Model Reduction and Approximation: Theory and Algorithms

433 Pages·2017·7.07 MB·English

by  Peter Benner

Model Reduction
and Approximation
Computational Science & Engineering
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Editor-in-Chief
Donald Estep Chen Greif J. Nathan Kutz
Colorado State University University of British Columbia University of Washington
Jan S. Hesthaven Ralph C. Smith
Editorial Board Ecole Polytechnique Fédérale de North Carolina State University
Daniela Calvetti Lausanne
Charles F. Van Loan
Case Western Reserve University
Johan Hoffman Cornell University
Paul Constantine KTH Royal Institute of Technology
Karen Willcox
Colorado School of Mines
David Keyes Massachusetts Institute of
Omar Ghattas Columbia University Technology
University of Texas at Austin
Series Volumes
Benner, Peter, Cohen, Albert, Ohlberger, Mario, and Willcox, Karen, Editors, Model Reduction and
Approximation: Theory and Algorithms
Kuzmin, Dmitri and Hämäläinen, Jari, Finite Element Methods for Computational Fluid Dynamics:
A Practical Guide
Rostamian, Rouben, Programming Projects in C for Students of Engineering, Science, and Mathematics
Smith, Ralph C., Uncertainty Quantification: Theory, Implementation, and Applications
Dankowicz, Harry and Schilder, Frank, Recipes for Continuation
Mueller, Jennifer L. and Siltanen, Samuli, Linear and Nonlinear Inverse Problems with Practical
Applications
Shapira, Yair, Solving PDEs in C++: Numerical Methods in a Unified Object-Oriented Approach,
Second Edition
Borzì, Alfio and Schulz, Volker, Computational Optimization of Systems Governed by Partial
Differential Equations
Ascher, Uri M. and Greif, Chen, A First Course in Numerical Methods
Layton, William, Introduction to the Numerical Analysis of Incompressible Viscous Flows
Ascher, Uri M., Numerical Methods for Evolutionary Differential Equations
Zohdi, T. I., An Introduction to Modeling and Simulation of Particulate Flows
Biegler, Lorenz T., Ghattas, Omar, Heinkenschloss, Matthias, Keyes, David, and van Bloemen Waanders, Bart,
Editors, Real-Time PDE-Constrained Optimization
Chen, Zhangxin, Huan, Guanren, and Ma, Yuanle, Computational Methods for Multiphase Flows
in Porous Media
Shapira, Yair, Solving PDEs in C++: Numerical Methods in a Unified Object-Oriented Approach
Edited by
PETER BENNER MARIO OHLBERGER
Max Planck Institute for Dynamics Universität Münster
of Complex Technical Systems Münster, Germany
Magdeburg, Germany
ALBERT COHEN KAREN WILLCOX
Université Pierre et Marie Curie Massachusetts Institute of Technology
Paris, France Cambridge, Massachusetts
Model Reduction
and Approximation
Theory and Algorithms
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2017 by the Society for Industrial and Applied Mathematics
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Library of Congress Cataloging-in-Publication Data
Names: Benner, Peter, editor. | Cohen, Albert, 1965- editor. | Ohlberger,
Mario, editor. | Willcox, Karen, editor.
Title: Model reduction and approximation : theory and algorithms / edited by
Peter Benner, Max Planck Institute for Dynamics of Complex Technical
Systems, Magdeburg, Germany, Albert Cohen, Université Pierre et Marie
Curie, Paris, France, Mario Ohlberger, Universität Münster, Münster,
Germany, Karen Willcox, Massachusetts Institute of Technology, Cambridge,
Massachusetts.
Description: Philadelphia : Society for Industrial and Applied Mathematics,
[2017] | Series: Computational science and engineering ; 15 | Includes
bibliographical references and index.
Identifiers: LCCN 2017012782 (print) | LCCN 2017016389 (ebook) | ISBN
9781611974829 (ebook) | ISBN 9781611974812 (print)
Subjects: LCSH: Mathematical models. | Mathematical optimization. |
Computational complexity.
Classification: LCC QA401 (ebook) | LCC QA401 .M3979 2017 (print) | DDC
511.3/4--dc23
LC record available at https://lccn.loc.gov/2017012782
is a registered trademark.
List of Contributors
AthanasiosC.Antoulas SerkanGugercin AnthonyNouy
RiceUniversity DepartmentofMathematics EcoleCentraledeNantes
Houston,TX77005 VirginiaTech DepartmentofComputerSci-
USA Blacksburg,VA24061 enceandMathematics,GeM
USA 1ruedelaNoe
UlrikeBaur 44321Nantes
MaxPlanckInstitutefor France
BernardHaasdonk
DynamicsofComplex
UniversityofStuttgart
TechnicalSystems MarioOhlberger
InstituteofAppliedAnalysis
Sandtorstr.1 UniversityofMünster
andNumericalSimulation
39106Magdeburg Institute for Computational
Pfaffenwaldring57
Germany andAppliedMathematics
70569Stuttgart
Einsteinstr.62
Germany
ChristopherBeattie 48149Münster
DepartmentofMathematics Germany
VirginiaTech ChristianHimpe
Blacksburg,VA24061 UniversityofMünster IvanOseledets
USA Institute for Computational SkolkovoInstituteofScience
andAppliedMathematics andTechnology
PeterBenner
Einsteinstr.62 Moscow143026
MaxPlanckInstitutefor
48149Münster Russia
DynamicsofComplex
Germany
TechnicalSystems
RonaldDeVore
Sandtorstr.1
TexasA&MUniversity
39106Magdeburg A.CosminIonita
CollegeStation,TX77843
Germany TheMathWorks,Inc.
USA
Natick,MA01760-2098
TobiasBreiten USA StefanVolkwein
Karl-Franzens-Universität UniversityofKonstanz
Graz SandaLefteriu DepartmentofMathematics
InstituteforMathematicsand EcoledesMines andStatistics
ScientificComputing 59508Douai Universitätsstr.10
Heinrichstraße36 France 78457Konstanz
8010Graz Germany
Austria
ImmanuelMartini
UniversityofStuttgart
MartinGubisch
Instituteof AppliedAnalysis
UniversityofKonstanz
andNumericalSimulation
DepartmentofMathematics
Pfaffenwaldring57
andStatistics
70569Stuttgart
Universitätsstr.10
Germany
78457Konstanz
Germany
v

