On the Selection of a Good Shape Parameter for RBF Approximation and Its Application for Solving

131 Pages·2017·6.77 MB·English

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Dissertations
Summer 8-1-2015
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Lei-Hsin Kuo
University of Southern Mississippi
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TheUniversityofSouthernMississippi
ONTHESELECTIONOFAGOODSHAPEPARAMETERFORRBF
APPROXIMATIONANDITSAPPLICATIONSFORSOLVINGPDES
by
LEI-HSINKUO
AbstractofaDissertation
SubmittedtotheGraduateSchool
ofTheUniversityofSouthernMississippi
inPartialFulfillmentoftheRequirements
fortheDegreeofDoctorofPhilosophy
August2015
ABSTRACT
ONTHESELECTIONOFAGOODSHAPEPARAMETERFORRBF
APPROXIMATIONANDITSAPPLICATIONSFORSOLVINGPDES
byLEI-HSINKUO
August2015
MeshlessmethodsutilizingRadialBasisFunctions(RBFs)areanumericalmethodthat
requirenomeshconnectionswithinthecomputationaldomain. Theyareusefulforsolving
numerousreal-worldengineeringproblems. Overthepastdecades,afterthe1970s,several
RBFshavebeendevelopedandsuccessfullyappliedtorecoverunknownfunctionsandto
solvePartialDifferentialEquations(PDEs).
However,someRBFs,suchasMultiquadratic(MQ),Gaussian(GA),andMatérnfunc-
tions,containafreevariable,theshapeparameter, c. Becausecexertsastronginfluenceon
theaccuracyofnumericalsolutions,muchefforthasbeendevotedtodevelopingmethodsfor
determining shape parameters which provide accurate results. Most past strategies, which
haveutilizedatrail-and-errorapproachorfocusedonmathematicallyprovenvaluesforc,
remaincumbersomeandimpracticalforreal-worldimplementations.
Thisdissertationpresentsanewmethod,Residue-ErrorCrossValidation(RECV),which
can be used to select good shape parameters for RBFs in both interpolation and PDE
problems. TheRECVmethodmapstheoriginaloptimizationproblemofdefiningashape
parameter into a root-finding problem, thus avoiding the local optimum issue associated
withRBFinterpolationmatrices,whichareinherentlyill-conditioned.
Withminimalcomputationaltime,theRECVmethodprovidesshapeparametervalues
whichyieldhighlyaccurateinterpolations. Additionally,whenconsideringsmallerdatasets,
accuracyandstabilitycanbefurtherincreasedbyusingtheshapeparameterprovidedbythe
RECVmethodastheupperboundofthecintervalconsideredbytheLOOCVmethod. The
RECV method can also be combined with an adaptive method, knot insertion, to achieve
accuracyuptotwoordersofmagnitudehigherthanthatachievedusingHaltonpoints.
iii
COPYRIGHT BY
LEI-HSIN KUO
2015
The University of Southern Mississippi
 ON THE SELECTION OF A GOOD SHAPE PARAMETER FOR RBF 
APPROXIMATION AND ITS APPLICATION FOR SOLVING PDES
by
Leihsin Kuo
A Dissertation
Submitted to the Graduate School
of The University of Southern Mississippi
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
Approved:
______________________________________
Dr. Ching­Shyang Chen, Committee Chair
Professor, Mathematics
______________________________________
Dr. James V. Lambers, Committee Member
Associate Professor, Mathematics
______________________________________
Dr. Haiyan Tian, Committee Member
Associate Professor, Mathematics
______________________________________
Dr. Huiqing Zhu, Committee Member
Assistant Professor, Mathematics
______________________________________
Dr. Karen S. Coats
Dean of the Graduate School
August 2015
ACKNOWLEDGMENTS
Iwouldliketogratefullythankmyadvisor,Dr. C.S.Chenforhissupport,and instruc-
tive guidance during my graduate studies at the University of Southern Mississippi. He
encouragedmetoexploretheresearchwhichIaminterestedin,andadvisedmetocreate
my own original idea for my dissertation. His advise was paramount in providing a well
roundedexperiencealongmylong-termcareergoals. Additionally,Igreatlyappreciatedmy
committeemembersforspendingtimeoneditingthisdissertation.
I take this opportunity to record our sincere thanks to all the faculty members of the
DepartmentofMathematicsfortheirencouragementandhelp.
ManythanksgotoDr. JosephKolibalattheUniversityofNewHaven,whotaughtme
tothinkasamathematicianandledmetounderstandthedifferencebetweenmathematics
andengineering.
