Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday

670 Pages·2009·14.32 MB·English

by  Ronald A. DeVore

Multiscale, Nonlinear and Adaptive Approximation
Ronald A. DeVore (cid:2) Angela Kunoth
Editors
Multiscale, Nonlinear
and Adaptive Approximation
Dedicated to Wolfgang Dahmen
on the Occasion of his 60th Birthday
Editors
RonaldA.DeVore AngelaKunoth
DepartmentofMathematics InstitutfürMathematik
TexasA&MUniversity UniversitätPaderborn
CollegeStation,TX77840 WarburgerStr.100
USA 33098Paderborn
[email protected] Germany
[email protected]
ISBN978-3-642-03412-1 e-ISBN978-3-642-03413-8
DOI10.1007/978-3-642-03413-8
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009935698
MathematicsSubjectClassification(2000):41-XX,65-XX
(cid:2)c Springer-VerlagBerlinHeidelberg2009
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DedicatedtoWolfgangDahmenonthe
Occasionofhis60thBirthday
Preface
Ontheoccasionofhis60thbirthdayinOctober2009,friends,collaborators,and
admirersofWolfgangDahmenhaveorganizedthisvolumewhichtouchesonvari-
ousofhisresearchinterests.Thisvolumewillprovideaneasytoreadexcursioninto
many important topics in applied and computational mathematics. These include
nonlinear and adaptive approximation, multivariate splines, subdivision schemes,
multiscale and wavelet methods, numerical schemes for partial differential and
boundaryintegralequations,learningtheory,andhigh-dimensionalintegrals.
CollegeStation,Texas,USA RonaldA.DeVore
Paderborn,Germany AngelaKunoth
June2009
vii
Acknowledgements
WearedeeplygratefultoDr.MartinPetersandThanh-HaLeThifromSpringerfor
realizingthisbookprojectandtoFrankHolzwarthfortechnicalsupport.
ix
Contents
Introduction:WolfgangDahmen’smathematicalwork ................ 1
RonaldA.DeVoreandAngelaKunoth
1 Introduction.............................................. 1
2 Theearlyyears:Classicalapproximationtheory................ 2
3 Bonn,Bielefeld,Berlin,andmultivariatesplines ............... 2
3.1 Computeraidedgeometricdesign.................... 3
3.2 Subdivisionandwavelets ........................... 4
4 Waveletandmultiscalemethodsforoperatorequations.......... 5
4.1 Multilevelpreconditioning.......................... 5
4.2 Compressionofoperators........................... 5
5 Adaptivesolvers .......................................... 6
6 Constructionandimplementation............................ 7
7 Hyperbolicpartialdifferentialequationsandconservationlaws ... 8
8 Engineeringcollaborations ................................. 9
9 Thepresent .............................................. 9
10 Finalremarks............................................. 10
PublicationsbyWolfgangDahmen(asofsummer2009)............... 10
Thewaythingswereinmultivariatesplines:Apersonalview........... 19
CarldeBoor
1 Tensorproductsplineinterpolation........................... 19
2 Quasiinterpolation ........................................ 20
3 MultivariateB-splines ..................................... 21
4 Kergininterpolation ....................................... 23
5 TherecurrenceformultivariateB-splines ..................... 25
6 Polyhedralsplines......................................... 27
7 Boxsplines .............................................. 28
8 SmoothmultivariatepiecewisepolynomialsandtheB-net ....... 31
References..................................................... 34
xi
xii Contents
On the efficientcomputationof high-dimensionalintegralsand the
approximationbyexponentialsums ................................ 39
DietrichBraessandWolfgangHackbusch
1 Introduction.............................................. 39
2 Approximation of completely monotone functions by
exponentialsums ......................................... 41
3 Rationalapproximationofthesquarerootfunction ............. 43
3.1 Heron’salgorithmandGauss’arithmetic-geometricmean 43
3.2 Heron’smethodandbestrationalapproximation........ 44
3.3 Extensionoftheestimate(19) ....................... 48
3.4 Anexplicitformula................................ 49
α
4 Approximationof1/x byexponentialsums .................. 50
4.1 Approximationof1/xonfiniteintervals .............. 50
4.2 Approximationof1/xon[1,∞)...................... 51
4.3 Approximationof1/xα,α>0 ...................... 54
5 Applicationsof1/xapproximations.......................... 55
5.1 Abouttheexponentialsums......................... 55
5.2 Applicationinquantumchemistry.................... 55
5.3 Inversem√atrix .................................... 56
6 Applicationsof1/ xapproximations ........................ 58
6.1 Basicfacts ....................................... 58
6.2 Applicationtoconvolution.......................... 58
6.3 Modificationforwaveletapplications................. 60
6.4 ExpectationvaluesoftheH-atom .................... 60
7 Computationofthebestap√proximation ....................... 62
8 Rationalapproximationof xonsmallintervals ............... 63
9 Thearithmetic-geometricmeanandellipticintegrals............ 65
10 Adirectapproachtotheinfiniteinterval ...................... 67
11 Sincquadraturederivedapproximations ...................... 68
References..................................................... 73
Adaptiveandanisotropicpiecewisepolynomialapproximation ......... 75
AlbertCohenandJean-MarieMirebeau
1 Introduction.............................................. 75
1.1 Piecewisepolynomialapproximation ................. 75
1.2 Fromuniformtoadaptiveapproximation.............. 77
1.3 Outline .......................................... 79
