Spline functions - computations methods.

426 Pages·2015·30.403 MB·English

by  Schumaker, Larry L.

Spline Func t ions
Spline Func t ions
COMPUTATIONAL METHODS
Larry L. Schumaker
Vanderbilt University
Nashville, Tennessee
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2015 by the Society for Industrial and Applied Mathematics
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Schumaker, Larry L., 1939-
Spline functions : computational methods / Larry L. Schumaker, Vanderbilt University, Nashville,
Tennessee.
pages cm. -- (Other titles in applied mathematics ; 142)
Includes bibliographical references and index.
ISBN 978-1-611973-89-1
1. Spline theory. 2. Numerical calculations. I. Title.
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is a registered trademark.

Contents
Preface ix
1 UnivariateSplines 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 B-splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 ComputingwiththeB-form . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Interpolationwithunivariatesplines . . . . . . . . . . . . . . . . . . 8
1.5 Hermiteinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Interpolationwithshapecontrol . . . . . . . . . . . . . . . . . . . . 16
1.7 Quasi-interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Discreteleast-squaresapproximation . . . . . . . . . . . . . . . . . . 22
1.9 Theeffectofnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.10 Penalizedleast-squaresapproximation . . . . . . . . . . . . . . . . . 26
1.11 Choosingoptimalknotsforleast-squaressplines . . . . . . . . . . . . 29
1.12 Two-pointboundary-valueproblems . . . . . . . . . . . . . . . . . . 32
1.13 Errorbounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.14 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.15 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2 Tensor-ProductSplines 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Tensor-productB-splines . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3 ComputingwiththetensorB-form . . . . . . . . . . . . . . . . . . . 52
2.4 Lagrangeinterpolationonagrid . . . . . . . . . . . . . . . . . . . . 57
2.5 Hermiteinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.6 Tensor-productquasi-interpolation . . . . . . . . . . . . . . . . . . . 64
2.7 Tensor-productdiscreteleast-squares . . . . . . . . . . . . . . . . . . 66
2.8 Theeffectofnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.9 Penalizedleast-squaresapproximation . . . . . . . . . . . . . . . . . 69
2.10 Errorbounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.11 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.12 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 ComputingwithTriangulations 79
3.1 Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
vi Contents
p
h
p
a. 3.2 Triangulationswithhangingvertices . . . . . . . . . . . . . . . . . . 79
s
oj 3.3 Storingandplottingtriangulations . . . . . . . . . . . . . . . . . . . 80
s/
al 3.4 Additionaldatastructurefortriangulations . . . . . . . . . . . . . . . 81
n
ur 3.5 Starsofverticesandtriangles . . . . . . . . . . . . . . . . . . . . . . 84
o
g/j 3.6 Barycentriccoordinates . . . . . . . . . . . . . . . . . . . . . . . . . 84
r 3.7 Findingthetrianglecontainingapoint . . . . . . . . . . . . . . . . . 85
o
m. 3.8 Constructingatriangulationwithgivenvertices . . . . . . . . . . . . 86
a
si 3.9 Gridgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
w.
3.10 Triangulationsofgriddeddataonarectangle . . . . . . . . . . . . . 89
w
w 3.11 Refiningatriangulation . . . . . . . . . . . . . . . . . . . . . . . . . 90
p:// 3.12 Adjustingtriangulationsbyedgeswapping . . . . . . . . . . . . . . . 91
htt 3.13 Eulerrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
ee 3.14 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
s
ht; 3.15 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
g
yri 4 ComputingwithSplines 97
p
o 4.1 Bernsteinbasispolynomials . . . . . . . . . . . . . . . . . . . . . . 97
c
r 4.2 Domainpointsonatriangle . . . . . . . . . . . . . . . . . . . . . . . 98
o
e 4.3 EvaluatingpolynomialsinB-form . . . . . . . . . . . . . . . . . . . 99
s
n
e 4.4 DerivativesofpolynomialsinB-form . . . . . . . . . . . . . . . . . 100
c
M li 4.5 Integralsandinnerproductsofpolynomials . . . . . . . . . . . . . . 101
4.6 Integralsofgeneralfunctionsovertriangles . . . . . . . . . . . . . . 102
A
SI 4.7 Plottingapolynomialpatch . . . . . . . . . . . . . . . . . . . . . . . 104
o 4.8 Domainpointsonatriangulation . . . . . . . . . . . . . . . . . . . . 105
ct t 4.9 ThespaceS0((cid:2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
e d
bj 4.10 Evaluatingsplinesandtheirderivatives . . . . . . . . . . . . . . . . . 109
u
n s 4.11 Integralsandinnerproductsofsplines . . . . . . . . . . . . . . . . . 110
o 4.12 Renderingasplinesurface . . . . . . . . . . . . . . . . . . . . . . . 110
buti 4.13 Spacesofsmoothsplines . . . . . . . . . . . . . . . . . . . . . . . . 113
stri 4.14 Someusefulmacro-elementspaces . . . . . . . . . . . . . . . . . . . 120
di 4.15 Degreeraising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
e
R 4.16 Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
05. 4.17 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
2
0. 4.18 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4
4.
