Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design

492 Pages·2019·9.883 MB·English

by  Houman Owhadi, Clint Scovel

OPERATOR-ADAPTED WAVELETS, FAST SOLVERS,
AND NUMERICAL HOMOGENIZATION
Although numerical approximation and statistical inference are traditionally covered as
entirely separate subjects, they are intimately connected through the common purpose of
makingestimationswithpartialinformation.Thisbookexplorestheseconnectionsfroma
gameanddecisiontheoreticperspective,showinghowtheyconstituteapathwaytodevel-
oping simple and general methods for solving fundamental problems in both areas. It
illustrates these interplays by addressing problems related to numerical homogenization,
operator-adapted wavelets, fast solvers, and Gaussian processes. This perspective reveals
much of their essential anatomy and greatly facilitates advances in these areas, thereby
appearing to establish a general principle for guiding the process of scientific discovery.
This book is designed for graduate students, researchers, and engineers in mathematics,
appliedmathematics,andcomputerscience,andparticularlyresearchersinterestedindraw-
ingonanddevelopingthisinterfaceamongapproximation,inference,andlearning.
houman owhadi isProfessorofAppliedandComputationalMathematicsandControl
and Dynamical Systems in the Department of Computing and Mathematical Sciences of
theCaliforniaInstituteofTechnology. HeearnedanM.Sc.fromthe E´colePolytechnique
in 1994 and was a high civil servant in the Corps des Ponts et Chausse´es until 2001. He
earnedhisPh.D.inprobabilitytheoryfromtheE´colePolytechniqueFe´de´raledeLausanne
in 2001 under the supervision of Ge´rard Ben Arous and joined the Centre National de la
Recherche Scientifique (CNRS) during the same year following a postdoctorate position
at Technion. He moved to the California Institute of Technology in 2004. Owhadi serves
asanassociateeditoroftheSIAMJournalonNumericalAnalysis,theSIAM/ASAJournal
onUncertaintyQuantification,theInternationalJournalofUncertaintyQuantification,the
Journal of Computational Dynamics, and Foundations of Data Science. He is one of the
maineditorsoftheSpringerHandbookofUncertaintyQuantification.Hisresearchinterests
concerntheexplorationofinterplaysamongnumericalapproximation,statisticalinference,
andlearningfromagametheoreticperspective,especiallythefacilitation/automationpossi-
bilitiesemergingfromtheseinterplays.Owhadiwasawardedthe2019GermundDahlquist
PrizebytheSocietyforIndustrialandAppliedMathematics.
clint scovel is Research Associate in the Computing and Mathematical Sciences
DepartmentattheCaliforniaInstituteofTechnology,aftera26-yearcareeratLosAlamos
NationalLaboratory,includingfoundationalresearchinsymplecticalgorithmsandmachine
learning.HereceivedhisPh.D.inmathematicsfromtheCourantInstituteofMathematics
atNewYorkUniversityin1983underthesupervisionofHenryMcKean.
TheCambridgeMonographsonAppliedandComputationalMathematicsseriesreflectsthecrucial
role of mathematical and computational techniques in contemporary science. The series publishes
expositionsonallaspectsofapplicableandnumericalmathematics,withanemphasisonnewdevel-
opmentsinthisfast-movingareaofresearch.
State-of-the-artmethodsandalgorithmsaswellasmodernmathematicaldescriptionsofphysical
andmechanicalideasarepresentedinamannersuitedtograduateresearchstudentsandprofessionals
alike.Soundpedagogicalpresentationisaprerequisite.Itisintendedthatbooksintheserieswillserve
toinformanewgenerationofresearchers.
Acompletelistofbooksintheseriescanbefoundat
www.cambridge.org/mathematics.
Recenttitlesincludethefollowing:
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Operator-Adapted Wavelets, Fast Solvers, and
Numerical Homogenization
From a Game Theoretic Approach to Numerical Approximation
and Algorithm Design
HOUMAN OWHADI
CaliforniaInstituteofTechnology
CLINT SCOVEL
CaliforniaInstituteofTechnology
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
79AnsonRoad,#06–04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108484367
DOI:10.1017/9781108594967
©HoumanOwhadiandClintScovel2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2019
PrintedintheUnitedKingdombyTJInternationalLtd.,Padstow,Cornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Owhadi,Houman,author.|Scovel,Clint,1955–author.
Title:Operator-adaptedwavelets,fastsolvers,andnumericalhomogenization:fromagametheoreticapproach
tonumericalapproximationandalgorithmdesign/HoumanOwhadi(CaliforniaInstituteofTechnology),
ClintScovel(CaliforniaInstituteofTechnology).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Series:Cambridgemonographsonappliedandcomputationalmathematics;35
Identifiers:LCCN2019007312|ISBN9781108484367(hardback)
Subjects:LCSH:Approximationtheory.|Estimationtheory.|Mathematicalstatistics.
Classification:LCCQA221.O942019|DDC515–dc23
LCrecordavailableathttps://lccn.loc.gov/2019007312
ISBN978-1-108-48436-7Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
ForAreen,Julien,Kailo,andAne´

