t/G;i T G;i eQ .t/. 744 P. Stechlinski and X. Liu
By Assumption 1, Z 0 2eT .t/Pi ŒF.t; yt / F.t; xt / 2ke.t/kkPi k #1 ke.t/k C #2 ke.t C s/kds :
At the switching and impulsive moment t D tk ,
VikC1 .e.tk // eT .tk /.I C Eik C EQ ik .tk //T Pik .I C Eik C EQ ik .tk //e.tk / eT .tk /Œ.I C Eik /T .P1 1 i "E;ik E;ik E;ik / .I C Eik / e.tk / T
C eT .tk /Œ"1 E;ik E;ik E;ik e.tk /; T
R0by Lemma 5. Then, using eT .tCs/Pi e.tCs/ds sup s0 eT .tCs/Pi e.tCs/and the well-known fact that for any positive definite matrix P 2 Rnn , symmetricmatrix Q 2 Rnn , and x 2 Rn , min .P1 Q/xT Px xT Qx max .P1 Q/xT Px, itfollows that v.t/ WD V .e.t// satisfies the conditions of Proposition 1 with bi WD ˇi ,ai WD ˛i , hi WD 0, di WD ıi . To prove case (i), note that P .t0 ; t/ D k 1 for t 2 Œtk1 ; tk /, and tk tk1 implies that t t0 k for t 2 Œtk1 ; tk /. Equation (5) implies that v.t/ kvt0 k exp .k 1/ ln ı C C T C .t0 ; t/ T .t0 ; t/ (7) kvt0 k .1=ı/ exp .t t0 / ln.ı/= C C T C .t0 ; t/ T .t0 ; t/ kvt0 k .1=ı/ exp .t t0 / ln.ı/= C .C /T .t0 ; t/ kvt0 k .1=ı/ exp .ln.ı/= C C /.t t0 /=M
for some M > 0 since suptt0 f.t t0 /=T .t0 ; t/g is finite. Robust exponentialsynchronization of the drive and response systems follows. Since tk tk1implies that t t0 k for t 2 Œtk1 ; tk /, case (ii) is proved by similar argumentsused to show case (i). To prove case (iii), RS is implied from Eq. (5) since, for t 2 Œtk1 ; tk /, 2 3 Xk1 Xv.t/ kvt0 k exp 4 ln ıij C i Tfig .t0 ; t/5 jD1 i2P u
kvt0 k exp Œln ıi1 C i1 C ln ıi2 C i2 C : : : C ln ıik1 C ik1 C ik :
That is, v.t/ c k where c WD kvt0 k = mini2P ıi . Case (iv) is proved similarly. Beginning from Eq. (7), case (v) implies that, for t 2 Œtk1 ; tk /,
v.t/ kvt0 k .ı/N0 expŒ.t t0 / ln.ı/=tavg C .C /T .t0 ; t/ kvt0 k .ı/N0 expŒ.ln.ı/=tavg C C /.t t0 /=M ; Robust Synchronization of Distributed-Delay Systems via Hybrid Control 745
implying RS. For case (vi), Eq. (5) implies that, for j 2 N,
v.t0 C j!/ 2 3 X jm X X kvt0 k exp 4 ln ıi C i Tfig .t0 ; t0 C j!/ i TQ fig .t0 ; t0 C j!/5 iD1 i2P u i2P s 2 3 X m X X D kvt0 k exp 4j ln ıi C j i Tfig .t0 ; t0 C !/ j i TQ fig .t0 ; t0 C !/5 iD1 i2P u i2P s 2 0 13 Xm X X kvt0 k exp 4j @ ln ıi C i i i .i /A5 : iD1 i2P u i2P s
Since v.t/ is also bounded on any compact interval, the result follows.
3 Example
Consider system (3) with n D 2, P D f1; 2; 3; 4g, t0 D 0, distributed delay D0:1, 0 1 2 0:5 AD ; CD ; 1 0 0 1:4 3:1 0 1:8 0:1 7:1 1:3 15 2H1 D ; H2 D ; H3 D ; H4 D ; 0 3:1 0:1 1:6 1:3 22:3 2 8:8 1:05 1:1 0:4 0:4 0:1 0:1 0:12 0:13 G1 D ; G2 D ; G3 D ; G4 D ; 1:11 1:04 0:3 0:1 0:1 0:1 0:3 0:1 0:1 0:1 0:4 0 0:2 0 0:3 0:1E1 D ; E2 D ; E3 D ; E4 D : 0:2 0:1 0:01 0:4 0 0:2 0:1 0:3
For 2 PC, let F. / D .0; 0:1 ln.cosh. 1 .0////. Let A;i D C;i D H;i DG;i D E;i D I, A;i D C;i D H;i D G;i D E;i D 0:01I, %A;i .t/ D %C;i .t/ D%H;i .t/ D %G;i .t/ D %E;i .t/ D cos.5t/I. Suppose the initial conditions are given byxt0 .s/ D .14; 20/ and y0 .s/ D .22; 33/ for s 0, and suppose that theswitching rule is periodic with 1 D 0:4, 2 D 0:4, 3 D 0:2, and 4 D 1 (i.e.,! D 2). 746 P. Stechlinski and X. Liu
(a) (b)
400 100 x (t) 1 x1(t) 300 5 σ(t) x2(t) 80 x2(t) 4 y1(t) (u=0) y (t) 200 1 3 y2(t) (u=0) y (t) 60 2 2 100 1 40 0 0 0 2 4 Time
−100 20
−200 0
−300 −20 −400 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time Time
Fig. 1 Simulations of drive system (1) and response system (2). (a) Trajectories without hybridcontrol. (b) Trajectories with hybrid control
Choose 10 10 0:354 0:020 0:253 0:037P1 D ; P2 D ; P3 D ; P4 D ; 01 01 0:020 0:204 0:037 0:374
and "A;i D "H;i D "C;i D "H;i D "E;i D 10 for i 2 P, #1 D 0:1, #2 D 0. Then,ˇ1 D 0:021, ˇ2 D 0:006, ˇ3 D 0:060, ˇ4 D 0:059, ˛1 D 5:602, ˛2 D 6:812,˛3 D 12:866, ˛4 D 11:867, D 1:768, ı1 D 1:070, ı2 D 0:825, ı3 D 2:087,ı4 D 1:028. Hence 1 D 5:623, 2 D 6:818, 3 D 12:806, 4 D 11:808, sothat P s D f3; 4g and P u D f1; 2g. Choose 1 D 11, 2 D 12, to get
4 X X X ln ıi C i i i .i / D 5:485; iD1 i2P u i2P s
which implies robust synchronization of the drive and response systems by Theo-rem 1. See Fig. 1 for an illustration.
4 Conclusions
The sets P s and P u in Theorem 1 represent the synchronizing and desynchronizingmodes, respectively, while the conditions ı < 1 and ı > 1 represent impulsivecontrol and possible impulsive perturbations, respectively. Each theorem conditionestablishes threshold conditions involving the mode activation times, mode syn-chronization/desynchronization effects, and impulsive effects, for various classesof switching rules (i.e., dwell-time (i)–(iv), average dwell-time (v), and periodic Robust Synchronization of Distributed-Delay Systems via Hybrid Control 747
switching (vi)). The results found are robust to model uncertainties, nonlinearperturbations, and impulsive perturbations. One possible future direction is toconsider the robust hybrid synchronization from an optimal control point of view.
Acknowledgements This research was financially supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC).
References
1. Khadra, A., Liu, X., Shen, X.: Application of impulsive synchronization to communication security. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 50(3), 341 (2003) 2. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520 (2004) 3. Li, C., Liao, X., Yang, X., Huang, T.: Impulsive stabilization and synchronization of a class of chaotic delay systems. J. Nonlinear Sci. 15(4), 043 (2005) 4. Li, P., Cao, J., Wang, Z.: Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Phys. A: Stat. Mech. Appl. 373, 261 (2007) 5. Lakshmikantham, V., Rao, M.R.M.: Theory of Integro-Differential Equations. Gordon and Breach Science Publishers, Amsterdam (1995) 6. Burton, T.: Volterra Integral and Differential Equations. Elsevier B.V., Amsterdam (2005) 7. Liu, J., Liu, X., Xie, W.C.: Delay-dependent robust control for uncertain switched systems with time-delay. Nonlinear Anal. Hybrid Syst. 2(1), 81 (2008) 8. Liberzon, D.: Switching in Systems and Control. Birkhauser, Boston (2003) 9. Davrazos, G., Koussoulas, N.T.: A review of stability results for switched and hybrid systems. In: Proceedings of 9th Mediterranean Conference on Control and Automation, Dubrovnik (2001)10. Evans, R.J., Savkin, A.V.: Hybrid Dynamical Systems. Birkhauser, Boston (2002)11. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49(4), 545 (2007)12. Hespanha, J.P., Morse, A.S.: Stability of switched systems with average dwell-time. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, vol. 3, p. 2655 (1999)13. Guan, Z.H., Hill, D., Shen, X.: On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans. Autom. Control 50(7), 1058 (2005)14. Guan, Z.H., Hill, D., Yao, J.: A hybrid impulsive and switching control strategy for synchro- nization of nonlinear systems and application to Chua’s chaotic circuit. Int. J. Bifurc. Chaos 16(1), 229 (2006)15. Alwan, M.S., Liu, X.: On stability of linear and weakly nonlinear switched systems with time delay. Math. Comput. Model. 48(7–8), 1150(2008)16. Zhang, Y., Liu, X., Shen, X.: Stability of switched systems with time delay. Nonlinear Anal. Hybrid Syst. 1(1), 44 (2007)17. Yang, C., Zhu, W.: Stability analysis of impulsive switched systems with time delays. Math. Comput. Model. 50(7–8), 1188 (2009)18. Zhu, W.: Stability analysis of switched impulsive systems with time delays. Nonlinear Anal. Hybrid Syst. 4(3), 608 (2010)19. Niamsup, P.: Stability of time-varying switched systems with time-varying delay. Nonlinear Anal. Hybrid Syst. 3(4), 631 (2009)20. Wang, X., Li, X.: A generalized Halanay inequality with impulse and delay. Adv. Dev. Technol. Int. 1(2), 9 (2012) 748 P. Stechlinski and X. Liu
21. Wang, Y., Xie, L., de Souza, C.E.: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 19, 139 (1992)22. de Souza, C.E., Li, X.: Delay-dependent robust H-infinity control of uncertain linear state- delayed systems. Automatica 35, 1313 (1999)23. Chen, W.H., Zheng, W.X.: Robust stability and H-infinity control of uncertain impulsive systems with time-delay. Automatica 45, 109 (2009)24. Xiang, Z., Chen, Q.: Robust reliable control for uncertain switched nonlinear systems with time delay under asynchronous switching. Appl. Math. Comput. 216(3), 800 (2010)25. Liu, X., Zhong, S., Ding, X.: Robust exponential stability of impulsive switched systems with switching delays: a Razumikhin approach. Commun. Nonlinear Sci. Numer. Simul. 17, 1805 (2012)26. Ballinger, G., Liu, X.: Existence and uniqueness results for impulsive delay differential equations. Dyn. Contin. Discret. Impuls. Syst. 5(3), 579 (1999)27. Liu, J., Liu, X., Xie, W.C.: Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica 47, 899 (2011)28. Hien, L.V., Phat, V.N.: Exponential stabilization for a class of hybrid systems with mixed delays in state and control. Nonlinear Anal. Hybrid Syst. 3(3), 259 (2009) Regularization and Numerical Integrationof DAEs Based on the Signature Method
Andreas Steinbrecher
Abstract Modeling and simulation of dynamical systems often leads todifferential-algebraic equations (DAEs). Since direct numerical integration of DAEsin general leads to instabilities and possibly non-convergence of numerical methods,a regularization is required. We present three approaches for the regularization ofDAEs that are based on the Signature method. Furthermore, we present a softwarepackage suited for the proposed regularizations and illustrate its efficiency on twoexamples.
1 Introduction
Modeling and simulation of dynamical systems is an important issue in manyindustrial applications and often leads to systems of differential-algebraic equations(DAEs). In this paper, we consider nonlinear DAEs of the form
F.t; z; zP/ D 0; (1)
with sufficiently smooth functions F W I Rn Rn ! Rm and z W I ! Rn , with acompact time interval I R. In general, DAEs are not suited for direct numerical integration due to theso called hidden constraints, which restrict the solution but are not explicitlystated as equations. Hidden constraints can be determined by a certain numberof differentiation of (parts of) the DAE. This number of differentiations gives aclassification of difficulties in the treatment of DAEs, see [2, 4, 5, 12]. To guarantee a stable and robust numerical integration, a regularization of DAEsis required. The basic idea of an approach to obtain a regularization for DAEs isto consider the original system together with a sufficient number of its derivatives.Then a system with the same solution set as the original system can be derived thatnow contains all (hidden) constraints explicitly as algebraic equations. Then it can
A. Steinbrecher ()Institute of Mathematics MA 4-5, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germanye-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 749J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_67 750 A. Steinbrecher
be guaranteed that all (hidden) constraints are satisfied within the numerical solutionand instabilities or drift from the solution manifold are avoided. In most modeling and simulation tools, the current state-of-the-art to regularizeDAEs is to use some kind of structural analysis based on the sparsity pattern ofthe system, to obtain necessary information for a regularization. The advantage ofa structural analysis in comparison to classical algebraic regularizations is that fastalgorithms based on graph theory can be applied. Usually, the Pantelides algorithm[9] in combination with the Dummy Derivative Method [6] is used. In this paper,we present regularization approaches based on the Signature method (˙-method)[10]. In Sect. 3, we discuss regularization techniques that are based on the informationprovided by the ˙-method which is reviewed in Sect. 2. Two of the regular-ization techniques yield overdetermined systems of DAEs. Then, in Sect. 4 weshortly discuss the software package QUALIDAES for the numerical integrationof (overdetermined) quasi-linear DAEs. In Sect. 5 we illustrate the applicability ofQUALIDAES in combination with the three proposed regularized formulations.
2 Signature Method: Structural Analysis of DAEs
The ˙-method, see [7, 8, 10] and also [11], can be applied to regular nonlinearDAEs of arbitrary high order p of the form F.t; z; zP; : : : ; z. p/ / D 0. We denote by Fithe ith component of the vector-valued function F and by zj the jth component ofthe vector z. Then, the ˙-method consists of the following steps:1. Building the signature matrix ˙ D Œij i;jD1;:::;n with ( highest order of derivative of zj in Fi , ij D 1 if zj does not occur in Fi .
