Scattered Data Approximation

348 Pages·2004·0.4 MB·English

by  Holger Wendland

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CAMBRIDGEMONOGRAPHSON
APPLIEDANDCOMPUTATIONAL
MATHEMATICS
SeriesEditors
P.G.CIARLET,A.ISERLES,R.V.KOHN,M.H.WRIGHT
17 Scattered Data Approximation
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1. APracticalGuidetoPseudospectralMethods,BengtFornberg
2. DynamicalSystemsandNumericalAnalysis,A.M.StuartandA.R.Humphries
3. LevelSetMethodsandFastMarchingMethods,J.A.Sethain
4. TheNumericalSolutionofIntegralEquationsoftheSecondKind,KendallE.Atkinson
5. OrthogonalRationalFunctions,AdhemarBultheel,PabloGonza´lez-Vera,ErikHendiksen,and
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7. GeometryandTopologyforMeshGenerationHerbertEdelsfrunner
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9. High-OrderMethodsforIncompressibleFluidFlow,M.O.Deville,P.F.FischerandE.H.Mund
10. PracticalExtrapolationMethods,AvramSidi
11. GeneralizedRiemannProblemsinComputationalFluidDynamics,MataniaBen-Artziand
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12. RadialBasisFunctions:TheoryandImplementations,MartinD.Buhmann
13. IterativeKrylovMethodsforLargeLinearSystems,HenkA.vanderVorst
14. SimulatingHamiltonianDynamics,BenLeimkuhler&SebastianReich
15. CollocationMethodsforVolterraIntegralandRelatedFunctionalEquations,HermannBrunner
16. TopologyforComputing,AfraJ.Zomorodian
Scattered Data Approximation
HOLGER WENDLAND
Institutfu¨rNumerischeundAngewandteMathematik
Universita¨tGo¨ttingen
CAMBRIDGEUNIVERSITYPRESS
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Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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© Cambridge University Press 2005
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relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
Firstpublishedinprintformat 2004
ISBN-13 978-0-511-26432-0 eBook(EBL)
ISBN-10 0-511-26432-1 eBook(EBL)
ISBN-13 978-0-521-84335-5 hardback
ISBN-10 0-521-84335-9 hardback
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Contents
Preface pageix
1 Applicationsandmotivations 1
1.1 Surfacereconstruction 1
1.2 Fluid–structureinteractioninaeroelasticity 4
1.3 Grid-freesemi-Lagrangianadvection 6
1.4 Learningfromsplines 7
1.5 Approximationandapproximationorders 13
1.6 Notation 15
1.7 Notesandcomments 16
2 Haarspacesandmultivariatepolynomials 18
2.1 TheMairhuber–Curtistheorem 18
2.2 Multivariatepolynomials 19
3 Localpolynomialreproduction 24
3.1 Definitionandbasicproperties 24
3.2 Normingsets 26
3.3 Existenceforregionswithconecondition 28
3.4 Notesandcomments 34
4 Movingleastsquares 35
4.1 Definitionandcharacterization 35
4.2 Localpolynomialreproductionbymovingleastsquares 40
4.3 Generalizations 43
4.4 Notesandcomments 44
5 Auxiliarytoolsfromanalysisandmeasuretheory 46
5.1 Besselfunctions 46
5.2 Fouriertransformandapproximationbyconvolution 54
5.3 Measuretheory 60
v
vi Contents
6 Positivedefinitefunctions 64
6.1 Definitionandbasicproperties 64
6.2 Bochner’scharacterization 67
6.3 Radialfunctions 78
6.4 Functions,kernels,andothernorms 82
6.5 Notesandcomments 84
7 Completelymonotonefunctions 85
7.1 Definitionandfirstcharacterization 86
7.2 TheBernstein–Hausdorff–Widdercharacterization 88
7.3 Schoenberg’scharacterization 93
7.4 Notesandcomments 96
8 Conditionallypositivedefinitefunctions 97
8.1 Definitionandbasicproperties 97
8.2 AnanalogueofBochner’scharacterization 103
8.3 ExamplesofgeneralizedFouriertransforms 109
8.4 Radialconditionallypositivedefinitefunctions 113
8.5 Interpolationbyconditionallypositivedefinitefunctions 116
8.6 Notesandcomments 117
9 Compactlysupportedfunctions 119
9.1 Generalremarks 119
9.2 Dimensionwalk 120
9.3 Piecewisepolynomialfunctionswithlocalsupport 123
9.4 Compactlysupportedfunctionsofminimaldegree 127
9.5 Generalizations 130
9.6 Notesandcomments 132
10 Nativespaces 133
10.1 Reproducing-kernelHilbertspaces 133
10.2 Nativespacesforpositivedefinitekernels 136
10.3 Nativespacesforconditionallypositivedefinitekernels 141
10.4 Furthercharacterizationsofnativespaces 150
10.5 Specialcasesofnativespaces 156
10.6 Anembeddingtheorem 167
10.7 Restrictionandextension 168
10.8 Notesandcomments 170
11 Errorestimatesforradialbasisfunctioninterpolation 172
11.1 Powerfunctionandfirstestimates 172
11.2 Errorestimatesintermsofthefilldistance 177
Contents vii
11.3 Estimatesforpopularbasisfunctions 183
11.4 SpectralconvergenceforGaussiansand(inverse)multiquadics 188
11.5 Improvederrorestimates 191
11.6 Sobolevboundsforfunctionswithscatteredzeros 194
11.7 Notesandcomments 204
12 Stability 206
12.1 Trade-offprinciple 208
12.2 Lowerboundsforλ 209
min
12.3 Changeofbasis 215
12.4 Notesandcomments 222
13 Optimalrecovery 223
13.1 Minimalpropertiesofradialbasisfunctions 223
13.2 Abstractoptimalrecovery 226
13.3 Notesandcomments 229
14 Datastructures 230
14.1 Thefixed-gridmethod 231
14.2 kd-Trees 237
14.3 bd-Trees 243
14.4 Rangetrees 246
14.5 Notesandcomments 251
15 Numericalmethods 253
15.1 Fastmultipolemethods 253
15.2 ApproximationofLagrangefunctions 265
15.3 Alternatingprojections 270
15.4 Partitionofunity 275
15.5 Multilevelmethods 280
15.6 Agreedyalgorithm 283
15.7 Concludingremarks 287
15.8 Notesandcomments 287
16 Generalizedinterpolation 289
16.1 OptimalrecoveryinHilbertspaces 289
16.2 Hermite–Birkhoffinterpolation 292
16.3 SolvingPDEsbycollocation 296
16.4 Notesandcomments 306
17 Interpolationonspheresandothermanifolds 308
17.1 Sphericalharmonics 308
viii Contents
17.2 Positivedefinitefunctionsonthesphere 310
17.3 Errorestimates 314
17.4 Interpolationoncompactmanifolds 316
17.5 Notesandcomments 321
References 323
Index 334