Contents
ListofFigures xi
ListofTables xv
ListofAlgorithms xvii
Preface xix
I Sampling-Based Methods 1
1 ProperOrthogonalDecomposition forLinear-Quadratic
OptimalControl 3
MartinGubischandStefanVolkwein
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 ThePODmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Reduced-ordermodelingforevolutionproblems . . . . . . . . . . . 23
1.4 Thelinear-quadraticoptimalcontrolproblem . . . . . . . . . . . . . 34
1.5 Numericalexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2 ATutorialonReducedBasisMethods 65
BernardHaasdonk
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.2 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3 Stationaryproblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4 Instationaryproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.5 Extensionsandoutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3 TheTheoreticalFoundationofReducedBasisMethods 137
RonaldA.DeVore
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.2 EllipticPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.3 Parametricellipticequations . . . . . . . . . . . . . . . . . . . . . . . . 140
3.4 Evaluatingnumericalmethods . . . . . . . . . . . . . . . . . . . . . . . 142
3.5 Comparingwidthsandentropiesof(cid:2) withthoseof(cid:3) . . . . . 148
(cid:3)
3.6 Widthsofourtwomodelclasses . . . . . . . . . . . . . . . . . . . . . . 151
vii
viii Contents
3.7 Numericalmethodsforparametricequations . . . . . . . . . . . . . 155
3.8 NonlinearmethodsinRBs . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
II Tensor-BasedMethods 169
4 Low-Rank Methods for High-Dimensional Approximation and Model
OrderReduction 171
AnthonyNouy
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.2 Tensorspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.3 Low-rankapproximationoforder-twotensors. . . . . . . . . . . . . 178
4.4 Low-rankapproximationofhigher-ordertensors . . . . . . . . . . . 183
4.5 Greedyalgorithmsforlow-rankapproximation . . . . . . . . . . . . 190
4.6 Low-rankapproximationusingsamples . . . . . . . . . . . . . . . . . 196
4.7 Tensor-structuredparameter-dependentorstochasticequations. . 200
4.8 Low-rankapproximationforequationsintensorformat . . . . . . 210
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5 ModelReductionforHigh-Dimensional ParametricProblemsby
TensorTechniques 227
IvanOseledets
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
5.2 Theconceptoftensorformats . . . . . . . . . . . . . . . . . . . . . . . 228
5.3 Canonicalformat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
5.4 Tuckerformat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.5 SVD-basedtensorformats . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.6 TTformat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.7 OptimizationalgorithmsinTTformat . . . . . . . . . . . . . . . . . 240
5.8 Dynamicallow-rankapproximation . . . . . . . . . . . . . . . . . . . 243
5.9 Black-boxapproximationoftensors. . . . . . . . . . . . . . . . . . . . 245
5.10 QuantizedTTformat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.11 Numericalillustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
III System-TheoreticMethods 259
6 ModelOrderReductionBasedonSystemBalancing 261
PeterBennerandTobiasBreiten
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6.2 BTforLTIsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
6.3 Balancing-relatedmodelreduction . . . . . . . . . . . . . . . . . . . . 266
6.4 BTforgeneralizedsystems . . . . . . . . . . . . . . . . . . . . . . . . . 269
6.5 Numericalsolutionoflinearmatrixequations. . . . . . . . . . . . . 275
6.6 Numericalexamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
6.7 Conclusionsandoutlook . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Contents ix
7 ModelReductionbyRationalInterpolation 297
ChristopherBeattieandSerkanGugercin
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7.2 Modelreductionviaprojection . . . . . . . . . . . . . . . . . . . . . . 298
7.3 Modelreductionbyinterpolation . . . . . . . . . . . . . . . . . . . . . 301
7.4 Interpolatoryprojectionsfor(cid:2) optimalapproximation. . . . . . 309
2
7.5 Modelreductionwithgeneralizedcoprimerealizations . . . . . . . 316
7.6 Realization-independentoptimal(cid:2) approximation. . . . . . . . . 320
2
7.7 Interpolatorymodelreductionofparametricsystems . . . . . . . . 322
7.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
8 TheLoewnerFrameworkforModelReduction 335
AthanasiosC.Antoulas,SandaLefteriu,andA.CosminIonita
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
8.2 TheLoewnerframeworkforlinearsystems . . . . . . . . . . . . . . 341
8.3 Reduced-ordermodelingfromdata . . . . . . . . . . . . . . . . . . . . 368
8.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
9 ComparisonofMethodsforParametricModelOrderReduction 377
Ulrike Baur, Peter Benner, Bernard Haasdonk, Christian Himpe, Immanuel
Martini,andMarioOhlberger
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
9.2 MethodsforPMOR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
9.3 Performancemeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.4 Expectations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
9.5 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.6 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
Index 409