2. Finding a highest value transversal (HVT) of ˙, i.e., a transversal T D f.1; j1 /; .2; j2 /; : : : ; .n; jn /g,Pwhere . j1 ; : : : ; jn / is a permutation of .1; : : : ; n/, with maximal value Val.T/ WD .i;j/2T ij .3. Computing the offset vectors c D Œci iD1;:::;n and d D Œdj jD1;:::;n with ci ; dj 2 N0 such that
dj ci ij for all i; j D 1; : : : ; n; and dj ci D ij for all .i; j/ 2 T: (2)
4. Forming the ˙-Jacobian J D ŒJij i;jD1;:::;n with 8 < @Fi .ij / if dj ci D ij ; Jij WD @zj : 0 otherwise. Regularization and Numerical Integration of DAEs Based on the Signature Method 751
5. Building the reduced derivative array 2 3 2 3 Fi .t; Z / F 1 .t; Z / 6 7 dt Fi .t; Z / 7 d 6 :: 7 6 F .t; Z / D 4 : 5D 0 with F i .t; Z / D 6 :: 7 for i D 1; : : :n; 4 : 5 F n .t; Z / dci dtci Fi .t; Z / (3) h i with Z T D z1 zP1 : : : z1.d1 / : : : zn zPn : : : zn.dn / :6. Success check: if F .t; Z / D 0,Pconsidered locally as an algebraic system, has a solution .t ; Z / 2 I RnC iD1 di , and J is nonsingular at .t ; Z /, then n
.t ; Z / is a consistent point and the method succeeds.If the ˙-method succeeds, the DAE (1) is called structurally regular with thestructural index S defined as S WD maxi ci if all dj > 0 or S WD maxi ci C 1if some dj D 0. Note that the success check of the ˙-method is performed locally ata fixed point .t ; Z /, such that the result may hold only locally in a neighborhood.The HVT as well as the offset vectors c and d are not uniquely defined. However,there exists a unique element-wise smallest solution, the so-called canonical offsets.These are uniquely determined and independent of the chosen HVT, see [10]. The reduced derivative array (3) contains all necessary equations to extract allhidden constraints by use of analytical manipulations. This forms the basis for all inthe following proposed regularization approaches.Example 1 Let us consider the simple pendulum (Fig. 1) of mass m > 0, length` > 0 under gravity g, see also [10, 11]. The state of the pendulum is describedby the position x and y, the velocities v and w of the mass point, and the Lagrangemultiplier . The system equations are given by 2 3 2 3 F1 .z; zP/ xP v 6 F .z; zP/ 7 6 7 6 2 7 6 yP w 7 6 7 6 7 0 D F.z; zP/ D 6 F3 .z; zP/ 7 D 6 mvP C 2x 7 (4) 6 7 6 7 4 F4 .z; zP/ 5 4 mwP C 2y C mg 5 F5 .z; zP/ x2 C y2 `2 (5) Twhere z D x y v w . The signature matrix ˙ for (4) and the canonical offsetvectors c and d are given by 2 3 2 3 2 3 1 1 0 1 1 1 2 61 1 0 17 617 627 6 1 7 6 7 6 7 6 7 6 7 6 7 ˙ D6 0 1 1 1 0 7; c D 6 0 7; d D 6 1 7; (6) 6 7 6 7 6 7 41 0 1 1 0 5 405 415 0 0 1 1 1 2 0 752 A. Steinbrecher
Fig. 1 Simple pendulum Y g X
m (x(t),y(t))
where the two possible HVTs are marked by light and dark gray boxes. Thecorresponding ˙-Jacobian J and the reduced derivative array (3) takes the form 2 3 xP v 2 3 6 xR vP 7 1 0 1 0 0 6 7 6 P y w 7 60 1 0 1 0 7 6 7 6 7 6 R y wP 7 6 7 6 7 JD6 0 0 m 0 2x7 ; 0 D F .t; Z / D 6 mvP C 2x 7 (7) 6 7 6 7 40 0 0 m 2y5 6 mwP C 2y C mg 7 6 7 2x 2y 0 0 0 6 x2 C y2 `2 7 4 5 2xPx C 2yPy 2xRx C 2Px2 C 2yRy C 2Py2 with Z T D x xP xR y yP yR v vP w wP . Since det.J/ D 4m.x2 C y2 / D 4m`2 ¤ 0,the ˙-Jacobian J is nonsingular at every consistent point, and the ˙-methodsucceeds with S D maxi ci C 1 D 3. G
3 Regularization Approaches for DAEs
In the following, we discuss three different regularization approaches for struc-turally regular DAEs (1) with m D n. These regularization approaches mainly usethe reduced derivative array (3) provided by the ˙-method and differ in its degreeof analytical preprocessing and possible degree of automation of the approach. Regularization and Numerical Integration of DAEs Based on the Signature Method 753
3.1 Regularization via State Selection PThe reduced derivative array (3) consists of M D i ci Cn equations in n unknownsz1 ; : : : ; zn .P Thus, to obtain the same number of equations and unknowns, we have tointroduce i ci new variables. For that we choose an HVT T from the set of all existing HVTs. Then for eachj D 1; : : : ; n we have a unique i with .i; j/ 2 T , and a regularization can be obtained . C1/ .d /from the reduced derivative array (3) by replacing the derivatives zj ij ; : : :; zj i C1with the algebraic variables !j ij ; : : :; !jdi if di > ij . We obtain the structurallyextended (StE) formulation
S .t; z; zP; !/ D 0 (8)
h i C1with ! T D !1T !nT and !jT WD !j ij !jdi 2 Rci if dj > ij or !j WD Œ 2R0 if dj D ij as new algebraic variables. Note that dj D ij C ci by (2). The obtained structurally extended formulation (8) is of increased size, notunique, since it depends on the chosen HVT T , and in general it is only validlocally in a neighborhood of a consistent point. Furthermore, it may lead to aninappropriate regularization, if an inappropriate HVT T is used (see Example 2).To choose a suitable HVT, i.e., one that is valid in a preferably large neighborhoodof a consistent Q point, we define the (local) weighting coefficient for each HVT Tas T WD .i;j/2T jJij j and choose one with largest value T , i.e., we choose T as an HVT of ˙ with T D maxT .T /. In [11] it is shown that the structurallyextended system (8) is (locally) a regular system of structural index S 1 if T is chosen as described above. This formulation has locally the same set of solutionsfor the original unknowns z as the original DAE (1). Due to the local validity, thedetermination of a new regularized formulation during the (numerical) integrationmay be necessary. In this case, the integration has to be interrupted and to berestarted with the newly regularized model equations. Unfortunately, this leads toan increase in simulation time and influences the obtained precision negatively. This approach completely can be automated using automatic or symbolicdifferentiation tools. Therefore, an analytical preprocessing is not necessary. In the proposed method, the selection of variables for which derivatives arereplaced is directly prescribed by the HVT T and the offset vectors. Once the ˙-method is done, this selection of variables is easy to achieve and requires no furthernumerical computations. In contrast, in the Dummy Derivative Method, see [6]. Asa result, the two approaches might result in different regularized systems.Example 2 The system (4) is regularPand of structural index S D 3. The reducedderivative array (7) consists of M D ci C n D 9 equations in n D 5 unknowns. 754 A. Steinbrecher
For the HVT T D T1 marked by the light gray boxes in the signature matrix (6)(assuming that T1 D 4x2 T2 D 4y2 ), we have to introduce new algebraicvariables as in the following table:
For j with i replace derivatives with new algebraic variables . C1/ .d / C1 d (s.t. .i; j/ 2 T ) zj ij ; : : :; zj j !j ij ; : : :; !j j .1/ .2/ 1 5 z1 ; z1 !11 ; !12 .2/ 2 2 z2 !22 .1/ 3 1 z3 !31 4 4 ; ; 5 3 ; ; Then we construct the structurally extended formulation as 2 3 !11 v 6 !12 !31 7 6 7 2 3 6 7 2 13 6 yP w 7 x !1 6 !22 wP 7 6 7 6 7 6y7 6 !2 7 6 7 6 7 6 7 S 1 .t; z; zP; !/ D 6 m!31 C 2x 7 D 0 with z D 6 v 7; ! D 6 12 7: 6 7 6 7 4 !2 5 6 mwP C 2y C mg 7 4w5 6 7 !31 6 x2 C y2 `2 7 6 7 4 2x!11 C 2yPy 5 2 1 2 2 2 2x!1 C2.!1 / C2y!2 C2Py (9)
For the HVT T D T2 marked by the dark gray boxes in the signature matrix (6)(assuming that T2 D 4y2 T1 D 4x2 ), we have to introduce new algebraicvariables as in the following table:
For j with i replace derivatives with new algebraic variables . C1/ .d / C1 d (s.t. .i; j/ 2 T ) zj ij ; : : :; zj j !j ij ; : : :; !j j .2/ 1 1 z1 !12 .1/ .2/ 2 5 z2 ; z2 !2 ; !22 1
3 3 ; ; .1/ 4 2 z4 !41 5 4 ; ; Regularization and Numerical Integration of DAEs Based on the Signature Method 755
Then we construct the structurally extended formulation as 2 3 xP v 6 !12 vP 7 6 7 2 3 6 !21 w 7 2 23 6 7 x !1 6 2 !2 !4 1 7 6 7 6 7 6y7 6 !1 7 6 7 6 7 6 27 S 2 .t; z; zP; !/ D 6 mvP C 2x 7 D 0 with z D 6 v 7; ! D 6 2 7: 6 7 6 7 4 !2 5 6 m!41 C 2y C mg 7 4w5 6 7 !41 6 x2 C y2 `2 7 6 7 4 2xPx C 2y!21 5 2 2 2 1 2 2x!1 C2Px C2y!2 C2.!2 / (10)
Applying the ˙-method to (9) and (10), we obtain a structurally regular systemwith structural index S D 1 as long as x ¤ 0 while (10) forms a structurally regularsystem with structural index S D 1 as long as y ¤ 0. G
3.2 Regularization via Algebraic Derivative Arrays
As we have seen in the previous section, the structurally extended regularization isonly valid locally. Therefore, it may be necessary to switch to another structurallyextended formulation within a numerical integration (also known as dynamic stateselection). This influences the time integration negatively. As a remedy in thissection we propose an extended regularization. As mentioned above, the reduced derivative array (3) contains all necessaryequations to obtain all hidden constraints. Now we replace all derivatives of all . j/ jvariables by algebraic variables, i.e., we replace zi by !i for i D 1; : : :; n andj D 1; : : :; di , in the reduced derivative array to obtain the algebraic derivative array Œ!i1 ; : : :; !idi T if di > 1; 0 D F .t; z; !/ D F .t; z1 ; !1 ; : : : ; zn ; !n / with !i D Œ if di D 0:
From this it is possible to extract the set of all hidden constraints in an algebraic waywhich, in particular, can be done automatically within the numerical integration. In combination with the original DAE (1) we obtain the algebraic derivativearray (ADA) formulation F.t; z; zP/ 0 D A .t; z; zP; !/ D (11) F .t; z; !/
which forms an overdetermined DAE for z and ! and has the same set of solutionsfor the original unknowns z as the original DAE (1). The obtained algebraic 756 A. Steinbrecher
derivative array formulation is of more increased size as the structurally extendedformulation but globally valid such that the regularized formulation has to bedetermined only once before the numerical integration. This approach also canbe automated using automatic or symbolic differentiation tools. Therefore, ananalytical preprocessing is not necessary.Example 3 Let us use the model equations of the simple pendulum to illustratethe regularization based on the algebraic derivative array formulation. The reducedderivative array is given in (7). Replacing all occurring derivatives with algebraicvariables, i.e., xP ! !11 , xR ! !12 , yP ! !21 , yR ! !22 , vP ! !31 , and wP ! !41 , leads tothe algebraic derivative array 2 3 !11 v 6 !12 !31 7 6 7 6 1 !2 w 7 6 7 6 2 !2 !4 1 7 6 7 6 7 0 D F .t; z; !/ D 6 m!31 C 2x 7: (12) 6 7 6 1 m!4 C 2y C mg 7 6 7 6 2 x Cy L 2 2 7 6 7 4 1 2.x!1 C y!2 / 1 5 2.x!12 C.!11 /2 Cy!22 C.!21 /2 /
Together with the original DAE (4), we get the algebraic derivative array formula-tion (11) with F.t; z; zP/ given in (4) and F .t; z; !/ given in (12). This correspondsto an overdetermined set of DAEs containing 14 equations for 11 unknowns. G
3.3 Regularization via Overdetermined Formulation
In the previous sections, we did obtain regularizations that increased in its size morethen necessary. Therefore, in the numerical integration more computational time isneeded than necessary. As mentioned in Sect. 2 the reduced derivative array (3) allows the determinationof the set of (hidden) constraints
0 D H.t; z/
in an analytical preprocessing. In combination with the original DAE (1), we obtainthe set F.t; z; zP/ 0 D O.t; z; zP/ D ; H.t; z/ Regularization and Numerical Integration of DAEs Based on the Signature Method 757
which forms the regularized overdetermined (OVD) formulation for z and whichhas the same set of solutions as the original DAE (1). The obtained overdeterminedformulation is of increased number of equations for the same unknown variables asin the original DAE. For more details see [12].Example 4 For the simple pendulum, see Example 1, we get the set of (hidden)constraints as 2 3 x2 C y2 `2 0 D H.t; z/ D 4 2xv C 2yw 5 (13) v 2 Cw2 m2 .x2 C y2 /yg
and consequently the regularization via overdetermined formulations is determinedby (1), (13) as a set containing eight equations for five unknowns. G
4 The Software Package QUALIDAES
The software package QUALIDAES (QUAsi LInear DAE Solver) is suited for thenumerical integration of (overdetermined) quasi-linear DAEs of the form " # " # E.z; t/ k.z; t/ differential part; zP D 0 G.z; t/ algebraic constraints:
QUALIDAES is implemented in FORTRAN. The discretization scheme is the 3-stage Runge-Kutta RADAU IIa of order 5 and uses adapted decomposition methodsw.r.t. the overdeterminedness to solve the (hidden) constraints (numerically) pre-cisely, while the differential part is solved in an ‘approximative sense’. Further features of QUALIDAES are variable step size control, continuousoutput, and check and correction of the initial values with respect to its consistency.The specification of the system DAEs can be done in FORTRAN source code oralternatively in MODELICA using a simple MODELICA-parser. For MODELICAa Matlab interface is provided, see [1]. For more information to QUALIDAES werefer to [13].