5 5 Macro-elementInterpolationMethods 135
1
2. 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8
o 5.2 TheC1 Powell–Sabininterpolant . . . . . . . . . . . . . . . . . . . . 137
9 t 5.3 TheC1 Powell–Sabin-12interpolant . . . . . . . . . . . . . . . . . . 141
1
7/ 5.4 TheC1 Clough–Tocherinterpolant . . . . . . . . . . . . . . . . . . . 143
0
2/ 5.5 TheC1 quinticArgyrisinterpolant . . . . . . . . . . . . . . . . . . . 146
1
d 5.6 TheC2 Wanginterpolant . . . . . . . . . . . . . . . . . . . . . . . . 149
e
d 5.7 Acomparisonofthemethods . . . . . . . . . . . . . . . . . . . . . . 150
a
nlo 5.8 Rateofconvergenceforlesssmoothfunctions . . . . . . . . . . . . . 151
w 5.9 Interpolationondomainswithholes . . . . . . . . . . . . . . . . . . 153
o
D 5.10 Interpolationatbadlyspacedpoints . . . . . . . . . . . . . . . . . . 154
5.11 Scaleinvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Contents vii
p
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a. 5.12 Theroleoftheminimalangleinerrorbounds . . . . . . . . . . . . . 157
s
oj 5.13 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
s/
al 5.14 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
n
r
u
o
g/j 6 ScatteredDataInterpolation 161
or 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
m. 6.2 InterpolationwithS0((cid:2)) . . . . . . . . . . . . . . . . . . . . . . . . 162
a 1
si 6.3 Minimalenergyinterpolatingsplines . . . . . . . . . . . . . . . . . . 166
ww. 6.4 AlocalC0 cubicsplinemethod . . . . . . . . . . . . . . . . . . . . . 173
w 6.5 AlocalC1 Clough–Tochermethod . . . . . . . . . . . . . . . . . . . 175
p:// 6.6 Estimatingderivativesfromscattereddata . . . . . . . . . . . . . . . 178
htt 6.7 Localtwo-stagemethodsbasedonderivativeestimation . . . . . . . . 185
e
e 6.8 Acomparisonofthemethods . . . . . . . . . . . . . . . . . . . . . . 190
s
ht; 6.9 Scaleinvariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
g 6.10 Nonrectangulardomainsandholes . . . . . . . . . . . . . . . . . . . 195
ri
py 6.11 Interpolatingatalargenumberofdatapoints . . . . . . . . . . . . . 198
o
c 6.12 Theerrorasafunctionofmeshsize . . . . . . . . . . . . . . . . . . 202
r
o 6.13 Non-Delaunaytriangulations . . . . . . . . . . . . . . . . . . . . . . 203
e
ns 6.14 Scattereddatainterpolationwithradialbasisfunctions . . . . . . . . 204
e
c 6.15 Almostinterpolationwithtensor-productsplines . . . . . . . . . . . . 208
M li 6.16 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A 6.17 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
I
S
o
ct t 7 ScatteredDataFitting 219
e 7.1 Least-squaresfitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
bj
u 7.2 Least-squaresfittingwithC0 splines . . . . . . . . . . . . . . . . . . 223
s
n 7.3 Least-squaresfittingwithC1 splines . . . . . . . . . . . . . . . . . . 226
o
uti 7.4 Least-squaresfittingwithC2 splines . . . . . . . . . . . . . . . . . . 231
b
ri 7.5 Scaleinvarianceofleast-squaresfitting . . . . . . . . . . . . . . . . . 233
dist 7.6 Acomparisonofthemethods . . . . . . . . . . . . . . . . . . . . . . 234
Re 7.7 Theeffectofnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
5. 7.8 Penalizedleast-squaresfitting . . . . . . . . . . . . . . . . . . . . . . 236
0
2 7.9 Penalizedleast-squaresfittingwithmacro-elementsplinespaces . . . 239
0.
4 7.10 Nonscaleinvarianceofpenalizedleast-squaresfitting . . . . . . . . . 243
4.