2 Piecewiseconstantone-dimensionalapproximation............. 80
2.1 Uniformpartitions................................. 81
2.2 Adaptivepartitions ................................ 83
2.3 Agreedyrefinementalgorithm ...................... 85
3 Adaptiveandisotropicapproximation ........................ 87
3.1 Localestimates ................................... 88
3.2 Globalestimates .................................. 90
3.3 Anisotropicgreedyrefinementalgorithm ............. 91
Contents xiii
3.4 Thecaseofsmoothfunctions. ....................... 95
4 Anisotropicpiecewiseconstantapproximationonrectangles .....100
4.1 Aheuristicestimate................................100
4.2 Arigourousestimate...............................103
5 Anisotropicpiecewisepolynomialapproximation ..............108
5.1 Theshapefunction ................................108
5.2 Algebraicexpressionsoftheshapefunction ...........109
5.3 Errorestimates....................................111
5.4 Anisotropicsmoothnessandcartoonfunctions .........112
6 Anisotropicgreedyrefinementalgorithms.....................116
6.1 Therefinementalgorithmforpiecewiseconstantson
rectangles ........................................119
6.2 Convergenceofthealgorithm .......................121
6.3 Optimalconvergence ..............................124
6.4 Refinementalgorithmsforpiecewisepolynomialson
triangles .........................................128
References.....................................................134
Anisotropicfunctionspaceswithapplications........................ 137
ShaiDekelandPenchoPetrushev
1 Introduction..............................................137
2 AnisotropicmultiscalestructuresonRn.......................139
2.1 Anisotropicmultilevelellipsoidcovers(dilations)ofRn .139
2.2 Comparison of ellipsoid covers with nested
triangulationsinR2................................142
3 Buildingblocks...........................................143
3.1 Constructionofamultilevelsystemofbases ...........143
3.2 Compactlysupporteddualsandlocalprojectors ........145
3.3 Two-level-splitbases...............................146
3.4 Globaldualsandpolynomialreproducingkernels.......148
3.5 Constructionofanisotropicwaveletframes ...........151
3.6 Discretewaveletframes ............................154
3.7 Two-level-splitframes .............................155
4 AnisotropicBesovspaces(B-spaces).........................156
4.1 B-spacesinducedbyanisotropiccoversofRn ..........156
4.2 B-spacesinducedbynestedmultileveltriangulationsof
R2 ..............................................158
4.3 ComparisonofdifferentB-spacesandBesovspaces ....159
5 Nonlinearapproximation...................................160
6 MeasuringsmoothnessviaanisotropicB-spaces................162
7 Application to preconditioning for elliptic boundary value
problems ................................................164
References.....................................................166
xiv Contents
Nonlinearapproximationanditsapplications........................ 169
RonaldA.DeVore
1 Theearlyyears ...........................................169
2 Smoothnessandinterpolationspaces .........................171
2.1 Theroleofinterpolation............................172
3 Themaintypesofnonlinearapproximation ...................174
3.1 n-termapproximation ..............................174
3.2 Adaptiveapproximation ............................178
3.3 Treeapproximation................................178
3.4 Greedyalgorithms.................................181
4 Imagecompression........................................185
5 RemarksonnonlinearapproximationinPDEsolvers ...........187
6 Learningtheory...........................................189
6.1 Learningwithgreedyalgorithms.....................193
7 Compressedsensing.......................................195
8 Finalthoughts ............................................199
References.....................................................199
Univariatesubdivisionandmulti-scaletransforms:Thenonlinearcase .. 203
NiraDynandPeterOswald
1 Introduction..............................................203
2 Nonlinearmulti-scaletransforms:Functionalsetting............210
2.1 Basicnotationandfurtherexamples ..................210
2.2 Polynomialreproductionandderivedsubdivision
schemes .........................................215
2.3 Convergenceandsmoothness .......................217
2.4 Stability .........................................223
2.5 Approximationorderanddecayofdetails .............227
3 Thegeometricsetting:Casestudies ..........................230
3.1 Geometry-basedsubdivisionschemes.................231
3.2 Geometricmulti-scaletransformsforplanarcurves .....240
References.....................................................245
RapidsolutionofboundaryintegralequationsbywaveletGalerkin
schemes ....................................................... 249
HelmutHarbrechtandReinholdSchneider
1 Introduction..............................................249
2 Problemformulationandpreliminaries .......................252
2.1 Boundaryintegralequations.........................252
2.2 Parametricsurfacerepresentation ....................253
2.3 Kernelproperties..................................255
3 Waveletbasesonmanifolds.................................256
3.1 Waveletsandmultiresolutionanalyses ................256
3.2 Refinementrelationsandstablecompletions ...........258
3.3 Biorthogonalsplinemultiresolutionontheinterval......259
3.4 Waveletsontheunitsquare .........................261