5 Numerical Results
In the following, we illustrate the efficiency of QUALIDAES with the proposedregularization techniques on two examples. The first one is the simple pendulum,and the second one is a mass-spring-chain with path constraints. 758 A. Steinbrecher
Efficiency (absolute error vs. simulation time)
QUALIDAES(ADA) 10 2 absolut error ERR=||numsol-refsol|| 2
QUALIDAES(StE) DASSL(DI1)
10 0
10 -2 QUALIDAES(OVD)
RADAU5(GGL) 10 -4 10 -1 10 0 10 1 simulation time
Fig. 2 Efficiency (simulation time vs. absolute error for different prescribed tolerances)
The numerical integrations are done on an AMD Phenom(tm) II X6 1090T,3210 MHz, 16 GB RAM, openSuSE 13.1 (Linux 3.11.10), GNU Fortran compilergcc version 4.8.1, no compiler options.Example 5 The model equations of the simple pendulum, see Example 1, are statedin (4) and have structural index S D 3 and differentiation index d D 3. The massand length are given by m D 1 and ` D 1 while we use the gravitational accelerationg D 13:7503716373294544. In this case the exact solution has a period of 2 s whichallows the comparison of the accuracy every period. The simulation is done for the time domain I D Œ0 s; 2000 s with initial values Tz.0/ D 1 0 0 0 0 . To show the efficiency (see Fig. 2) of the numerical integrationwith QUALIDAES in comparison we use the widely used solvers RADAU5, with theGear-Gupta-Leimkuler (GGL) formulation [3] and DASSL with the d-index-1 (DI1)formulation consisting the first four equations of (4) and the last equation of (13). The numerical solution QUALIDAES(OVD) offers the best efficiency, i.e., lesscomputational effort and small absolute error of the numerical solution. Thisnumerical solution is slightly more efficient as the numerical result obtained withRADAU5(GGL) followed by the numerical solutions of QUALIDAES(ADA) andQUALIDAES(StE). The large time consumption for the integration of the StE-formlies in its local validity as discussed in Example 2. Within one period of 2 s it has tobe switched between (9) and (10) four times. Furthermore, due to the increasedsize of the ADA-form and the StE-form both are less efficient than the OVD- Regularization and Numerical Integration of DAEs Based on the Signature Method 759
Fig. 3 Mass-spring-chain F Control
p m 3
Path p m 2
p m 1
F Control
form. The numerical integration using DASSL is not successful at all due to thestability properties of BDF methods, the occurring drift off effect by use of the DI1-form, where the constraints on position level and on velocity level are lost, and thelarge time domain I. Nevertheless, the maximally obtained precision is excellent forQUALIDAES(OVD) and QUALIDAES(ADA). G
Example 6 In the following we are interested in a path following problem of a mass-spring-chain as in Fig. 3. On the first and the last body the force F is applied whilethe center body has to follow a prescribed path (14g). We have pi and vi as theposition and velocity of body i D 1; 2; 3 and F as unknown variables. The modelequations are given by
pP 1 D v1 ; (14a) pP 2 D v2 ; (14b) pP 3 D v3 ; (14c) mvP 1 D c. p1 p2 / C F; (14d) mvP 2 D c. p1 p2 / c. p2 p3 /; (14e) mvP 3 D c. p2 p3 / C F; (14f) 0 D p2 sin.t/ (14g) 760 A. Steinbrecher
Efficiency (absolute error vs. simulation time) 2 10 DASSL(DI1)
absolut error ERR=||numsol-refsol|| 2 10 0
RADAU5(DI1) 10 -2
QUALIDAES(ADA) 10 -4
QUALIDAES(StE) -6 10
QUALIDAES(OVD)
10 -1 simulation time
Fig. 4 Efficiency (simulation time vs. absolute error for different prescribed tolerances)
with structural index S D 5 and differentiation index d D 5. Therefore, we havehidden constraints up to level 4, i.e., up to the 4th derivative of (several) modelequations are necessary to determine all hidden constraints. We use m D 1 andc D 1=6 with the exact solution p2 .t/ D sin.t/, v2 .t/ D cos.t/, p1 .t/ D p3 .t/ D2 sin.t/, v1 .t/ D v3 .t/ D 2 cos.t/, F.t/ D 3 sin.t/=2. The simulation is done forthe time do-main I D Œ0 s; 500 s . In the same style as in the last example, the efficiency of the numerical solutionsis illustrated in Fig. 4. Again the numerical solution QUALIDAES(OVD) offers the best efficiencyfollowed by the numerical solution QUALIDAES(StE). In contrast to the previousexample, the efficiency using the StE-form is better since the StE-form is valid forthe whole time domain and has not to be switched. Furthermore, the numerical solu-tion QUALIDAES(ADA) offers a good efficiency. The efficiency of RADAU5(DI1)is similar, but the obtained precision is not as good as for the previous results. Thereason is that we used the DI1-form, where the hidden constraints of level 4 areincluded instead of (14g). Therefore, the (hidden) constraints up to level 3 are lost,and the solution drifts away from these lost constraints. Nevertheless, the maximallyobtained precision is excellent for QUALIDAES with all regularizations proposed inSect. 3, i.e., for the StE-form, the ADA-form, and the OVD-form. G Regularization and Numerical Integration of DAEs Based on the Signature Method 761
6 Summary
The aim of this article was to discuss several approaches for the regularization ofdifferential-algebraic equations that benefit numerical integration. For this purpose,we have proposed three different regularization approaches which end in thestructurally extended formulation, the algebraic derivative array formulation, andthe regularized overdetermined formulation. All these regularization approachesare based on the signature method, which was reviewed in Sect. 2, and two ofthem require numerical integration methods suited for overdetermined differential-algebraic equations. We also briefly introduced the software package QUALIDAESand illustrated its efficiency for two examples.
Acknowledgements This work has been supported by the European Research Council throughAdvanced Grant MODSIMCONMP.
References
1. Altmeyer, R., Steinbrecher, A.: Regularization and Numerical Simulation of Dynamical Systems Modeled with Modelica. Preprint 29-2013, Institut für Mathematik, Berlin (2013) 2. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. Classics in Applied Mathematics, vol. 14. SIAM, Philadel- phia (1996) 3. Gear, C.W., Leimkuhler, B., Gupta, G.K.: Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12–13, 77–90 (1985) 4. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II – Stiff and Differential- Algebraic Problems, 2nd edn. Springer, Berlin (1996) 5. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich, 2006. 6. Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy variables. SIAM J. Sci. Comput. 14, 677–692 (1993) 7. Nedialkov, N., Pryce, J., Tan, G.: DAESA – a Matlab tool for structural analysis of DAEs: soft- ware. Technical report CAS-12-01-NN, Department of Computing and Software, McMaster University, Hamilton (2012) 8. Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (i): computing Taylor coefficients. BIT Numer. Math. 45(3), 561–591 (2005) 9. Pantelides, C.C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9(2), 213–231 (1988)10. Pryce, J.D.: A simple structural analysis method for DAEs. BIT 41(2), 364–394 (2001)11. Scholz, L., Steinbrecher, A.: Regularization of DAEs based on the signature method. BIT Numer. Math. 1–22 (2015). doi:10.1007/s10543-015-0565-x12. Steinbrecher, A.: Numerical solution of quasi-linear differential-algebraic equations and industrial simulation of multibody systems. Ph.D. thesis, Technische Universität Berlin (2006)13. Steinbrecher, A.: QUALIDAES: a software package for the numerical integration of quasi- linear differential-algebraic equations. Technical report, Institut für Mathematik, Technische Universität Berlin, Berlin (in preparation) Symbolic-Numeric Methods for ImprovingStructural Analysis of Differential-AlgebraicEquation Systems
Guangning Tan, Nedialko S. Nedialkov, and John D. Pryce
Abstract Systems of differential-algebraic equations (DAEs) are generated rou-tinely by simulation and modeling environments, such as MapleSim and those basedon the Modelica language. Before a simulation starts and a numerical method isapplied, some kind of structural analysis is performed to determine which equationsto be differentiated, and how many times. Both Pantelides’s algorithm and Pryce’s˙-method are equivalent in the sense that, if one method succeeds in findingthe correct index and producing a nonsingular Jacobian for a numerical solutionprocedure, then the other does also. Such a success occurs on many problems ofinterest, but these structural analysis methods can fail on simple, solvable DAEs andgive incorrect structural information including the index. This article investigates ˙-method’s failures and presents two symbolic-numeric conversion methods for fixingthem. Both methods convert a DAE on which the ˙-method fails to a DAE on whichthis SA may succeed.
1 Introduction
We consider differential-algebraic equations (DAEs) of the general form
fi . t; the xj and derivatives of them / D 0; i D 1; : : : ; n; (1)
G. Tan ()School of Computational Science and Engineering, McMaster University, 1280 Main Street West,Hamilton, ON, L8S 4L8, Canadae-mail: [emailprotected]. NedialkovDepartment of Computing and Software, McMaster University, 1280 Main Street West, Hamilton,ON, L8S 4L8, Canadae-mail: [emailprotected]. PryceSchool of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, Wales, UKe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 763J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_68 764 G. Tan et al.
where the xj .t/; j D 1; : : : ; n are state variables that are functions of an independentvariable t, usually regarded as time. Pryce’s structural analysis (SA), the ˙-method [10], determines for (1) itsstructural index, number of degrees of freedom (DOF), variables and derivatives thatneed initial values, and the constraints of the DAE. These SA results can help decidehow to apply an index reduction algorithm [3], perform a regularization process[12], or design a solution scheme for a Taylor series method [5–7]. The ˙-methodis equivalent to Pantelides’s algorithm [9]: they both produce the same structuralindex [10, Theorem 5.8], which is an upper bound for the differentiation index, andoften both indices are the same [10]. The ˙-method succeeds on many problems of practical interest, producing anonsingular System Jacobian. However, this SA can fail—hence Pantelides’s algo-rithm can fail as well—on some simple, solvable DAEs, producing an identicallysingular System Jacobian. We investigate such SA’s failures and present two symbolic-numeric conversionmethods for fixing them. After each conversion, provided some conditions aresatisfied, the value of the signature matrix is guaranteed to decrease. We conjecturethat such a decrease should result in a better problem formulation of a DAE, so thatthe SA may produce a nonsingular System Jacobian and hence succeed. Section 2 summarizes the ˙-method. Section 3 describes SA’s failures. Section 4presents our two conversion methods, each of which is illustrated with an exampletherein. Section 5 presents concluding remarks. This article is a succinct version of the technical report [13]; the reader is referredto it for more detailed explanations and examples.
2 Summary of the ˙ -Method
This SA method [10] constructs for a DAE (1) an n n signature matrix ˙, whose.i; j/ entry ij is either an integer 0, the order of the highest derivative to whichvariable xj occurs in equation fi , or 1 if xj does not occur in fi . A highest-value transversal (HVT) of ˙ is a set T of n positions .i; j/ with oneposition in each row and each column, such that the sum of the corresponding entriesis maximized. This sum is the value of ˙, written Val.˙/. If Val.˙/ is finite, thenthe DAE is structurally well posed (SWP); otherwise it is structurally ill posed. We assume henceforth the SWP case. Using the HVT, we find n equation and nvariable offsets c1 ; : : : ; cn and d1 ; : : : ; dn , respectively, which are integers satisfying
ci 0 for all i; dj ci ij for all i; j with equality on a HVT . (2)
These offsets are valid but not unique. There exists an elementwise smallest solutionof (2) termed the canonical offsets [10]. Symbolic-Numeric Methods for Improving Structural Analysis of DAEs 765
We associate with given valid offsets an n n System Jacobian J, defined as ( .ij / @fi =@xj if dj ci D ij ; and Jij D (3) 0 otherwise .
If J is nonsingular at a consistent point as defined in [5], then we say the ˙-methodsucceeds and there exists (locally) a unique solution through this point [10]. In the success case, the canonical offsets determine the structural index and thenumber of DOF, defined as ( 1 if minj dj D 0 S D max ci C i 0 otherwise X X X and DOF D Val.˙/ D ij D dj ci : .i;j/2T j i
(DOF means “degrees of freedom”, while DOF is the corresponding number.)Example 1 We illustrate1 the above concepts with the simple pendulum DAE ofdifferentiation index 3.
0 D f1 D x00 C x x y ci x y f1 "2 0ı # 0 f1 " 1 x# 0 D f2 D y00 C y g ˙ D f2 2 0 0 ı J D f2 1 y ı f3 0 0 2 f3 2x 2y 0 D f3 D x2 C y2 L2 dj 2 2 0
The state variables are x; y, and ; G is gravity and L > 0 is the length of thependulum. There are two HVTs of ˙, marked with and ı, respectively. A blankin ˙ denotes 1, and a blank in J denotes 0. Since det.J/ D 2.x2 C y2 / D 2L2 ¤ 0, the System Jacobian is nonsingular,and the SA succeeds. The structural index is S D mini ci C 1 D 2 C 1 D 3 (becauseminj dj D d3 D 0),Pwhich equals P the differentiation index. The number of DOF isDOF D Val.˙/ D d j j c i i D 4 2 D 2.
1 When we present a DAE example, we show its signature matrix ˙, canonical offsets ci and dj ,and the associated System Jacobian J. 766 G. Tan et al.
3 Structural Analysis’s Failure
We say that the ˙-method fails, if a DAE (1) has a finite Val.˙/ and an identicallysingular J. In the failure case, the SA reports Val.˙/ as an “apparent” DOF but nota meaningful one. In this article, we focus on the case where such an identically singular J isstructurally nonsingular. That is, there exists a HVT T of ˙ such that Jij isgenerically nonzero for all .i; j/ 2 T.Example 2 We illustrate a failure case with the following DAE2 in [1, p. 23].
0 D f1 D x0 C ty0 g1 .t/ x y ci x y f 1 1 0 f 1 t ˙D 1 JD 1 0 D f2 D x C ty g2 .t/ f2 0 0 1 f2 1 t dj 1 1
The SA fails since det.J/ D 0. Now J is identically singular but structurallynonsingular: on a HVT marked by , each of J11 D 1 and J22 D t is genericallynonzero. One simple fix is to replace f1 by f 1 D f1 C f20 , which results in the systembelow; cf. [3, Example 5].
0 D f 1 D y C g1 .t/ g02 .t/ x y ci x y ˙ D f1 0 0 J D f1 1 0 D f2 D x C ty g2 .t/ f2 0 0 0 f2 1 t dj 0 0
Now that det.J/ D 1 ¤ 0, the SA succeeds. Notice Val.˙ / D 0 < 1 D Val.˙/. Another simple fix is to introduce a new variable z D x C ty and eliminate x in f1and f2 .
0 D f 1 D y C z0 g1 .t/ y z ci y z f 0 1 0 f 1 1 ˙D 1 JD 1 0 D f2 D z g2 .t/ f2 0 1 f2 1 dj 0 1
For this resulting DAE, det.J/ D 1 ¤ 0, and the SA succeeds. After solving for yand z, we can obtain x D z ty. This fix also gives Val.˙ / D 0 < 1 D Val.˙/. The above two manipulations illustrate the two conversion methods presented inSect. 4, respectively.