Contents
Preface pagexiii
Acknowledgments xiv
ReadingGuide xv
1 Introduction 1
1.1 StatisticalNumericalApproximation 1
1.2 TheGameTheoreticPerspective 4
1.3 IntheSettingofSobolevSpaces 7
1.4 UncertaintyQuantificationandProbabilisticNumerics 19
1.5 StructureoftheBook 20
PartI TheSobolevSpaceSetting 23
2 SobolevSpaceBasics 25
2.1 TheSobolevSpace 25
2.2 TheOperatorandItsCorrespondingEnergyNorm 27
3 OptimalRecoverySplines 34
3.1 Information-BasedComplexity 34
3.2 OptimalRecovery 35
3.3 VariationalPropertiesofOptimalRecoverySplines 36
4 NumericalHomogenization 38
4.1 AShortReviewofClassicalHomogenization 38
4.2 TheNumericalHomogenizationProblem 47
4.3 IndicatorandDiracDeltaFunctionsasφ 51
i
4.4 Accuracy 54
4.5 ExponentialDecay 54
4.6 LocalPolynomialsasφ 58
i,α
vii
viii Contents
4.7 AShortReviewoftheLocalizationProblem 59
4.8 A Short Review of Optimal Recovery Splines in Numerical
Analysis 61
5 Operator-AdaptedWavelets 63
5.1 AShortReview 63
5.2 OverviewoftheConstructionofOperator-AdaptedWavelets 65
5.3 Non-adaptedPrewaveletsasφ(k) 66
i
5.4 Operator-AdaptedPrewavelets 73
5.5 MultiresolutionDecompositionofHs((cid:4)) 74
0
5.6 Operator-AdaptedWavelets 76
5.7 UniformlyBoundedConditionNumbers 79
5.8 MultiresolutionDecompositionofu ∈ Hs((cid:4)) 81
0
5.9 InterpolationMatrixR(k−1,k) 84
5.10 TheDiscreteGambletDecomposition 86
5.11 LocalPolynomialsasφ(k) 88
i
6 FastSolvers 90
6.1 AShortReview 90
6.2 TheGambletTransformandSolve 92
6.3 Sparse and Rank-Revealing Representation of the Green’s
Function 94
6.4 NumericalIllustrationsoftheGambletTransformandSolve 95
6.5 TheFastGambletTransform 99
PartII TheGameTheoreticApproach 103
7 GaussianFields 105
7.1 GaussianRandomVariable 105
7.2 GaussianRandomVector 106
7.3 GaussianSpace 108
7.4 ConditionalCovarianceandPrecisionMatrix 109
7.5 GaussianProcess 112
7.6 GaussianMeasureonaHilbertSpace 113
7.7 GaussianFieldonaHilbertSpace 115
7.8 Canonical Gaussian Field on (Hs((cid:4)),(cid:3)·(cid:3)) in Dual Pairing
0
with(H−s((cid:4)),(cid:3)·(cid:3)∗) 116
7.9 Degenerate Noncentered Gaussian Fields on Hs((cid:4)) in Dual
0
PairingwithH−s((cid:4)) 118
8 OptimalRecoveryGamesonHs((cid:4)) 119
0
8.1 ASimpleFiniteGame 119
8.2 ASimpleOptimalRecoveryGameonRn 122