Certified Reduced Basis Methods for Parametrized Partial Differential Equations

139 Pages·2016·2.253 MB·English

by  Jan S Hesthaven, Gianluigi Rozza, Benjamin Stamm

SPRINGER BRIEFS IN MATHEMATICS
Jan S. Hesthaven
Gianluigi Rozza
Benjamin Stamm
Certified Reduced
Basis Methods
for Parametrized
Partial Differential
Equations
123
SpringerBriefs in Mathematics
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More information about this series at http://www.springer.com/series/10030
Jan S. Hesthaven Gianluigi Rozza
(cid:129)
Benjamin Stamm
fi
Certi ed Reduced Basis
Methods for Parametrized
Partial Differential Equations
123
Jan S.Hesthaven Benjamin Stamm
EPFL-SB-MATHICSE-MCSS Laboratoire Jacques-Louis Lions
EcolePolytechnique Fédérale deLausanne SorbonneUniversités,UPMCUnivParis06,
EPFL CNRS
Lausanne Paris Cedex 05
Switzerland France
Gianluigi Rozza
SISSAMathLab
International Schoolfor AdvancedStudies
Trieste
Italy
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN978-3-319-22469-5 ISBN978-3-319-22470-1 (eBook)
DOI 10.1007/978-3-319-22470-1
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Preface
During the past decade, reduced order modeling has attracted growing interest in
computationalscienceandengineering.Itnowplaysanimportantroleindelivering
high-performancecomputing(andbridgingapplications)acrossindustrialdomains,
from mechanical to electronic engineering, and in the basic and applied sciences,
including neurosciences, medicine, biology, chemistry, etc. Such methods are also
becoming increasingly important in emerging application domains dominated by
multi-physics, multi-scale problems as well as uncertainty quantification.
This book seeks to introduce graduate students, professional scientists, and
engineers to a particular branch in the development of reduced order modeling,
characterized by the provision of reduced models of guaranteed fidelity. This is a
fundamentaldevelopmentthatenablestheusertotrusttheoutputofthemodeland
balancetheneedsforcomputationalefficiencyandmodelfidelity.Thetextdevelops
theseideasbypresentingthefundamentalswithagraduallyincreasingcomplexity;
comparisons are made with more traditional techniques and the performance
illustratedbymeansofafewcarefullychosenexamples.Thebookdoesnotseekto
replace review articles on the topics (such as [1–5]) but aims to widen the per-
spectives on reduced basis methods and to provide an integrated presentation. The
text begins with a basic setting to introduce the general elements of certified
reduced basis methods for elliptic affine coercive problems with linear compliant
outputs and then gradually widens the field, with extensions to non-affine,
non-compliant, non-coercive operators, geometrical parametrization and time-
dependent problems.
Wewouldliketopointoutsomeoriginalingredientsofthetext.Chapter3guides
the reader through different sampling strategies, providing a comparison between
classictechniquesbasedonsingularvaluedecomposition(SVD),properorthogonal
decomposition(POD),andgreedyalgorithms.Inthiscontextitalsodiscussesrecent
results on a priori convergence in the context of the concept of the Kolmogorov
N-width[6].Chapter4containsathoroughdiscussionofthecomputationoflower
boundsforstabilityfactorsandacomparativediscussionofthevarioustechniques.
Chapter 5 focuses on the empirical interpolation method (EIM) [7], which is
emerging as a standard element to address problems exhibiting non-affine
vii
viii Preface
parametrizations and nonlinearities. It is our hope that these last two chapters will
provide a useful overview of more recent material, allowing readers who wish to
address more advanced problems to pursue the development of reduced basis
methods for applications of interest to them. Chapter 6 offers an overview of a
numberofmoreadvanceddevelopmentsandisintendedmoreasanappetizerthanas
a solution manual.
Throughout the text we provide some illustrative examples of applications in
computational mechanics to guide readers through the various topics. All of the
mainalgorithmicelementsareoutlinedbygraphicalboxestoassistthereaderinhis
or her efforts to implement the algorithms, emphasizing a matrix notation. An
appendix with mathematical preliminaries is also included.
This book is loosely based on a Reduced Basis handbook available online [8],
and we thank the co-author of this handbook, our colleague Anthony T. Patera