2 In the original formulation, the driving functions are f1 ; f2 . Here they are renamed g1 ; g2 . Symbolic-Numeric Methods for Improving Structural Analysis of DAEs 767
4 Conversion Methods
We present two conversion methods for systematically fixing SA’s failures. The firstmethod is based on replacing an existing equation by a linear combination of someequations and derivatives of them. We call this method the linear combination (LC)method and describe it in Sect. 4.1. The second method is based on substitutingnewly introduced variables for some expressions and enlarging the system. We callthis method the expression substitution (ES) method and describe it in Sect. 4.2. We only present the main features of these methods; the reader is referred to [13]for more details and especially the proofs of Theorems 1 and 2 below. Given a DAE (1), we assume it has a finite Val.˙/ and a System Jacobian J that isidentically singular but structurally nonsingular. We also assume the equations in (1)are sufficiently differentiable, so that our methods fit into the ˙-method theory. After a conversion, we obtain a system with signature matrix ˙ and SystemJacobian J. If Val.˙/ is finite and J is still identically singular, then we canperform another conversion using either of the methods, provided the correspondingconditions are satisfied. Suppose a sequence of conversions produces a solvableDAE with Val.˙ / 0 and a generically nonsingular J. Given the fact that eachconversion reduces some Val.˙/ by at least one, the total number of conversionsdoes not exceed the value of the original signature matrix. If the resulting system is structurally ill posed after a conversion, that is,Val.˙ / D 1, then we say the original DAE is ill posed [13].
4.1 Linear Combination Method
Let u be a nonzero n-vector function in the co*kernel co*ker.J/ of J, that is, JT u isidentically zero. We consider J and u as functions of t, the xj ’s and derivatives ofthem. Let xj ; u denote the order of the highest derivative to which xj occurs in theexpressions defining the function u, or 1 if xj does not occur in u at all. Denote ˚ I D f i j ui 6 0 g; c D min ci ; and L D i 2 I j ci D c : (4) i2I
Here ui 6 0 means that ui is generically nonzero. The LC method is based on thefollowing theorem.Theorem 1 If xj ; u < d j c for all j D 1; : : : ; n ; (5) 768 G. Tan et al.
and we replace an fl with l 2 L by X .ci c/ fl D ui fi ; (6) i2I
then Val.˙/ < Val.˙/, where ˙ is the signature matrix of the resulting DAE. We call (5) the condition for applying the LC method.Example 3 We illustrate this method with the following problem:
0 D f1 D x01 C x3 0 D f3 Dx1 x2 C g1 .t/ 0 D f2 D x02 C x4 0 D f4 Dx1 x4 C x2 x3 C x1 C x2 C g2 .t/ ;
where g1 and g2 are given driving functions.
x1 x2 x3 x4 c i x1 x2 x3 x4 f1 2 1 0 30 f1 2 1 1 3 1 0 70 17 ˙D 26 JD 26 f f 1 f3 4 0 0 51 f3 4 x2 x1 5 f4 0 0 0 0 0 f4 x2 x1 dj 1 1 0 0
Here Val.˙/ D 1. A shaded entry ij in ˙ marks a position .i; j/ where dj ci >ij 0 and hence Jij D 0 by the formula (3) for J. The SA fails here since J isidentically singular (but structurally nonsingular). T ˚ We choose u ˚D x2 ; x 1 ; 1; 1 2 co*ker.J/. Then (4) becomes I D 1; 2; 3; 4 ,c D 0, and L D 1; 2; 4 . The condition (5) holds since
.x1 ; u/ D 0 < 1 0 D d1 c; .x2 ; u/ D 0 < 1 0 D d2 c ; .x3 ; u/ D 1 < 0 0 D d3 c; .x4 ; u/ D 1 < 0 0 D d4 c :
Selecting l D 4 2 L for example, we replace f4 by X .ci c/ f4 D ui fi D x2 f1 C x1 f2 C f30 f4 D x1 x2 C g01 .t/ g2 .t/ : i2I
The resulting DAE is 0 D . f1 ; f2 ; f3 ; f 4 / with the SA results as follows.
x1 x2 x3 x4 c i x1 x2 x3 x4 f1 2 1 0 30 f1 2 1 1 3 1 0 70 17 ˙D 26 JD 26 f f 1 f3 4 0 0 51 f3 4 x2 x1 5 f 4 0 0 1 f 4 1 1 dj 1 1 0 0 Symbolic-Numeric Methods for Improving Structural Analysis of DAEs 769
Now Val.˙ / D 0 < 1 D Val.˙/. The SA succeeds at all points where det.J/ Dx2 x1 ¤ 0. From (4) and (6), we can recover the replaced equation fl by P .c c/ ı fl D f l i2Inflg ui fi i ul :
Provided ul ¤ 0 for all t in the interval of interest, it is not difficult to show that theoriginal DAE and the resulting one have the same solution (if there exists one); see[13, § 5.3].
4.2 Expression Substitution Method
Let v be a nonzero n-vector function in the kernel of J; write v 2 ker.J/. Denote ˚ JD j j vj 6 0 ; s D jJj ; ˚ (7) MD i j dj ci D ij for some j 2 J ; and c D max ci : i2M
We choose an l 2 J, and introduce s 1 new variables
.dj c/ vj .dl c/ ˚ yj D xj x for all j 2 J n l : (8) vl l
In each fi with i 2 M, we vj .d c/ .cci / substitute yj C xl l vl (9) . / ˚ for every xj ij with ij D dj ci and j 2 J n l :
Denote by f i the equations that result from these substitutions, and write f i D fi fori … M. Using (8), we append to these f i ’s the equations
.dj c/ vj .dl c/ ˚ 0 D gj D yj C xj x for all j 2 J n l (10) vl l
that prescribe the substitutions. Hence the resulting enlarged system consists of ˚ equations 0 D f 1 ; : : : ; f n and 0 D gj for all j 2 J n l ˚ in variables x1 ; : : : ; xn and yj for all j 2 J n l :
Key to applying the ES method is the following theorem. 770 G. Tan et al.
Theorem 2 Let J, s, M, and c be as defined in (7). Assume ( < dj c if j 2 J xj ; v and dj c 0 for all j 2 J : (11) dj c otherwise ,
For any l 2 J, if we• introduce .s 1/ new variables as defined in (8),• perform substitutions in fi for all i 2 M by (9), and• append the equations gj in (10),then Val.˙/ < Val.˙/, where ˙ is the signature matrix of the resulting DAE. We call (11) the conditions for applying the ES method.Example 4 We illustrate the ES method with the artificially constructed DAE below.
0 00 0 D f1 D x1 C ex1 x2 x2 C h1 .t/ x1 x2 ci x1 x2 f 1 2 0 f ˛ ˛x2 ˙ D f1 0 1 1 J D f1 1 x2 0 D f2 D x1 C x2 x02 C x22 C h2 .t/ 2 2 dj 1 2
0 00Here h1 and h2 are given driving functions, and ˛ D ex1 x2 x2 . Obviously det.J/ D0 and the SA fails. Suppose we choose v D .x2 ; 1/T 2 ker.J/. Then (7) becomes ˚ ˚ J D 1; 2 ; s D jJj D 2; M D 1; 2 ; and c D max ci D 1 : i2M
We can apply the ES method as the conditions (11) hold:
.x1 ; v/ D1 1 D 1 1 1D d1 c 1; d1 c D 1 1 D 0 0 ; .x2 ; v/ D 0 0 D 2 1 1D d2 c 1; d2 c D 2 1 D 1 0 : ˚ ˚ For example, we choose l D 2 2 J. Now J n l D 1 . Using (8) and (10), weintroduce a new variable .d c/ v1 .d2 c/ .11/ x2 .21/ y1 D x1 1 x2 D x1 x D x1 C x2 x02 ; v2 1 2
and append the equation 0 D g1 D y1 C x1 C x2 x02 . Then we substitute .y1 x2 x02 /0for x01 in f1 to obtain f 1 , and substitute y1 x2 x02 for x1 in f2 to obtain f 2 . The resultingDAE and its SA results are shown below.
0 02 0 D f 1 D x1 C ey1 Cx2 C h1 .t/ 2x1 x2 y1 3 ci " x1 x2 y1 # f1 0 1 1 0 f1 1 2x02 ˇ ˇ 0 D f 2 D y1 C x22 C h2 .t/ ˙ D f2 4 0 0 5 1 J D f2 2x2 1 g1 0 1 0 0 g1 1 x2 0 D g1 D y1 C x1 C x2 x02 dj 0 1 1 Symbolic-Numeric Methods for Improving Structural Analysis of DAEs 771
0 02Here ˇ D ey1 Cx2 . Now Val.˙/ D 1 < 2 D Val.˙/. The SA succeeds at all pointswhere det.J/ D 2ˇ.x2 C x02 / x2 ¤ 0. From the steps of applying the ES method, we can “undo” the expressionsubstitutions to recover the original DAE. Similar to the LC method, the ES methodalso guarantees that, provided vl ¤ 0 for all t in some interval of interest, the originalDAE and the resulting one have equivalent solutions (if any) in a natural sense; cf.[13, § 6.3]. Our experience suggests that it is effective to attempt the LC method first, andif its condition (5) is violated, then we try the ES method. Also, it is desirable tochoose an l 2 L [resp. l 2 J] in the LC [resp. ES] method, such that an ul [resp. vl ]never becomes zero. For example, it can be a nonzero constant, x21 C 1, or 2 C cos x2 .Such a choice of l guarantees that the resulting DAE is “equivalent” to the originalone—that is, they always have the same solution, if there exists one. We show below that the LC method does not succeed on the DAE in Example 4because the condition (5) is not satisfied. T 0 00 Choose u D 1; ˛ 2 co*ker.J/, where ˛ D ex1 x2 x2 . Then (4) becomes ˚ ˚ ˚ ˚ ID i j ui 6 0 D 1; 2 ; c D min ci D 0; and L D i 2 I j ci D c D 1 : i2I
Since x1 and x2 occur of orders 1 and 2 in u, respectively, we have
.x1 ; u/ D 1 6< 1 0 D d1 c and .x2 ; u/ D 2 6< 2 0 D d2 c :
Hence (5) is not satisfied. Choosing l D 1 2 I for example and replacing f1 by f 1 D u1 f1 C u2 f20 D x1 C h1 .t/ C ˛ 1 C x01 C x2 x002 C .x02 /2 C 2x2 x02 C h02 .t/ results in the DAE 0 D f 1 ; f2 .
x1 x2 ci x1 x2 ˙ D f1 1 2 0 J D f1 ˛ ˛x2 f2 0 1 1 f2 1 x2 dj 1 2
Here D x01 C x2 x002 C .x02 /2 C 2x2 x02 C h02 .t/. The SA fails still, since J is identicallysingular. Now Val.˙ / D Val.˙/ D 2.
5 Conclusions
We proposed two symbolic-numeric conversion methods for improving the ˙-method. They convert a DAE with a finite Val.˙/ and an identically singular (butstructurally nonsingular) System Jacobian to a DAE that may have a generically 772 G. Tan et al.
nonsingular System Jacobian. A conversion guarantees that both DAEs have (atleast locally) the same solution. The conditions for applying these methods can bechecked automatically, and the main result of a conversion is Val.˙ / < Val.˙/,where ˙ is the signature matrix of the resulting DAE. An implementation of these methods requires(1) computing a symbolic form of a System Jacobian J,(2) finding a vector in co*ker.J/ [respectively ker.J/],(3) checking the LC condition (5) [respectively ES conditions (11)], and(4) generating the equations for the resulting DAE.These symbolic computations may seem expensive. However, for DAEs whose Jcan be permuted into a block-triangular form [8, 11], we can locate the diagonalblocks that are singular and then apply our methods to these blocks only, instead ofworking on the whole DAE. This approach is presently under development. We combine MATLAB’s Symbolic Math Toolbox [14] with our structural analysissoftware DAESA [8, 11], and have built a prototype code that applies our conversionmethods automatically. We aim to incorporate them in a future version of DAESA. With our prototype code, we have applied our methods on numerous DAEs. Theyare either arbitrarily constructed to be “SA-failure cases” for our investigations, orborrowed from the existing literature: Campbell and Griepentrog’s robot arm [2], thetransistor amplifier and the ring modulator [4], and the linear constant coefficientDAE in [12]. Our conversion methods succeed in fixing all these solvable DAEs;see [13] and especially Appendix B thereof. We believe that our assumptions andconditions are reasonable for practical problems, and that these methods can helpmake the ˙-method more reliable. Finally, we conjecture that reducing Val.˙/ tends to give a better formulation ofa DAE from the SA’s perspective and then a nonsingular System Jacobian.
Acknowledgements The authors acknowledge with thanks the financial support for this research:GT is supported in part by the Ontario Research Fund, Canada, NSN is supported in part by theNatural Sciences and Engineering Research Council of Canada, and JDP is supported in part bythe Leverhulme Trust, the UK.
References
1. Brenan, K., Campbell, S., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996) 2. Campbell, S.L., Griepentrog, E.: Solvability of general differential-algebraic equations. SIAM J. Sci. Comput. 16(2), 257–270 (1995) 3. Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14(3), 677–692 (1993) 4. Mazzia, F., Iavernaro, F.: Test set for initial value problem solvers. Technical report 40, Department of Mathematics, University of Bari (2003). http://pitagora.dm.uniba.it/~testset/ 5. Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (I): computing Taylor coefficients. BIT Numer. Math. 45(3), 561–591 (2005) Symbolic-Numeric Methods for Improving Structural Analysis of DAEs 773
6. Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (II): computing the system Jacobian. BIT Numer. Math. 47(1), 121–135 (2007) 7. Nedialkov, N.S., Pryce, J.D.: Solving differential-algebraic equations by Taylor series (III): the DAETS code. JNAIAM J. Numer. Anal. Ind. Appl. Math. 3, 61–80 (2008) 8. Nedialkov, N.S., Pryce, J.D., Tan, G.: Algorithm 948: DAESA—a Matlab tool for structural analysis of differential-algebraic equations: software. ACM Trans. Math. Softw. 41(2), 12:1– 12:14 (2015). doi:10.1145/2700586 9. Pantelides, C.: The consistent initialization of differential-algebraic systems. SIAM. J. Sci. Stat. Comput. 9, 213–231 (1988)10. Pryce, J.D.: A simple structural analysis method for DAEs. BIT Numer. Math. 41(2), 364–394 (2001)11. Pryce, J.D., Nedialkov, N.S., Tan, G.: DAESA—a Matlab tool for structural analysis of differential-algebraic equations: theory. ACM Trans. Math. Softw. 41(2), 9:1–9:20 (2015). doi:10.1145/268966412. Scholz, L., Steinbrecher, A.: A combined structural-algebraic approach for the regularization of coupled systems of DAEs. Technical report 30, Reihe des Instituts für Mathematik Technische Universität Berlin, Berlin (2013)13. Tan, G., Nedialkov, N., Pryce, J.: Symbolic-numeric methods for improving structural analysis of differential-algebraic equation systems. Technical report, Department of Computing and Software, McMaster University, 1280 Main St. W., Hamilton, ON, L8S4L8, Canada (2015). 84pp. Download link:http://www.cas.mcmaster.ca/cas/0reports/CAS-15-07-NN.pdf14. The MathWorks, Inc.: Matlab symbolic math toolbox (2015). https://www.mathworks.com/ help/symbolic/index.html Pinning Stabilization of Cellular NeuralNetworks with Time-Delay Via DelayedImpulses
Kexue Zhang, Xinzhi Liu, and Wei-Chau Xie
Abstract This paper studies the pinning stabilization problem of time-delaycellular neural networks. A new pinning delayed-impulsive controller is proposedto stabilize the networks. Based on a Razumikhin-type stability result, a globalexponential stability of time-delay cellular neural networks with delayed impulses isconstructed. It is shown that the global exponential stabilization of delayed cellularneural networks can be effectively realized by controlling a small portion of units inthe networks via delayed impulses. Numerical simulations are provided to illustratethe theoretical result.