IamgratefulforbeinginvitedasavisitingscholaratNationalTaiwanUniversity(NTU)
byDr. D.L.YounginMayandJune2012. BecauseofDr. D.L.Young’sfinancialsupport,
IhadtheopportunitytostudytheHoubolttimemarchingschemefortheWaveequations.
Furthermore,IwouldliketoacknowledgethefinancialsupportgivenbyDr. Y.C.Hon
at City University of Hong Kong (CityU), from June through August 2012. This support
providedmetheopportunitytoexploretheresearchofshapeparameterforRBFs.
I also want to thank Dr. C. H. Tsai, Dr. C. M. Fan, Dr. G. M. Yao, Dr. E. Cenek, Dr.
T. H. Hsu, Dr. F. F. Dou, Corwin A. Stanford, Ryanne McNeese, Alex Cibotarica, Thir
Raj Dangal, BalaramKhatri-Ghimire, Anup Lamichhane, MeganRichardson, Jaeyoun Oh
Roberts, Tulsi Upadhyay, Suanrong Chen,Miao Yu, Vika Veunvi, Koshal Dahal, Daniel
Lanterman,HibaNaccache,WeiZhao,andthemembersofthelabinCityUandNTU.Your
truefriendshipisararetreasureofmylife.
Most importantly, I would like to thank my family. None of this would have been
possiblewithouttheirunconditionalloveandsupport.
v
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LISTOFILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . . . . . viii
LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LISTOFABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction 1
1.2 Outline 3
2 LITERATUREREVIEW-ABACKGROUNDOFRBFINTERPOLATION 5
2.1 DefiningRadialBasisFunctions(RBFs) 5
2.2 RBFinterpolation 6
2.3 RBFInterpolationMatrixInvertibility 10
2.4 TheTrade-OffPrinciple 14
2.5 Leave-One-OutCrossValidation(LOOCV) 17
2.6 LOOCVVersusPreviousMethods: ANumericalComparison 18
3 DEVELOPINGTHERECVMETHODFORSHAPEPARAMETERINRBF
INTERPOLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 ResidueErrorvs. OptimalShapeParameter 23
3.2 RECVvs. LOOCV 27
3.3 ModifiedKnotInsertion(MKI) 34
4 NUMERICALEXPERIMENTSUTILIZINGTHERECVMETHODINRBF
INTERPOLATIONPROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 NumericalComparisonofRECVcandCombinedc 45
4.2 GeneralizabilityofRECVShapeParameters 52
4.3 ModifiedKnotInsertion 56
5 UTILIZING THE RECV METHOD WITH KBC MESHLESS METHODS
TOSOLVETIME-INDEPENDENTPDEPROBLEMS . . . . . . . . . . . . 70
5.1 Kernal-BasedCollocationMethods 70
vi
5.2 SuitabilityoftheKBCMethodUtilizingRECVcandMKIAdaptivePoints 74
6 UTILIZING THE RECV METHOD WITH THE HOUBOLT METHOD IN
TIME-DEPENDENTWAVEEQUATIONS . . . . . . . . . . . . . . . . . . . 93
6.1 TheWaveEquation 93
6.2 TheHouboltMethod 94
6.3 CombiningtheKBCMethodsandtheHalboltMethod 95
6.4 NumericalResults 96
7 CONCLUSIONSANDFUTUREWORK . . . . . . . . . . . . . . . . . . 104
7.1 Conclusions 104
7.2 FutureWork 106
APPENDIX
A DerivativesandParticularSolutionsofMQ . . . . . . . . . . . . . . . . . 108
A.1 DerivativesforKansa’sMethod 108
A.2 ParticularSolutionforMAPS 109
B AdditionalComputerProgram . . . . . . . . . . . . . . . . . . . . . . . 110
B.1 Hardy’sshapeparameter 110
B.2 Franke’sshapeparameter 111
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
vii
LIST OF ILLUSTRATIONS
Figure
2.1 Comparison ofthe accuracy ofMQ RBF approximation andMatlab’sbuilt-in
v4method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Somecommonlyusedradialbasisfunctions,φ(r)inR2. . . . . . . . . . . . . 13
2.3 GARBFwithdifferentshapeparametervalues. . . . . . . . . . . . . . . . . . 13
2.4 StationaryMQRBFinterpolation. Thetotalnumber ofinterpolationpointsare
fixedat100(solidline)and400(dashline). . . . . . . . . . . . . . . . . . . . 15
2.5 Non-stationary MQ interpolation. Shape parameter are fixed at c=1 (solid
line)andc=5(dashline). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Maximum approximation error, E , versus shape parameter, c, for different
m
numbers of points, N, for the MQ approximation described in Example 2.2.1.