(MIT),forhisencouragement,support,andadviceduringthewritingofthebook.It
benefitsfromourlong-lastingcollaborationwithhimandhismanyco-workers.We
wouldliketoacknowledgeallthosecolleagueswhocontributedatvariouslevelsin
thepreparationofthismanuscriptandtherelatedresearch.Inparticular,we would
like to thank Francesco Ballarin and Alberto Sartori for preparing representative
tutorialsandthenewopen-sourcesoftwarelibraryavailableasacompaniontothis
book at http://mathlab.sissa.it/rbnics. An important role, including the provision of
useful feedback, was also played by our very talented and motivated students
attending regular doctoral and master classes at EPFL and SISSA (and ICTP),
tutorials in Minneapolis and Savannah, and several summer/winter schools on the
topic in Paris, Cortona, Hamburg, Udine (CISM), Munich, Sevilla, Pamplona,
Barcelona, Torino, and Bilbao.
Lausanne, Switzerland Jan S. Hesthaven
Trieste, Italy Gianluigi Rozza
Paris, France Benjamin Stamm
June 2015
References
1. C. Prudhomme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera, G. Turinici,
Reliablereal-timesolutionofparametrizedpartialdifferentialequations:reduced-basisoutput
boundmethods.J.FluidsEng.124,70–80(2002)
2.A.Quarteroni,G.Rozza,A.Manzoni,Certifiedreducedbasisapproximationforparametrized
PDEandapplications.J.MathInd.3(2011)
3.G.Rozza,FundamentalsofreducedbasismethodforproblemsgovernedbyparametrizedPDEs
andapplications,inCISM Lecturesnotes“SeparatedRepresentationandPGDbasedmodel
reduction:fundamentalsandapplications”(SpringerVienna,2014)
Preface ix
4. G. Rozza, P. Huynh, N.C. Nguyen, A.T. Patera, Real-Time Reliable Simulation of Heat
Transfer Phenomena, in ASME, Heat Transfer Summer Conference collocated with the
InterPACK09 and 3rd Energy Sustainability Conferences, American Society of Mechanical
Engineers(2009),pp.851–860
5.G.Rozza,P.Huynh,A.T.Patera,Reducedbasisapproximationandaposteriorierrorestimation
foraffinelyparametrizedellipticcoercivepartialdifferentialequations:Applicationtotransport
andcontinuummechanics.Arch.Comput.MethodsEng.15,229–275(2008)
6.P.Binev,A.Cohen,W.Dahmen,R.DeVore,G.Petrova,P.Wojtaszczyk,Convergencerates
forgreedyalgorithmsinreducedbasismethods.SIAMJ.Math.Anal.43,1457–1472(2011)
7. M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An empirical interpolation method:
applicationtoefficientreduced-basisdiscretizationofpartialdifferentialequations.C.R.Math.
339,667–672(2004)
8. A.T. Patera, G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for
ParametrizedPartialDifferentialEquations,CopyrightMIT2007,MITPappalardoGraduate
MonographsinMechanicalEngineering,http://www.augustine.mit.edu,2007
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Historical Background and Perspectives. . . . . . . . . . . . . . . . 2
1.2 About this Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Software Libraries with Support for Reduced Basis
Algorithms and Applications . . . . . . . . . . . . . . . . . . . . . . . 6
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Parametrized Differential Equations . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Parametrized Variational Problems . . . . . . . . . . . . . . . . . . . 15
2.1.1 Parametric Weak Formulation. . . . . . . . . . . . . . . . . 16
2.1.2 Inner Products, Norms and Well-Posedness
of the Parametric Weak Formulation . . . . . . . . . . . . 16
2.2 Discretization Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Toy Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Illustrative Example 1: Heat Conduction Part 1. . . . . 20
2.3.2 Illustrative Example 2: Linear Elasticity Part 1. . . . . 22
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Reduced Basis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 The Solution Manifold and the Reduced Basis
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Reduced Basis Space Generation. . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Proper Orthogonal Decomposition (POD). . . . . . . . . 32
3.2.2 Greedy Basis Generation . . . . . . . . . . . . . . . . . . . . 34
3.3 Ensuring Efficiency Through the Affine Decomposition. . . . . 37
3.4 Illustrative Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.1 Illustrative Example 1: Heat Conduction Part 2. . . . . 39
3.4.2 Illustrative Example 2: Linear Elasticity Part 2. . . . . 41
3.5 Summary of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xi