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

670 Pages·2009·14.321 MB·English

by  Ronald A. DeVore, Angela Kunoth (auth.), Ronald DeVore, Angela Kunoth (eds.)

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