1 Introduction
Cellular neural networks (CNNs), introduced by Chua and Yang in [1, 2], haveattracted the attentions of several researchers in recent years. This is mainly dueto their broad applications in many areas, including image processing and patternrecognition (see, e.g., [2, 3] ), data fusion [4], odor classification [5], and solvingpartial differential equations [6]. In real-world applications, it is inevitable for the existence of time delay inthe processing and transmission of signals among units of CNNs. Hence, it ispractical to investigate CNNs with time-delay (DCNNs) (see, e.g., [7–12]). Stabilityof DCNNs, as a prerequisite for their applications, has been studied extensively inthe past decades, and various control methods have been introduced to stabilizeDCNNs, such as intermittent control [9], sliding mode control [10], impulsivecontrol [11], and sampled-data control [12]. Among these control algorithms, theimpulsive control method has been proved to be an effective approach to stabilizeDCNNs. The control mechanism of this method is to control the unit states of
K. Zhang () • X. LiuDepartment of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canadae-mail: [emailprotected]; [emailprotected]. XieDepartment of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON,N2L 3G1, Canadae-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 775J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_69 776 K. Zhang et al.
the CNN with small impulses, which are small samples of the state variables ofthe CNN, at a sequence of discrete moments. Since the time delay in unavoidablein sampling and transmission of the impulsive information is dynamical systems,many control problems of dynamical systems have been investigated via delayedimpulses in recent years, such as stabilization of stochastic functional systems [13]and synchronization of dynamical networks [14]. The regular impulsive control method to stabilize a CNN is to control eachunit of the network to tame the unit dynamics to approach a steady state (i.e.,equilibrium point). However, a neural network is normally composed of a largenumber of units, and it is expensive and infeasible to control all of them. Motivatedby this practical consideration, the idea of controlling a small portion of units,named pinning control, was introduced in [15, 16], and many pinning impulsivecontrol algorithms have been reported for many control problems of dynamicalnetworks (see, e.g., [17–23]) It is worth noting that no time delay is consideredin these pinning impulsive controllers proposed in the above literatures. However,it is natural and essential to consider the delay effects when processing the impulseinformation in the controller. Due to the cost effective advantage of impulsive control method and pinningcontrol strategy and the wide existence of time delay, it is practical to investigatethe pinning impulsive control approach that takes into account of delays. However,to our best knowledge, no such result has been reported for stabilization of DCNNs.Therefore, in this paper, we propose a novel pinning delayed-impulsive controllerfor the DCNNs. The remainder of this paper is organized as follows. In Sect. 2,we formulate the problem and introduce the pinning delayed-impulsive controlalgorithm. In Sect. 3, a global exponential stability result of the impulsive DCNNsis obtained. Then, in Sect. 4, a numerical example is considered to illustrate thetheoretical result. Finally, conclusions are stated in Sect. 5.
2 Preliminaries
Let N denote the set of positive integers, R the set of real numbers, RC the set ofnonnegative real numbers, and Rn the n-dimensional real space equipped with theEuclidean norm. For a; b 2 R with a < b and S Rn , we define n ˇ ˇ PC .Œa; b ; S/ D W Œa; b ! Sˇ .t/ D .tC /; for any t 2 Œa; b/I .t /
exists in S; for any t 2 .a; b I .t / D .t/ for all but o at most a finite number of points t 2 .a; b ;
where .tC / and .t / denote the right and left limit of function at t, respec-tively. For a given constant > 0, the linear space PC .Π; 0 ; Rn / is equippedwith the norm defined by jj jj D sups2Π;0 jj .s/jj, for 2 PC .Π; 0 ; Rn /. Pinning Stabilization Via Delayed Impulses 777
Consider the following CNN with delays:
xP D Cx.t/ C Af .x.t// C Bf .x.t 1 //; (1)
where x D .x1 ; x2 ; : : :; xn /T 2 Rn , and xi is the state of the ith unit; n denotesthe number of units in CNN (1); f .x.t// D . f1 .x1 .t//; f2 .x2 .t//; : : :; fn .xn .t///T andfj .xj .t// denotes the output of the jth unit at time t; A D Œaij nn and B D Œbij nn ;constants aij and bij represent the strengths of connectivity between units i and jat time t and t 1 , respectively; 1 corresponds to the transmission delay whenprocessing information from the jth unit; C D Diagfc1 ; c2 ; : : :; cn g and constant cidenotes the rate at which the ith unit will reset its potential when disconnected withother units of the network. Throughout this paper, we make the following assumptions:(A) f .0/ D 0 and there exists a constant L such that jj f .u/ f .v/jj Ljju vjj for all u; v 2 Rn . The Lipschtiz condition on the nonlinear activation function fi has been widelyconsidered due to its significance in the application of CNNs (see, e.g., [2, 24, 25]).Based on the assumption, system (1) admits the trivial solution. The objective of this paper is to design the following delay-dependent pinningimpulsive controller to exponentially stabilize CNN (1): P1 kD1 Œ1 xi .t/ C 2 xi .t 2 / ı.t tk /; i 2 Dlk ; Ui .t; xi / D (2) 0; i 62 Dlk ;
for i D 1; 2; : : :; n, where 1 and 2 are impulsive control gains to be determined; 2 > 0 denotes the time delay in controller Ui ; the impulsive instant sequenceftk g satisfies ftk g R, 0 t0 < t1 < : : : < tk < : : :, and limk!1 tk D 1;ı./ is the Dirac Delta function, and xi .tk / denotes the left limit of xi at time tk .Let l denote the number of units to be pinned at each impulsive instant, and theindex set Dlk is defined as follows: reorder states x1 .tk /, x2 .tk /; : : :; xn .tk / suchthat jxp1 .tk /j jxp2 .tk /j : : :jxpl .tk /j jxplC1 .tk /j : : : jxpn .tk /j, thendefine Dlk D fp1; p2; : : :; plg. The pinning impulsive control mechanism can beexplained as follows: at each impulsive instant, we only control l units that havelarger deviations from the trivial state than the remaining n l units. The definition of Dlk is borrowed from [26]. Since no delay has been consideredin the pinning impulsive controller in [26], the pinning algorithm introduced in [26]can be treated as a special case of our pinning delayed-impulsive control strategy(i.e., 2 D 0). The impulsive controlled CNN (1) can be written in the form of an impulsivesystem: 8 < xP .t/ D Cx.t/ C Af .x.t// C Bf .x.t 1 //; t 2 Œtk1 ; tk /; x .t / D 1 xi .tk / C 2 xi .tk 2 /; i 2 Dlk ; k 2 N; (3) : i k xt0 D ; 778 K. Zhang et al.
where D .1 ; 2 ; : : :; n /T and i 2 PC .Œa; b ; R/ is the initial function, xi .tk / D xi .tkC / xi .tk /, and xt0 is defined by xt0 .s/ D x.t0 C s/ for s 2 Œ ; 0 and D maxf 1 ; 2 g. Then the pinning impulsive stabilization problem of CNN (1)is transformed into the stability problem of impulsive system (3).Definition 1 The trivial solution of system (3) is said to be globally exponentiallystable (GES), if there exist constants ˛ > 0 and M 1 such that, for any initial dataxt0 D , the following inequality holds,
jjx.t; t0 ; /jj Mjjjj e˛.tt0 / ; for all t t0 :
3 Stabilization
In this section, we will use a Razumikhin-type stability criterion to constructverifiable conditions for the GES of impulsive CNN (3). First, a class of Lyapunovfunctions for impulsive systems is introduced.Definition 2 Function V W RC Rn ! RC is said to belong to the class 0 if thefollowing is true:(1) V is continuous in each of the sets Œtk1 ; tk / Rn , and for each x 2 Rn , t 2 Œtk1 ; tk /, and k 2 N, lim.t;y/!.tk ;x/ V.t; y/ D V.tk ; x/ exists;(2) V.t; x/ is locally Lipschiz in all x 2 Rn , and for all t t0 , V.t; 0/ 0. To establish a stability result for system (3), the following lemma is required.Lemma 1 Assume that there exist V 2 0 , and constants p; q; c; w1 ; w2 ; 1 > 0,and 2 0, such that (i) w1 jjxjjp V.t; x/ w2 jjxjjp , for t 2 RC and x 2 Rn ;(ii) for t 2 Œtk1 ; tk /, 2 PC .Œ ; 0 ; Rn /, and k 2 N,
V 0 .t; .0// cV.t; .0//;
whenever V.t C s; .s// < qV.t; .0// for all s 2 Π; 0 ;(iii) for k 2 N and 2 PC .Π; 0 ; Rn /,
V.tk ; .0/ C Ik .tk ; // 1 V.tk ; .0// C 2 sup fV.tk C s; .s//gI s2Π;0
1(iv) q > 1 C2 > ecd , where d D supk2N ftkC1 tk g.Then the trivial solution of system (3) is GES. Lemma 1 is a direct consequence of Theorem 3.1 in [27]. Next, verifiableconditions will be constructed for the GES of system (3) by utilizing a quadraticLyapunov function. Pinning Stabilization Via Delayed Impulses 779
Theorem 1 If the following inequality is satisfied
ecd < 1; (4) q l 2 jjBjjwhere D 1 nl C n j1 C 1 j C j2 j , c D 2 mini fci g C 2L.jjAjj C p / > 0,and d D supk2N ftkC1 tk g, then the trivial solution of system (3) is GES.Proof Consider the Lyapunov function V.x/ D xT x. Note that condition (i) ofLemma 1 is satisfied with w1 D w2 D 1 and p D 2. Taking the time-derivativealong solutions of (3) yields
P V.x/ D 2xT .t/Pxh.t/ i D 2xT .t/ Cx.t/ C Af .x.t// C Bf .x.t 1 // 2cmin C 2jjAjjL V.x.t// C 2jjBjjLjjx.t/jjjjx.t 1 /jj 2cmin C 2jjAjjL C jjBjjL"1 V.x.t// C "jjBjjLV.x.t 1 //; (5)
where cmin D mini fci g and constant " > 0. It can be seen from (4) that there existsa constant q > 0 such that
1 q> > ecN d ; (6) pwhere cN D 2 mini fci g C 2L.jjAjj C qjjBjj/. If V.x.t C s// < qV.x.t// for alls 2 Π; 0 , then we can obtain from (5) that P V.x/ 2cmin C 2jjAjjL C jjBjjL."1 C q"/ V.x.t//: (7)
Define h as a function of ": h."/ D 2cmin C 2jjAjjL C jjBjjL."1 C q"/. Then,for " > 0, function h attains its minimum value cN at " D p1q (that is, h0 ."/ D 0 at" D p1 ). Therefore, we can get from (7) that V.x/ q P cN V.x/. Given a constant > 0, letting N1 D 1 nl Œ1 .1 C /.1 C 1 /2 , then X .1 N1 / x2i .tk / .1 N1 /.n l/ min fx2i .tk /g i2Dlk i62Dlk
D lŒN1 .1 C /.1 C 1 /2 min fx2i .tk /g i2Dlk X ŒN1 .1 C /.1 C 1 /2 x2i .tk /; i2Dlk 780 K. Zhang et al.
i.e.,
X X X n 2 .1 C /.1 C 1 / x2i .tk / C x2i .tk / N1 x2i .tk /: i2Dlk i62Dlk iD1
Then, for t D tk , we have X X V.x.tk // D x2i .tk / C x2i .tk / i2Dlk i62Dlk X X D Œ.1 C 1 /xi .tk / C 2 xi .tk 2 / 2 C x2i .tk / i2Dlk i62Dlk X Œ.1 C /.1 C 1 /2 x2i .tk / C .1 C 1 /22 x2i .tk 2 / i2Dlk X C x2i .tk / i62Dlk X X .1 C /.1 C 1 /2 x2i .tk / C x2i .tk / i2Dlk i62Dlk
X n C .1 C 1 /22 x2i .tk 2 / iD1
N1 V.x.tk // C .1 C 1 /22 V.x.tk 2 // 1 V.x.tk // C 2 sup fV.x.tk C s//g; s2Π;0
where 1 D N1 , 2 D .1 C 1 /22 , and constant > 0 to be determined to minimizethe value of 1 C 2 . 2 p n N Let h./ D nl .1 C 1 /2 C 22 1 , then, for D j 1C1 j l , we have hN 0 ./ D 0, Ni.e., h./ attains its minimum for > 0. Hence,
l r l 2 D minf1 C 2 g D 1 C j1 C 1 j C j2 j : >0 n n
Based on the above discussion, we can conclude that all the conditions ofLemma 1 are satisfied. Thus, the trivial solution of system (3) is GES. t u Pinning Stabilization Via Delayed Impulses 781
Remark 1 Parameter is related to impulsive control gains 1 , 2 and the ratio l=n.It can be seen from (4) that, the fewer units are controlled at impulsive instants, themore frequently the impulsive controllers need to be added to the network.
4 Numerical Simulations
In this section, we will consider an example to demonstrate our theoretical result. Inorder to observe the pinning control process clearly, we will investigate a CNN withonly two units in the following example.Example 1 Consider CNN (1) with n D 2, c1 D c2 D 1, 1 D 1, 2 0:1 1:5 0:1 AD ; BD ; 5 3 0:2 2:5
and f .x/ D . f1 .x1 /; f2 .x2 //T with f1 ./ D f2 ./ D tanh./. The chaotic attractor ofCNN (1) is shown in Fig. 1.We consider two types of impulsive controllers:(1) l D 1, i.e., impulsive control one unit at each impulsive instant. Let tk tk1 D 0:03, 2 D 1, 1 D 0:868, and 2 D 0:2, then (4) is satisfied. Thus, Theorem 1 implies that the trivial solution of (3) is GES. See Fig. 2 for numerical simulations.(2) l D 2, i.e., impulsive control two units at each impulsive instant. Let tk tk1 D 0:08, and 2 , 1 , 2 are the same as those in the first scenario, then (4) is satisfied and Theorem 1 implies that the trivial solution of (3) is GES. Numerical results are shown in Fig. 3.The initial data in Figs. 2 and 3 is chosen the same as that in Fig. 1, and the reddot denotes the state x at initial time t D 0. The vertical (or horizontal) lines inFig. 2a represent the state jump of x2 (or x1 ) while the other state is unchanged.Since both units are controlled in Fig. 3a, no vertical and horizontal lines can beobserved. It can be seen from Fig. 2 that different unit may be controlled at differentimpulsive instants. This is consistent with our pinning algorithm of controlling theunit which has the largest state deviation with the equilibrium. However, it is morepractical to control one specific unit at all impulsive instants. Next, we apply thepinning impulsive controller to the first and second unit at all impulsive instantsrespectively, and numerical results are shown in Fig. 4a, b. The impulsive control 782 K. Zhang et al.