5 7.11 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
1
2. 7.12 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
8
o
9 t 8 ShapeControl 247
1
7/ 8.1 Nonnegativeinterpolation . . . . . . . . . . . . . . . . . . . . . . . . 247
0
2/ 8.2 Nonnegativeleast-squares . . . . . . . . . . . . . . . . . . . . . . . 251
1
d 8.3 Monotoneinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . 254
e
d 8.4 Monotoneleast-squares . . . . . . . . . . . . . . . . . . . . . . . . . 268
a
o 8.5 Convexinterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 270
nl
w 8.6 Convexleast-squares . . . . . . . . . . . . . . . . . . . . . . . . . . 283
o
D 8.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.8 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
viii Contents
p
h
p
a. 9 Boundary-ValueProblems 289
s
oj 9.1 Examplesofboundary-valueproblems . . . . . . . . . . . . . . . . . 289
s/
al 9.2 TheRitz–Galerkinmethod . . . . . . . . . . . . . . . . . . . . . . . 290
n
our 9.3 Solvingsecondorderboundary-valueproblemswithSd0((cid:2)) . . . . . . 294
g/j 9.4 Solvingsecondorderboundary-valueproblemsusingmacro-element
r spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
o
m. 9.5 Solvingsecondorderboundary-valueproblemsusingthespace
a
si S1,2((cid:2)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
w. 5
w 9.6 SolvingthebiharmonicequationwithC1 quinticsplines . . . . . . . 316
w
9.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
p:// 9.8 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
htt
ee 10 SphericalSplines 325
s
ht; 10.1 Sphericaltriangulations . . . . . . . . . . . . . . . . . . . . . . . . . 325
g 10.2 Sphericalpolynomials . . . . . . . . . . . . . . . . . . . . . . . . . 333
yri 10.3 ThespaceS0((cid:2))onasphericaltriangulation . . . . . . . . . . . . . 340
p d
co 10.4 Spacesofsmoothsphericalsplines . . . . . . . . . . . . . . . . . . . 344
or 10.5 Sphericalmacro-elementspaces . . . . . . . . . . . . . . . . . . . . 348
se 10.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
n
e 10.7 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
c
M li
11 ApplicationsofSphericalSplines 357
A
o SI 1111..21 HInetermrpiotelaitniotenrpwoiltahtiSo10n(w(cid:2)i)thm. .ac.ro.-e.le.m.en.t.sp.a.ce.s. .. .. .. .. .. .. .. .. .. .. .. 336517
ct t 11.3 Scattereddatainterpolationonthesphere . . . . . . . . . . . . . . . 369
e
bj 11.4 Least-squaresfittingwithsphericalsplines . . . . . . . . . . . . . . . 384
u
n s 11.5 Theeffectofnoise . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
o 11.6 Sphericalpenalizedleast-squares . . . . . . . . . . . . . . . . . . . . 393
buti 11.7 PDE’sonthesphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
stri 11.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
di 11.9 Historicalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
e
R
5. Bibliography 403
0
2
0.
4 ScriptIndex 407
4.
5
2.1 FunctionIndex 409
8
o
9 t SubjectIndex 411
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m. Preface
a
si
w.
w
w
p://
htt Spline functions have been studied intensively for over 50 years, and there are now
e manythousandsofpapersonthesubject. Thereasonforthisgreatinterestisthatsplines
e
ht; s haraevaesmwahneyredietsisirraebqleuifreeadtutoreasptphraotxmimaakteetahnemunkidneoawlntofoulnscftoiornacwomidpeuvtaartiioetnyalolyf.aTphpulisc,atthioeyn
g
yri havefoundapplicationsinApproximationTheory,NumericalAnalysis,Engineering,and
p
o a variety of other business, medical, and scientific areas. What makes splines especially
c
r attractiveisthefactthat
o
e
ns • they are easy to work with computationally, and in particular, there are stable and
e
c efficientalgorithmsforevaluatingthem,theirderivatives,andtheirintegrals;
M li
• thereareveryconvenientrepresentationswhichprovideastrongconnectionbetween
A
SI theshapeofasplineanditsassociatedcoefficients;
o
ct t • splinesarecapableofapproximatingsmoothfunctionswell,andtheexactrelation-
bje ship between the smoothness of a function and its order of approximation can be
su quantified.
n
o
uti Thetheoreticalaspectsofsplineshavebeenwelldocumentedintheliterature;seethe
b list of booksat the end of this book and the online referencelist. Some of them discuss
ri
st applications,andafewevendescribecomputationalmethods. Thepurposeofthisbookis
edi togiveadetailedtreatmentofsomeofthemostimportantcomputationalaspectsofsplines.
R
5. Itcanbeconsideredasacomplementtomytwoearlierbooksonsplines,neitherofwhich
0 dealtwithapplicationsorcomputations.Thefirst,SplineFunctions:BasicTheory,wasfirst
2
0. publishedin1978byWileyInterscience,anddealsonlywithunivariateandtensor-product
4
4. splines. Athirdeditionwithsupplementarymaterialandreferenceswaspublishedin2007
5
1 by Cambridge University Press. The second book, SplineFunctionsonTriangulations,
2.