Comparingthe cvaluespredictedby theHardy,Franke, andModified Franke
methods (see Table 2.3) to these results illustrates the relative strengths and
weaknessesoftheapproaches. . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 ErrorbehaviorforMQRBFinterpolation. . . . . . . . . . . . . . . . . . . . . 25
3.2 Maximum residue error, r , and maximum approximation error, E , for MQ
m m
approximationssolvedbydifferentprecisionsolvers. . . . . . . . . . . . . . . 26
3.3 Error behavior for an MQ RBF interpolation using N = 100, and N = 400
uniformlydistributedpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 4225Haltonpointsserveasgivendatasetintheunitsquaredomain,[0,1]2. . . 36
3.5 Comparison of the adaptive points generated by Franke’s knot insertion and
ModifiedKnotInsertion(MKI). . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Comparisonofboundaryerrorbehavior. . . . . . . . . . . . . . . . . . . . . . 38
3.7 Maximum approximation error, E , versus number of data points, N, when
m
usingHaltonpoints,RegularGridpoints,andMKIadaptivepoints. . . . . . . 39
3.8 Maximum approximation error, E , versus number of data points, N, for
m
Franke’sKnotInsertionandtheMKImethods. . . . . . . . . . . . . . . . . . . 39
4.1 UniforminterpolationPointsdistributedin[0,1]2. . . . . . . . . . . . . . . . . 43
4.2 ProfilesoftestfunctionsF1–F6. . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 ProfilesoftestfunctionsF7–F8. . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 Shape parameter and error-residue convergence with lower N values for test
functionsF1–F4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Shape parameter and error-residue convergence with lower N values for test
functionsF7–F8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6 Shape parameter and error-residue convergence with lower N values for test
functionsF1–F4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
viii
4.7 Shape parameter and error-residue convergence with lower N values for test
functionsF5–F8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.8 ComparisonofMQRBFinterpolationerrorbehaviorusingvariousN uniformly
distributedpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.9 Left: 10 initial randomly selected points. Right: 10,000 Halton points dis-
tributedintheunitsquaredomain[0,1]2. . . . . . . . . . . . . . . . . . . . . 56
4.10 1,600 adaptive points generated using the MKI method for test functions F1
throughF4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.11 1,600adaptivepointsgeneratedusingtheMKImethodfortestfunctionsF5–F8. 58
4.12 Various numbers of adaptive points, N, generated using the MKI method for
testfunctionF1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.13 DensityofadaptivepointsgeneratedusingtheMKImethod. . . . . . . . . . . 60
4.14 Error-Residue Convergence: MKI adaptive points versus Halton evenly dis-
tributedpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.15 Error-Residue Convergence: MKI adaptive points versus Halton evenly dis-
tributedpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.16 Maximum approximation error versus number of interpolation points using
MKIadaptivepointsorHaltonuniformlydistributedpoints. . . . . . . . . . . 64
4.17 Maximum approximation error versus number of interpolation points using
MKIadaptivepointsorHaltonuniformlydistributedpoints. . . . . . . . . . . 65
4.18 GeneralizabilityofunknownfunctionMKIadaptivepointsetsforotherunknown
fctionswithinthesamedomain. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.19 GeneralizabilityofunknownfunctionMKIadaptivepointsetsforotherunknown
fctionswithinthesamedomain. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.20 MKIadaptivepointsversusHaltonpoints: Convergencerateswheninterpolat-
ingMQs,MQderivatives,andMQintegrations. . . . . . . . . . . . . . . . . . 68
4.21 MKIadaptivepointsversusHaltonpoints: Convergencerateswheninterpolat-
ingMQs,MQderivatives,andMQintegrations. . . . . . . . . . . . . . . . . . 69
5.1 Theanalyticalsolution,u(see(5.20)),totheboundaryvalueproblemdescribed
inExample5.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Regular grid points with equally spaced boundary points, and MKI adaptive
pointsinthecomputationaldomain,Ω=Ω∪Γ,forExample5.2.1 . . . . . . . 76
5.3 ComparisonofboundaryerrorbehaviorforKansa’smethodusingeitherregular
grid interior points with equally spaced boundary points, or MKI adaptive
collocationpointsfortheboundaryvalueproblemdescribedinExample5.2.1. 77
5.4 Comparison of boundary error behavior for the MAPS using either regular
grid interior points with equally spaced boundary points, or MKI adaptive
collocationpoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Suitability of RECV c and MKI adaptive points with Kansa’s method for the
boundaryvalueproblemdescribedinExample5.2.1. . . . . . . . . . . . . . . 79
ix