1 x2
−1
−2
−3
−4
−5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x1
Fig. 1 Chaotic behavior of CNN (1) with the parameters given in Example 1. The initial data forthis simulation is .s/ D Œ1; 1 T for s 2 Œ 1 ; 0 , and the red dot denotes the state x at the initialtime t D 0
(a) 0.4 (b) 1 x x12 0.2 0.5 0 −0.2 0 x2
−0.4 x
−0.6 −0.5
−0.8 −1 −1 −1.2 −1.5 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x1 t
Fig. 2 Impulsive control one unit of CDN (1) at each impulsive instant. (a) Phase portrait.(b) State trajectories
gains 1 , 2 , and the impulsive sequence ftk g are chosen the same as those in Fig. 2.Figure 4 implies that stabilization cannot be realized via this type of pinning strategywith the given parameters, and more strict conditions may be required to guaranteethe stability which will be investigated in our future research. Pinning Stabilization Via Delayed Impulses 783
(a) 0 (b) 1 x −0.2 x12 0.5 −0.4
−0.6 0 x2
x −0.8 −0.5 −1 −1 −1.2
−1.4 −1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x t 1
Fig. 3 Impulsive control both units of CDN (1) at each impulsive instant. (a) Phase portrait.(b) State trajectories
(a) (b) 2.5 x 1.5 x 1 1 2 x x 2 2
1.5 1 1 0.5 0.5 0 0 x
−0.5 −1 −0.5 −1.5 −1 −2 −2.5 −1.5 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 t t
Fig. 4 Impulsive control one specific unit of CDN (1) through all the impulsive instants.(a) Control the first unit. (b) Control the second unit
5 Conclusion
We have studied the stabilization problem of cellular neural networks with time-delay. An impulsive controller that takes into account both the pinning controlalgorithm and time delays has been proposed. Sufficient conditions for the globalexponential stability of cellular neural networks with time-delay and delayedimpulses have been derived by using a Razumikhin-type stability result. Our resulthas shown that the delayed cellular neural networks can be exponentially stabilizedby pinning controlling a small portion of units at each impulsive instant. Numericalsimulations have been provided to demonstrate the effectiveness of our theoreticalresult. 784 K. Zhang et al.
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Yixuan Zhao and Pierre Patie
Abstract The purpose of this note is to carry out a convergence analysis of thespectral representation, derived recently in Patie and Savov (Spectral expansionsof non-self-adjoint generalized Laguerre semigroups, submitted, 2015), of somenon-self-adjoint Markovian semigroups related to spectrally negative ˛-stable Lévyprocesses conditioned to stay positive. More specifically, we start by performing anerror analysis for the spectral type series expansions. Moreover, these semigroupsare closely related to a class of invariant semigroups, whose speed of convergence toequilibrium has been studied in Patie and Savov (Spectral expansions of non-self-adjoint generalized Laguerre semigroups, submitted, 2015). Our second aim is tocarry out a numerical analysis on the convergence rate which is illustrated with twoexamples.
1 Introduction
The class of Lévy stable and related processes have attracted much attentionover the past decades. In particular, this class of processes have been shown tobe an appropriate statistical description for a large number of phenomena. Suchapplications include, e.g., the study of turbulence in physics [4], seismic seriesand earthquakes in geography [6], risk process in financial mathematics [10], andprimary sequences of protein-like copolymers in biomedical science [3]. Therefore,being able to explicitly represent the transition kernel and the solution to the Cauchyproblem associated to stable related semigroups has become a critical issue inmany areas. In our context, we consider a spectrally negative ˛-stable processZ D .Zt /t0 , with ˛ 2 .1; 2/ . It means that Z is a process with stationary andindependent increments, having no positive jumps, and its law is characterized by
Y. Zhao ()Cornell University ORIE Department, 288 Rhodes Hall, Cornell University, Ithaca, NY, USAe-mail: [emailprotected]. PatieCornell University ORIE Department, 206 Rhodes Hall, Cornell University, Ithaca, NY, USAe-mail: [emailprotected]
© Springer International Publishing Switzerland 2016 787J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6_70 788 Y. Zhao and P. Patie
its characteristic exponent which takes the form, for t > 0; <.z/ > 0, Z 0 ˛ jyj˛1 ln EŒe D z t D zZt .ezy 1 zy/ dy 1 .˛/
where throughout stands for the Gamma function. Next, let K 0 D .Kt0 /t0 be thesemigroup of the process Z killed upon entering the negative half-line. That is, forf 2 Bb .RC /, the set of bounded Borelian functions on RC D Œ0; 1/, and for anyt; x > 0, Kt0 is defined by
Kt0 f .x/ D Ex Πf .Zt /Ift<T0 g ;
where T0 D infft > 0I Zt < 0g. By [2, Section 3.2], p˛1 .x/ D x˛1 ; x > 0; isan excessive function for K 0 , i.e. p˛1 0 on RC and for all t > 0, Kt0 p˛1 .x/ "p˛1 .x/. Hence one may construct a semigroup K " D .Kt /t0 through Doob’s h-transform as follows,
Kt" f .x/ D p1 0 ˛1 .x/Kt .p˛1 f /.x/; x > 0;
which turns out to be the Feller semigroup of the ˛-stable process conditioned tostay positive, and will be referred to as ˛-SP+ throughout . It has been shown in [2]that the infinitesimal generator of K " , denoted by L" , can be expressed in terms ofan Erdélyi-Kober type operator in the form, for a smooth function f , Z d2 x y˛1 f .y/ dy L" f .x/ D x˛ D˛0C;1;1 f .x/ D x1˛ : dx2 0 .x y/ ˛1 .2 ˛/
It should be noted that K " is also a ˛1 -self-similar semigroup as in the definition of[5], i.e. Kt" f .cx/ D Kc"1=˛ t dc f .x/ for any c > 0, where dc f .x/ D f .cx/. Hence it iseasily seen that the semigroup K D .Kt /t0 defined by 1 Kt f .x/ D Kt" f ı p˛ .x ˛ /
is a 1-self-similar semigroup, as p˛ .x/ D x˛ is a homeomorphism of RC to RC .For sake of presentation, it will be more convenient to work with the semigroup K.According to a celebrated result by Lamperti [5], there exists a bijection betweenthe set of convolution semigroups on R and the one of positive self-similar Fellersemigroups. In particular, it has been shown, see [2, 8], that K is associated viathe Lamperti mapping to the semigroup of a spectrally negative Lévy process D.t /t0 , with Laplace exponent given, for z 2 CC D fz 2 CI <.z/ > 0g, by
.˛z C ˛/ log EŒez1 D .z/ D : (1) .˛z/ Convergence Analysis of the Spectral Expansion of Stable Related Semigroups 789
Another semigroup of interests is the so-called associated generalized Laguerre (gL)semigroup denoted by P D .Pt /t0 and defined as follows
Pt f .x/ D Ket 1 f ı det .x/; x > 0: (2)
Moreover, P admits a stationary measure with absolutely continuous density .x/dx,and by [9, Theorem 1.6], P can be extended uniquely to a strongly continuous 2contraction semigroup, also denoted 2 R 1by2 P, on the weighted Hilbert space L ./endowed with the norm jj f jj D 0 f .x/.x/dx. From Bertoin and Yor [1], the R1 n Qn kD1 .k/integer moments of are given by 0 x .x/dx D nŠ D
.˛nC˛/ .˛/nŠ , whichyields that its Mellin transform is Z 1 .˛s/ M .s/ D xs1 .x/dx D ; s 2 CC : (3) 0 .˛/ .s/
Hence standard Mellin inversion technique gives
X1 .1/nC1 sin. n ˛ / . ˛ C 1/ ˛n n .x/ D x ; (4) nD0 .˛/nŠ
which, by Stirling approximation, defines a function analytical on CC and by [9,Theorem 2.1], is a positive probability density on RC . It can be easily shown thatthe semigroup P is non-self-adjoint in L2 ./ and thus the spectral theorem cannot be applied. However, Patie and Savov [9] have recently developed an originalapproach to provide a spectral representation of the class of gL semigroups, whichenables us to write Pt f and Kt f as infinite series expansions under some mildconditions on f and t. In particular, we shall observe that when f is a power function,the series expansion is reduced to finitely many terms and thus can be evaluatedprecisely. Hence the first aim of this paper is to perform a numerical convergenceanalysis for the series expansion. Moreover, in [9], an interesting study on the speedof convergence of a gL semigroup to its equilibrium is presented. In particular,depending on the subspace of L2 ./ one considers, one can observe either thehypocoercivity phenomena, that is bounds of the form Cet or a non-classical boundof the form C.e2t 1/1=2 for some C > 0. Therefore, the second objective of thispaper is to carry out a numerical Ranalysis on the speed of convergence for Pt f to f 1in L2 ./ as t ! 1, where f D 0 f .x/.x/dx. Such numerical plots can serve asillustrations for the theoretical results in [9]. In this paper, we also study another class of semigroups called the Gauss-Laguerre semigroup, denoted by G D .Gt /t0 and defined as follows. For any 1˛ 2 .1; 2/ and ˇ 2 Œ1 ˛1 ; 1/, we introduce the Gauss-Laguerre operator definedfor a smooth function f and x > 0, by Z 1 sin..˛ 1// L˛;ˇ f .x/ D .d˛;ˇ x/f 0 .x/ C x f 00 .xy/g˛;ˇ .y/dy; (5) 0 790 Y. Zhao and P. Patie
.˛ˇC˛ˇ/where d˛;ˇ D .˛ˇC1ˇ/ and
.˛ 1/ ˇC 1 C1 1 g˛;ˇ .y/ D 1 y ˛1 2 F1 .˛ˇ C˛ ˇ; ˛I ˛ˇ C˛ ˇ C1I y ˛1 /; (6) ˇ C ˛1 C1
with 2 F1 the Gauss hypergeometric function. Then it is shown in [7, Theorem 1.1]that L˛;ˇ is the generator of a non-self-adjoint contraction Markov semigroup G inL2 .e˛;ˇ /, where
1 1 xˇC ˛1 1 ex ˛1 e˛;ˇ .x/ D ; x > 0; (7) ..˛ 1/ˇ C 1/
is its unique invariant measure. The motivation for studying this class of semigroupsis that while they share many similar properties with P, the expressions of theirspectral expansions are much simpler and easier to evaluate. In this paper, we willalso study the speed of convergence to equilibrium for G.