8 waswrittenwithM.-J.Lai, anddealswithbivariatesplinesdefinedontriangulationsand
19 to qteutraadhreadnrgaullpatairotnitsio,snpshienriRca3.lsItpwlinaessadlseofinpeudbloisnhtehdeinsp2h0e0re7,bayndCtarmivbarriiadtgeesUplniniveesrsdietyfinPerdesosn.
07/ Splinescanbeusedtosolveseveralofthemostbasicproblemsinnumericalanalysis.
2/ Thetopicsdiscussedhereinclude
1
d
de • interpolationinoneorseveralvariables,
a
o
nl • datafittinginoneorseveralvariables,
w
o
D • numericalquadrature, i.e., the computation of derivativesand integrals based only
ondiscretedata,
ix
x Preface
p
h
p
a. • the numerical solution of problems involving ordinary or partial differential equa-
s
oj tions.
s/
al
n Thegoalof thisbookis to notonlydescribe a varietyof methodsforattackingthese
r
u
o problemsusingsplines, buttoalsoexplainwhythemethodswork,andtogiveactualnu-
g/j mericalexamples.Theaimistodescribeeachmethodinenoughdetailtoallowthereader
r
o
m. to programithimself. However,to enhancethe learningprocess, andto encouragemore
a experimentationwithsplines,IhavepreparedaMATLAB(cid:2) packageSplinePakwhichin-
si
w. cludesimplementationsofallofthemethodsdiscussedinthebook. Thepackageincludes
w anextensivecollectionofp-filescontainingMATLABfunctionsforcarryingoutthevar-
w
p:// ious algorithmsin the book, a set of scripts providedas m-files for carrying out specific
htt tasks, andfinallya numberofdatafilesusedinthenumerousexamplesinthebook. The
e packageislicensedbyNashboroPress;seewww.splinepak.com.Itshouldbeempha-
e
s sized that the code in the package is not production code, and has not been designed to
ht; achievemaximalefficiency. Indeed,forlargescalecomputation,theusershouldconsider
g
ri codingthealgorithmsinanotherlanguage.
y
p Thebookisprimarilyaboutbivariatepolynomialsplinesontriangulations,butforthe
o
r c sake of completeness, and to provide numerical tools for comparison purposes, I have
o
e included chapters on univariate splines, tensor-product splines, and splines on spherical
s
n triangulations.Thematerialisorganizedasfollows.
e
c
M li Chapter1dealswithunivariatesplines.First,wediscussalgorithmsforevaluatingsplines
andtheirderivatives,forcomputingintegralsofsplines,andforplottingsplines. Thenwe
A
I discussseveralmethodsforsolvingthebasicLagrangeandHermiteinterpolationproblems
S
o inonevariable.Inaddition,quasi-interpolation,least-squares,penalizedleast-squares,and
ct t theuseofsplinesforsolvingtwo-pointboundary-valueproblemsaretreated.
e
bj Chapter2isdevotedtotheuseoftensor-productsplinestosolveinterpolationandapprox-
u
n s imationproblemsinthebivariatesetting. First,itisshownhowunivariatealgorithmscan
utio beusedtoevaluatetensor-productsplines, andtocomputetheirderivativesandintegrals.
b Thenwestudyinterpolation,quasi-interpolation,least-squares,andpenalizedleast-squares
stri methodsbasedontensor-productsplines.
di
e Chapter3explainshowtodealwithtriangulationscomputationally,andisacriticalprepa-
R
5. rationforthelaterchapters.Hereweexplainhowtoconstruct,store,andmanipulatetrian-
0 gulations.Refinementandedgeswappingalgorithmsarealsoincluded.
2
40. Chapter4describesindetailhowtostore,evaluate,andrendersplinesontriangulations.
4. Thealgorithmsdescribedhereare essentialtoolsfortheremainderofthe book. Thekey
5
1
2. underlyingconceptistheBernstein–Bézierrepresentationofsplines. Hereweexplainhow
o 8 tocomputewiththespaceSd0((cid:2))ofC0 splines,andwitharbitrarysubspacesofsmoother
9 t splines. Severalspecialmacro-elementspacestobeusedextensivelylaterarediscussedin
7/1 detail.
0
2/ Chapter 5 is the first chapterwhere real applicationsare addressed. Here we show how
1
d certainmacro-elementspacescanbeusedtosolveinterpolationproblemsinthebivariate
de setting. The focus is on Hermite interpolation at the vertices of a triangulation. I give
a
o numericalexamplesforeachofthemethods,andincludetheresultsofsomeexperiments
nl
w comparingthevariousinterpolationmethods.
o
D Chapter 6 treats the problemof scattered data fitting where no derivativeinformationis
available. Both local and global methods are discussed, beginning with minimal energy