2 The ˛-SP+ Semigroup and Its Associated gL Semigroup
2.1 Spectral Expansions and Convergence to Equilibrium
We start by introducing a few notations. For x 2 RC , let n X n .˛/kŠ k P n .x/ D .1/ k k x; (8) kD0 .˛k C ˛/ 1 .xn .x//.n/ 1 X .1/kC1 sin. k ˛ / .n C k ˛ C 1/ k W n .x/ D D x˛ ; (9) nŠ nŠ kD0 .˛/kŠ
where, for a smooth function f , f .n/ .x/ D d dxf .x/ Wn .x/ n n , and we let V n .x/ D .x/ , which, 2 L ./.Finally, we let ˛ to be the largest solution to the equation .1 by [9], is in 1 arcsin.// ˛1 cos ˛1 D 12 , and set .˛ 1/ 1 T˛ D log sin and T˛ ;˛ D max T˛ ; 1 C : 2 ˛
Then we have the following theorem. Convergence Analysis of the Spectral Expansion of Stable Related Semigroups 791
Theorem 1(a) For any f 2 L2 ./, x > 0 and t > T˛ ;˛ , we have in L2 ./,
1 X Pt f .x/ D ent h f ; V n i P n .x/; (10) nD0 1 X ˝ ˛ Kt f .x/ D .t C 1/n f ı d.tC1/ ; V n P n .x/: (11) nD0
Particularly, recalling that pm .x/ D xm , we have for any t > 0,
X m .˛m C ˛/ Kt pm .x/ D xm C xmn tn : (12) nD1 .˛.m n/ C ˛/nŠ
(b) There exists C > 0 and an integer k 0 such that for any f 2 L2 ./ and t > T˛ ;˛ , s .k/ k 1 kPt f f k C 2 2 kf f k : (13) e2.tT˛ ;˛ / 1
Moreover, when f D pm , we have, for all t > 0,
.˛m C ˛/ t kPt pm pm k e kpm epm ke ; (14) .˛/.mŠ/2
where e.x/ D ex ; x > 0.Remark 1 Note that (12) indeed coincides with the expression of the entiremoments given by Bertoin and Yor [1, Proposition 1], where the form of as intheir theorem is given by (1). 0Proof We first observe that limu!1 u˛ .u/ D limu!1 u˛ .˛uC˛/ .˛u/ D C˛ for 0some C˛ > 0, therefore the spectral expansion (10) follows from [9, Theo-rem 1.9(1)]. Next, by applying a simple deterministic time change on (2), we getthat for any f 2 L2 ./ and t > 0,
Kt f .x/ D Plog.tC1/ f ı d.tC1/ .x/;
which further yields (11) from (10). Moreover, by [9, Proposition 9.3], the Mellintransform of W n is, for s 2 CC ,
.1/n .s/ .1/n .˛s/ M Wn .s/ D M .s/ D : (15) .n C 1/ .s n/ .˛/ .n C 1/ .s n/ 792 Y. Zhao and P. Patie
Hence taking f D pm , we have
.1/n .t C 1/m .˛m C ˛/ hpm ı dtC1 ; V n i D .t C 1/m M Wn .m C 1/ D ; .˛/ .n C 1/ .m C 1 n/
which, since has a pole at any negative integers, vanishes when n m C 1.Then (11) becomes a finite series and easy algebra yields the representation (12).For the speed of convergence, (13) comes from [9, Theorem 1.9(3c)]. Moreover, by[9, Theorem 2.9], P intertwins with Q D .Qt /t0 , the classical Laguerre semigroup,which is self-adjoint on L2 .e/ with invariant measure e.dx/. More specifically, thereexists a multiplicative kernel such that Pt f D Qt f for any f 2 L2 .e/, and we .˛/.mŠ/2have for all m 2 N, pm .x/ D Qm mŠ .k/ pm .x/ D
.˛mC˛/ pm .x/. Hence we see that kD1 k .˛mC˛/1 pm .x/ D p .x/ .˛/.mŠ/2 m and by [9, Theorem 1.9(3a)], we have
.˛m C ˛/ t kPt pm pm k k1 pm e1 pm ke D e kpm epm ke .˛/.mŠ/2
which completes the proof. t u
2.2 Numerical Illustrations
In this section, we first provide the plots of em .N/, the relative truncation errorof (12) up to N terms which is formally defined by ˇP ˝ ˛ ˇ ˇ m n f ı d.tC1/ ; W n P n .x/ ˇˇ ˇ nDNC1 .t C 1/ em .N/ D ˇ Pm ˝ ˛ ˇ; ˇ nD0 .t C 1/ n f ı d .tC1/ ; W n P n .x/ ˇ
for N running from 1 up to m 1 > 0. Figures 1, 2, and 3 depict the plots for.m D 20; x D 1/; .m D 40; x D 1/; .m D 40; x D 5/, respectively. Eachplot consists of three experiments ˛ D 1:2; 1:5; 1:8., and in each figure, the leftpanel plots log em .N/ with respect to N while the right panel plots em .N/. It can beseen from the plots that em .N/ exhibits a super-exponential decay, and the relativeerror becomes negligible within only a few terms. Comparing Fig. 1 with Fig. 2,we observe that keeping the value of x constant, the convergence is slower forlarger m (for example, 6 terms are sufficient for a fairly good convergence whenm D 20, but not when m D 40), which is indeed expected since there are morenon-zero terms in the series summation (12) when m gets large. Comparing Fig. 2with Fig. 3, we also observe that the rate of convergence among different valuesof ˛’s depends on the value of x. When x D 1, ˛ D 1:2 converges the fastestand ˛ D 1:8 converges the slowest. But when x D 5, ˛ D 1:5 converges thefastest but ˛ D 1:2 becomes the slowest. Hence the relation of convergence rate Convergence Analysis of the Spectral Expansion of Stable Related Semigroups 793
Log Truncation Error m = 20, x = 1 Truncation Error 10 3 α = 1.2 α = 1.2 α = 1.5 α = 1.5 α = 1.8 α = 1.8 0 2.5
−10 2 loge(N)
−20
e(N) 1.5 −30
1 −40
0.5 −50
−60 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10
Number of terms N Number of terms N
Fig. 1 Truncation error plots for m D 20; x D 1; ˛ D 1:2; 1:5; 1:8
Log Truncation Error m = 40, x = 1 Truncation Error 20 40 α = 1.2 α = 1.2 α = 1.5 α = 1.5 α = 1.8 35 α = 1.8 0
30 −20
25log e(N)
−40 e(N) 20 −60 15
−80 10
−100 5
−120 0 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10
Number of terms N Number of terms N
Fig. 2 Truncation error plots for m D 40; x D 1; ˛ D 1:2; 1:5; 1:8
Log Truncation Error m = 40, x = 5 Truncation Error 20 3.5 α = 1.2 α = 1.2 α = 1.5 α = 1.5 α = 1.8 α = 1.8 0 3
−20 2.5log e(N)
−40 2 e(N)
−60 1.5
−80 1
−100 0.5
−120 0 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10
Number of terms N Number of terms N
Fig. 3 Truncation error plots for m D 40; x D 5; ˛ D 1:2; 1:5; 1:8 794 Y. Zhao and P. Patie
Log Invariance Error for α-SP+, m = 20 x 10 −4 Invariance Error −8 1.2 α = 1.2 α = 1.2 α = 1.5 α = 1.5 α = 1.8 α = 1.8 −10 1
−12 0.8log (t)
−14
(t) 0.6 −16
0.4 −18
0.2 −20
−22 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 t t
Fig. 4 Invariance error plots for m D 20; ˛ D 1:2; 1:5; 1:8
Log Invariance Error for α-SP+, m = 40 x 10 −8 Invariance Error −18 1.2 α = 1.2 α = 1.2 α = 1.5 α = 1.5 −20 α = 1.8 α = 1.8 1 −22 0.8 −24log (t)
(t) −26 0.6
−28 0.4 −30 0.2 −32
−34 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 t t
Fig. 5 Invariance error plots for m D 40; ˛ D 1:2; 1:5; 1:8
with respect to ˛ is inconclusive. Next, we provide some numerical plots on thespeed of convergence to equilibrium for the semigroup P. Figures 4 and 5 depictthe plots for .t/ D kPkpt pmmep pm k m ke with m D 20 and m D 40, respectively. Each plotconsists of three experiments ˛ D 1:2; 1:5; 1:8, and each experiment runs for t D 1to 10. In each figure, the left panel plots log .t/ while the right panel plots .t/.From the figures, it is easy to observe exponential decrease for each case. Under thesame conditions, by comparing ˛ D 1:2; 1:5; 1:8, we can see that the convergencebecomes faster as ˛ increases. Moreover, by comparing m D 20 with m D 40, wealso observe that the convergence rate increases as m grows, which can be attributedto the fact that mC1 is indeed the number of non-zero terms in the series summation.In other words, the convergence is faster using more terms for evaluation. Convergence Analysis of the Spectral Expansion of Stable Related Semigroups 795
3 Convergence to Equilibrium of the Gauss-Laguerre Semigroup
The spectral expansion of G can be found in [7] and in this section, we only statethe speed of convergence to equilibrium as in the following theorem.Theorem 2 Denoting T˛ D log.2˛1 1/, there exists a constant C˛ > 0 suchthat for all f 2 L2 .e˛;ˇ / and t > T˛ , r 1 kGt f e˛;ˇ f ke˛;ˇ C˛ k f e˛;ˇ f ke˛;ˇ : (16) e2.tT˛ / 1
In particular, when f D pm , we have for all t > 0,
..˛ 1/.m C ˇ/ C 1/ t kGt pm e˛;ˇ pm ke˛;ˇ e kpm epm ke : (17) mŠ ..˛ 1/ˇ C 1/
Proof By [9, Example 3.3], G corresponds to a spectrally negative Lévy process ˛;ˇ with Laplace exponent ˛;ˇ .u/ D u
..˛1/.uCˇ1/C1/ ..˛1/.uCˇ/C1/ , hence (16) follows from[9, Theorem 1.9 (3b)]. Moreover, as in the previous case, G also intertwins with theclassical Laguerre semigroup Q via an intertwining kernel denoted by ˛;ˇ , i.e. wehave Gt ˛;ˇ f D ˛;ˇ Qt f for all f 2 L2 .e/. Furthermore, by [7, Proposition 3.2] andits proof, we have for all m 2 N,
mŠ ..˛ 1/ˇ C 1/ ˛;ˇ pm .x/ D pm .x/: ..˛ 1/.m C ˇ/ C 1/
..˛1/.mCˇ/C1/Hence 1 ˛;ˇ pm .x/ D mŠ ..˛1/ˇC1/ pm .x/ and by [9, Theorem 1.9(3a)], we have
..˛ 1/.m C ˇ/ C 1/ tkGt pm e˛;ˇ pm ke˛;ˇ et k1 1 ˛;ˇ pm e˛;ˇ pm ke D e kpm epm ke : mŠ ..˛ 1/ˇ C 1/
This completes the proof of this proposition. t u Now we proceed to give the numerical plots for the convergence of Gt pm to kGt pm e˛;ˇ pm ke ˛;ˇe˛;ˇ pm . Figures 6 and 7 depict the plots .t/ D kpm epm ke with m D 20and m D 40 respectively. Each plot consists of three experiments .˛; ˇ/ D.1:2; 4/; .1:5; 1/; .1:8; 0/ and each experiment runs for t D 1 to 10. In eachfigure, the left panel plots log .t/ while the right panel plots .t/. It is easy to observe that in all cases, we still have exponential decay, whichverifies our bound (17). Moreover, we still observe faster convergence when m getslarger, which can be explained exactly as the previous ˛-SP+ example since forlarger value of m, there are more terms used in the series evaluation. 796 Y. Zhao and P. Patie
Log Invariance Error for Gauss-Lauguerre semigroup m = 20 x 10 14 Invariance Error 36 7 α = 1.2, β = −4 α = 1.2, β = −4 α = 1.5, β = −1 α = 1.5, β = −1 34 α = 1.8, β = 0 6 α = 1.8, β = 0
32 5log (t)
30 4
(t) 28 3
26 2
24 1
22 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6
t t
Fig. 6 Invariance error plots for m D 20; .˛; ˇ/ D .1:2; 4/; .1:5; 1/; .1:8; 0/
Log Invariance Error for Gauss-Lauguerre semigroup m= 40 x 10 40 Invariance Error 94 3.5 α = 1.2, β = −4 α = 1.2, β = −4 α = 1.5, β = −1 α = 1.5, β = −1 92 α = 1.8, β = 0 α = 1.8, β = 0 3 90 2.5 88log (t)
86 2
84 (t) 1.5
82 1 80 0.5 78
76 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 t t
Fig. 7 Invariance error plots for m D 40; .˛; ˇ/ D .1:2; 4/; .1:5; 1/; .1:8; 0/
Acknowledgements The authors would like to thank an anonymous referee for insightfulsuggestions which improved an earlier version of the paper.
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Absorbing boundaries, 22 Apparent permeability, 5Abstract Cauchy problem, 653 Apparent velocity (convective flux), 7Accelerated life testing experiments, 585 Aquaponic agriculture, 189Acoustics, 481 Asymptotic stability, 121, 130Acute Bee Paralysis Virus (ABPV), 299 Augmented body, 48Adaptive error control, 448 Augmented gyrostat, 60Adaptive mesh refinement, 461 Average absolute calibration error, 566Adjusted volatility swap strike, 569 Average dwell time, 355, 381, 746Adjusting the Heston drift, 565 Axial fan, 401–403, 407, 408Advection, 469Advection-diffusion PDE, 4Aeroacoustics, 505 Backward time, 159Aerodynamics, 493 BACOL, 460–462, 467Aerosol, 89–91, 95, 97, 98 BACOLI, 447–451, 453, 461, 462, 467Airfoil, 505 Banach fixed point theorem, 704Air volume fraction, 284 Barotropic vorticity equation, 37Algebraic derivative array Basic reproduction number, 255 formulation for differential-algebraic BEM, 493 equations, 755 Bessel function, 17 regularization of differential-algebraic Beta-plane, 37 equations via s, 755 BGK collision approximation, 470Algorithmic (or automatic) differentiation Bidomain models, 209 (AD), 425 Binary set, 691Almost block diagonal, 460 Biofilm, 267ALT with step-stress plan, 585 Biofilm detachment rate, 267Analog system, 701 Biot-Savart Law, 493Analysis of differential-algebraic equations Birth, 291, 293–295 signature method, 749 Blade Element Momentum Theory, 493Analytic description of the polyhedron, 692 Bounce-back, 473Analytic descriptions of the binary set, 699 Boundary conditions, 18, 447, 448, 450–453Analytics, 233 Dirichlet, 18Anomalous diffusion, 15 Neumann, 18ANSYS Fluent, 280 Brain mechanics, 224Anti-monotone, 602 Branch and Bound Polyhedral-SphericalApparent diffusivity, 7 Method, 690
© Springer International Publishing Switzerland 2016 799J. Bélair et al. (eds.), Mathematical and Computational Approaches in AdvancingModern Science and Engineering, DOI 10.1007/978-3-319-30379-6 800 Index
Brass instruments, 481 Consistent point, 715Brownian motion process, 369 Constraints of minimum number of failures,Brusselator, 641–643, 647, 648 591B-spline, 459–462, 467 Continuous approaches, 689Bubble column, 280 Continuous dependence on modeling, 654, 657Bubble injection, 277 Continuous dependence on parameter, 629Bubble plumes, 278 Continuous time random walk, 15Bubble velocity, 280, 283 Continuous-time GARCH model for stochasticBurner, 89–91, 93–98 volatility with delay, 563 Contractivity, 617 Control Theory, 687Cardiac electrophysiology, 209 Control variate method, 557Cauchy-Riemann operator, 658 ConversionCell model, 447, 448, 452–456 to explicit ODE, 722Cell-task method, 133–136, 138, 139 to implicit ODE, 720Cellular neural networks, 775 Convex differentiable extension of functionCerebrospinal fluid dynamics, 224 from a set, 691CFD, 389, 390, 392, 393, 395, 398, 505 Convex extension, 693CFD modelling, 277 Convexity adjustment, 569Chain of gyrostats, 59 Coordinate change, 142Chain of rigid bodies, 48 Coupled lattice Boltzmann, 209Chandrasekhar, 161 Coupled nonlinear parabolic/elliptic PDEs, 209Change of numeraire technique, 552 Covariance and correlation swaps, 571Channel radius, 4 Covariance matrix, 141Chaos game, 618 Critical pressure, 6Chaotic Lur’e system, 725 Critical temperature, 6Characteristic function, 553 C0 -semigroup, 121Chaste, 447, 453–456 Cytokine, 331Chemostat, 267Cholera, 290Circle inversion, 609 DAE, 450, 451, 665, 666, 714Circle inversion map, 610 consistent point, 715Circulation, 277, 285 semi-explicit index-1, 721Circ*mscribed surface, 699 Damping matrix, 101Civan’s transport model, 7 Darcy’s Law, 7Classical Heston model, 565 DASSL, 461, 467, 722Classical solution, 654 Data assimilation, 141Click graph, 602 Davey-Stewartson system, 27Closed formulas for variance and volatility DD scheme, 719 swaps price processes, 566 DD-specification vector, 714, 719Co-infection, 290, 291, 293, 295–297 DDspec, 719Coalition attacks, 597 Death, 291, 293–296Collage Theorem, 115 Defaultable bond, 526Collision, 469 Deformation, 141Collocation, 447, 449–451, 454, 455 Degrees of freedom (DOF), 714, 716COLROW, 460 Delay, 561, 565Combinatorial algorithms, 690 Delayed Heston model, 561, 565Combinatorial set, 694 Delayed Heston model for variance, 566Combinatorial sets inscribed into a sphere, 691 Delayed stochastic volatility, 563Combustion, 89–93, 96–98, 389–392, 398 Delayed-adjusted long-range variance, 568Competition instabilities, 641 Delayed-adjusted mean-reverting speed, 568Compound laser modes, 83 Delta function, 24Compressibility factor, 4 Demand for pollution control, 341Conformal invariance, 145 Demand markets, 538 Index 801
Derivative based global sensitivity measures, Error control, 460–462, 465, 467 194 Escape velocity, 163Detailed chemical kinetic, 89, 93 Escherichia coli, 245DGM, 481 ESI-CyDesign, 722Differential algebraic system, 641, 643 Euler method, 556Differential-algebraic equations (DAEs), 461, Euler-Bernouli, 101 763 Eulerian approach, 280 dummy derivatives, 750 Eutrophication, 277 dynamic state selection, 755 Evolutionary variational inequality problems, hidden constraints, 749 541 numerical integration of , 749 Exponential, 521 numerical integration of with Exponential stability, 355, 368 QUALIDAES, 757 Extended BACOLI, 447, 448, 452–456 regularization of , 749 structural index, 751Differentiation index, 665 FDE, 153Direct cover condition, 430 Feedback control, 725Discontinuous Galerkin, 435 Fekete point sets, 641, 642, 646, 647Discrete Boltzmann equation (DBE), 470 Finite difference scheme, 18Discrete and continuous time difference explicit, 18 equations, 629 implicit, 18Discrete logarithmic energy, 642, 646 Fishery, 342, 345, 347Disease elimination, 256 Fitting model, 588Disease intervention strategies, 256 Flame, 89–98Dissipative, 125 Flow conditions, 5Dual-Rotor, 493 Food-borne illness, 245Dummy derivative, 665–667, 669, 719 Fortran, 493 scheme, 719 Forward contract, 561 universal, 667, 669, 674 Found vector, 718Dummy derivatives Fourier spectrum, 40 for differential-algebraic equations, 750 Fox function, 17 regularization of differential-algebraic Fractals, 609 equations via , 753 Fractional Brownian motion, 571Dynamic feedback controller, 121 Fractional calculus, 15Dynamic friction, 153, 161 Fréchet space, 703Dynamic hedge ratio, 566 Free Vortex, 493Dynamic models, 224 FRPM, 691Dynamic state selection for differential- Fuel-injector, 89, 96, 97 algebraic equations, 755 Functional differential equation, 153 Functional representation of a discrete set, 689
Eigenvalues, 123Electrophysiology, 447, 448 Galois group, 621Elliptic PDEs, 113 Gas phase, 280˙-method, 713 Gaussian collocation, 459–461 standard solution scheme, 715 Gaussian interest rate model, 551 system Jacobian, 713 Geography, 241Entrainment, 153 Geometric configurations, 279Environmental Monitoring, 233 Geometrically complex, 469EPCA, 367 Geometry of similar motions of Hess tops, 54EPDCOL, 460 Global relative sensitivity, 195Epidemic model, 256 Globally exponentially stable, 778Equilibrium, 291, 292 Global solution on a sphere, 692Equivalent martingale measure, 552 GPR (Gain per Resource), 600 802 Index
GPR-Core property, 602 Index-pair, 716GPU, 435 Infinitesimal generator, 653Gravitation, 153 Injector, 89, 91, 95, 96, 98Gravitational current, 158 IntegrationGravitational field, 154 numerical of differential-algebraicGray-Scott, 641–643, 648 equations, 749Green’s function, 643, 649 numerical of differential-algebraicGreen’s matrix, 644, 645 equations with QUALIDAES, 757Ground beef, 245 Interface, 402, 405–408Grunwald-Letnikov derivative, 18 Interfacial forces, 280, 281, 283Gyrostat, 59, 60 Interferon, 331 Intersecting functional representation, 698 Intracranial pressure, 223H-adaptivity, 435 Intrinsic permeability, 5H1 , 367 Invalid traffic, 595Harmonic maps, 143 Inverse Laplace transform, 17Hausdorff distance, 617 Inverse problem, 102, 113Hazard rate estimator, 590 Ionic models, 209Heat transfer, 389, 390 Irreducible representation, 697Hedge ratio, 569 Isotropy subgroup, 83Hedge volatility swap, 565 ISS, 367, 370Helmholtz Theorem, 493 Itô integral, 369Helminths, 289 Itô Lemma, 370Hess solution, 51 Item, 718Hess top, 51 algebraic, 719Heston model, 549, 566 found, 718Hidden constraints, 665, 667 loop-closer, 719Hidden constraints of differential-algebraic state, 718 equations, 749HIV, 290 AIDS, 290 Jacobian matrix, 425Holomorphic semigroup, 653 Jordan triples, 103hom*oclinic orbit, 27hom*ology, 235Honeybee, 299 Kelvin-Voigt damping, 121Hopf bifurcation, 224 Knudsen number, 4Human-environment system, 341, 343, 349 Kozeny-Carman relation, 5Humans, 290, 291, 293, 294HVT, 666, 668, 715Hybrid control, 739 Lévy process, 565Hybrid parallelization, 139 Lévy-based stochastic volatility with delay,Hybrid system, 367, 687, 739 565Hyperbolic conservation laws, 435 Lagmuir isotherm, 6 Lagrange inversion, 162 Lang-Kobayashi rate equations, 81Ill-posed problem, 653 Laplace transform, 16Immunity, 290, 291, 293 Laplace-Beltrami, 643Impingement, 89, 95–98 Lattice Boltzmann method (LBM), 469Implicit ODE, 714, 716 Law of total probability, 15Impulse extension equation, 199 Lax-Wendroff scheme, 470Impulsive switched singular systems, 355 Least square method, 555Index (differentiation), 714 Levy-based stochastic volatility model, 571Index for differential-algebraic equations Liénard-Wiechert potentials, 153 structural , 751 Lifting Line Theory (LLT), 493 Index 803
Linear matrix inequality, 725 Negative false alarms, 603Linear problem over a discrete set, 692 Neutral differential delay equations, 629Liquid phase, 280 Newton’s law, 156Local radial point interpolation method Next-generation matrix, 256 (LRPIM), 471 Next-generation method, 291, 292Logarithm, 521 Non-coalition attacks, 597Loop-closer, 719 Non-gray, 389, 390, 393, 398Lotka-Volterra, 342 Non-Markovian continuous-time GARCHLower bound, 427 model, 567Lower order interpolant, 462 Nonlinear, 343, 345Lyapunov function, 367 Nonlinear wave propagation, 481Lyapunov method, 355 Normalization matrix, 102Lyapunov-Krasovkii functional, 725 NREL Phase VI, 493 Numerical integration of differential-algebraic equations, 749Magma, 623 with QUALIDAES, 757Malaria, 289–297Markets of Optional Processes, 523Martingale, 568 Offsets, 715Mass matrix, 101 canonical, 666, 668, 715Mathematical model, 189, 225 valid, 668, 672Mathlab Toolbox, 194 Online advertising, 595Matlab code, 19 Optimal robust designs, 587Mean free path, 4 Optional martingale deflator, 524Mellin transform, 17, 789 Optional processes, 520 inverse, 17 Optional semimartingale, 520Meshless, 469 Option pricing, 549Meshless lattice Boltzmann method Orbitally stable, 646 (MLLBM), 479 Ordinary differential equation model, 313Meshless local Petrov-Galerkin (MLPG), 470 Ordinary differential equations (ODE), 189,Metabolic heat generation rate, 168 450, 451, 665Metapopulation model, 256 Orthogonal matrix, 103Micro-jet, 89, 95–98 Overdetermined formulationMinimal circ*mscribed sphere, 692 regularization of differential-algebraicMixed functional representation, 698 equations via s, 756Modelling, 290 SIR model, 290Molecular-dynamics, 133 Parabolic, 447, 456Momentum exchange, 280 Paraffin-oil, 89–91, 93–98Monodomain model, 447–449, 452, 453, 456 Parallel speedup, 135Monte Carlo method, 556 Parameter estimation, 113Monte Carlo sampling methods, 194 Particle distribution function, 470Mosquitoes, 290, 291, 293, 294, 296 Past history, 565, 567Multi-factor stochastic volatility with delay, Path-dependent history, 561 563 Pathogen die-off, 309Multi-scale, 447–449, 452 Payoff of variance swap, 562Multiple Lyapunov functions, 355, 379, 380, Payoff of volatility swap, 562 387 PDE, 447, 448, 456Multi-scale system, 209 PDECOL, 460 Penalty method based on functional representations, 691Naive volatility swap strike, 569 Pendulum, 751Nano-particle, 89, 93 Pennes equation, 168Nearly incompressible flows, 469 Perforated domain, 113 804 Index
Periodic boundary condition, 401, 402, 408 Regularization of differential-algebraicPeriodic solutions, 629 equations, 749Permanent rotation of gyrostat, 61 via algebraic derivative arrays, 755Permeability, 476 via dummy derivatives, 753Persistent hom*ology, 235 via overdetermined formulation, 756Perturbation expansion, 162 via state selection, 753Pinning impulsive control, 777 algebraic derivative array formulation, 755Plot initial hedge ratio, 569 regularized overdetermined formulation,Point vortices, 642 757Poiseuille flow, 474 structurally extended formulation, 753Polar decomposition, 102 Regularized overdetermined formulation forPollutant, 89, 90, 98 differential-algebraic equations, 757Pollution epidemic, 341–344, 346–349 Relative efficiency, 593Polyhedral and spherical relaxations, 690 Relative invariant, 622Polyhedron, 689 Relativity, 153Population dispersal, 255 Reliability, 462, 465–467Population dynamics, 189 Reliable control, 380Porosity, 4 Reliable Controllers, 379, 381, 383, 385, 387Porosity-pressure correlation, 5 Reproduction number, 291, 292, 294, 295, 297Porous media, 469 Retarded field, 153Portfolio, 568 Retracts, 687Positive false alarms, 603 Riemann-Liouville derivative, 18Post-Newtonian, 153 Riemannian manifold, 142Pressure field, 7 Rock density, 3Price of variance swap, 562, 567 Rock permeability, 3Price of volatility swap, 562, 567 Rotating electrical machine, conjugate heatProjected dynamical system, 537 transfer, turbulence modelling,Projection onto a discrete set, 692 CFD., 413Proportional hazards models, 587 Rotational domain, 406, 407Pseudo-spectral numerical method, 39 Rotor disc, 402, 403, 405, 408PSM, 693 Row-sum lumping, 474 RTE, 389, 390, 392, 394, 395, 398
Quadratic function, 693 SA, 665, 668, 713QUALIDAES SA-friendly, 713, 716 numerical integration of differential- Schistosomiasis, 289–297 algebraic equations, 757 Schnakenberg, 641, 642Quasi-MLE method, 590 Schrödinger equation, 121Quasi-reversibility method, 654 Security, 561 Semigroup ˛-SP+, 788Radial basis functions (RBF), 471 Gauss-Laguerre, 789Radiation, 389–392, 395, 397, 398 generalized Laguerre, 789Rarefaction coefficient correlation, 5 self-similar, 788Razumikhin, 367 series expansion, 791Razumikhin-type stability criterion, 778 convergence, 792Reach control problem, 687 stationary distribution, 789Reaction-diffusion, 641 convergence, 791, 794, 795Real gas deviation factor, 6 Sensitivity, 291, 293–295, 297Recovery, 291, 293–295 Sensitivity analysis, 10, 189Refinement, 435 SensSB, 194Reflecting boundaries, 23 SEPCA, 367Regularization, 654, 661 Shale gas, 4 Index 805
Shale rock, 4 Stochastic volatility with delay and jumps, 564Shape functions, 471 Stock, 561Short cosets, 624 Strong pathwise duality, 533Siegmund duality, 530 Strongly predictable, 520Signature matrix, 668, 670, 715 Structural analysis, 665, 668, 713 method, 665, 666, 713 Structural analysis of differential-algebraicSignature method for differential-algebraic equations, 763 equations, 749 signature method, 749Similar motions of rigid bodies’ chain, 50 Structural index for differential-algebraicSimilar-clique property, 602 equations, 751Similarity graph, 602 Structurally extended formulation forSimple pendulum, 751 differential-algebraic equations, 753Singular impulsive system, 358 Succeed, 713, 716Singular perturbations, 629 Superconvergent interpolant, 462Singularly perturbed, 379, 380, 641 Supply markets, 538Singularly perturbed systems, 379, 381, 383, Swarming, 299 385, 387 Switched, 379–381, 383, 385, 387SIRS model, 255 Switched systems, 367, 737Skew-symmetric, 108 Synchronization, 725, 740SLW, 389, 390, 392, 395, 397, 398 System Jacobian, 666, 713Snails, 290, 291, 293–295Social learning, 342, 350Social misperception, 342 Target reproduction number, 256Social stigmatization, 342, 344 Test function, 471Software, 447–456 Three Lines Theorem, 660Soliton, 27 Tight gas, 3Solving scheme, 718 Time delay, 775Soot formation, 90, 91, 97 Time evolution problem, 702Sound propagation, 481 Time-censoring, 585Spatial error estimate, 461, 462 Time-changed Brownian motion, 568Spectral analysis, 125 Time-fractional advection-diffusion equation,Spectral mapping theorem, 108 16Spherical harmonics, 642 Time-fractional diffusion equation, 16Spot self-replication, 641 Time-scale tolerance, 203Spot dynamics, 642, 643, 648, 649 Tip clearance, 402, 404, 408, 410Spot patterns, 641–643, 649 Topological data analysis, 233Spot self-replication, 644 Topological obstruction, 687SSS, see Standard solution scheme Topology, 233Stabilization, 121 Tortuosity, 5Stable process, 787 Touching functional representation, 698Standard solution scheme, 713, 715 Touching set of smooth surfaces, 689Star bi-coloring, 427 Traffic similarity, 600State selection Trailing-edge (TE) noise, 505 regularization of differential-algebraic Translation group, 662 equations via , 753 Transmissibility, 293–295State variables, 714 Transport model, 4State vector, 718 Transversal, 666, 672, 715Stator, 401, 403–405, 407 highest-value, 715Stauduhar method, 621 Tropical cyclone, 35Steady state pressure field, 8 True model, 588Stiffness matrix, 101 Trumpet, 481Stochastic integral, 521 Tuberculosis, 290Stochastic monotonicity, 530 Turbulent, 89–93, 95–98Stochastic volatility, 549 Turing, 642 806 Index
Twisted cuboidal, 645 Varroa destrutor, 299Two-layer sets, 690 Vasicek model, 549Two-sided compressions, 427, 431 Vertex located set, 691Two-stage design, 587, 592 Volatility, 561, 563 Volatility or variance models with delay, 563 Volatility surface fitting, 565UBQP, 689 Volatility swap, 561, 568UGR - unconventional gas reservoirs, 7 Volatility swap fair strike, 568Unconstrained binary quadratic problem, 689 Volatility swap hedging, 568Unconventional reservoirs, 3, 4 Vortex, 35Uniform exponential regulator, 202 Vortex filament, 493Uniform motion, 157 Vortex Rossby wave, 36Uniformly entire function, 708Uniformly exponentially regulated in the mean, 202 Water flow streamlines, 282Unit hypercube, 692 Water quality, 278Unusual conditions, 520 Water velocity, 281–283Upward skip-free, 529 Wave mean-flow interaction, 36 Weak form, 471 Well-posedness, 123Vaccine, 289 Wiener process, 369Vanilla option price surface, 566 Wind energy, 493Vanilla options, 569 Wind turbine, 493Variance and volatility swaps hedging, 566 Witness complexes, 237Variance swap, 561, 568Variance swap fair strike, 568Variational inequality, 539 XFOIL, 493
