Matrix Preconditioning Techniques and Applications

601 Pages·2005·4.558 MB·English

by  Ke Chen

CAMBRIDGEMONOGRAPHSON
APPLIEDANDCOMPUTATIONAL
MATHEMATICS
SeriesEditors
M.J.ABLOWITZ,S.H.DAVIS,E.J.HINCH,A.ISERLES,
J.OCKENDON,P.J.OLVER
19 Matrix Preconditioning Techniques
and Applications
The Cambridge Monographs on Applied and Computational Mathematics
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Alsointhisseries:
1. APracticalGuidetoPseudospectralMethods,BengtFornberg
2. Dynamical Systems and Numerical Analysis, A. M. Stuart and A. R.
Humphries
3. LevelSetMethodsandFastMarchingMethods,J.A.Sethian
4. TheNumericalSolutionofIntegralEquationsoftheSecondKind,Kendall
E.Atkinson
5. OrthogonalRationalFunctions,AdhemarBultheel,PabloGonza´lez-Vera,
ErikHendiksen,andOlavNja˚stad
6. TheTheoryofComposites,GraemeW.Milton
7. GeometryandTopologyforMeshGenerationHerbertEdelsbrunner
8. Schwarz–ChristoffelMappingTofinA.DriscollandLloydN.Trefethen
9. High-Order Methods for Incompressible Fluid Flow, M. O. Deville, P. F.
FischerandE.H.Mund
10. PracticalExtrapolationMethods,AvramSidi
11. Generalized Riemann Problems in Computational Fluid Dynamics,
MataniaBen-ArtziandJosephFalcovitz
12. RadialBasisFunctions:TheoryandImplementations,MartinBuhmann
13. IterativeKrylovMethodsforLargeLinearSystems,HenkA.vanderVorst
14. Simulating Hamiltonian Dynamics, Benedict Leimkuhler and Sebastian
Reich
15. Collocation Methods for Volterra Integral and Related Functional
Equations,HermannBrunner
16. TopologyforComputing,AfraJ.Zomorodian
17. ScatteredDataApproximation,HolgerWendland
Matrix Preconditioning Techniques
and Applications
KE CHEN
ReaderinMathematics
DepartmentofMathematicalSciences
TheUniversityofLiverpool
  
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Dedicatedto
ZhuangandLeoLingYi
andthelovingmemoriesofmylateparentsWan-QingandWen-Fang
Indecidingwhattoinvestigate,howtoformulateideasandwhatproblems
tofocuson,theindividualmathematicianhastobeguidedultimatelyby
theirownsenseofvalues.Therearenoclearrules,orratherifyouonly
followoldrulesyoudonotcreateanythingworthwhile.
SirMichaelAtiyah(FRS,FieldsMedallist1966).What’sitall
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Contents
Preface pagexiii
Nomenclature xxi
1 Introduction 1
1.1 Directanditerativesolvers,typesofpreconditioning 2
1.2 Normsandconditionnumber 4
1.3 Perturbationtheoriesforlinearsystemsandeigenvalues 9
1.4 TheArnoldiiterationsanddecomposition 11
1.5 Clusteringcharacterization,fieldofvaluesand
(cid:1)-pseudospectrum 16
1.6 FastFouriertransformsandfastwavelettransforms 19
1.7 Numericalsolutiontechniquesforpracticalequations 41
1.8 Commontheoriesonpreconditionedsystems 61
1.9 GuidetosoftwaredevelopmentandthesuppliedMfiles 62
2 Directmethods 66
2.1 TheLUdecompositionandvariants 68
2.2 TheNewton–Schulz–Hotellingmethod 75
2.3 TheGauss–Jordandecompositionandvariants 76
2.4 TheQRdecomposition 82
2.5 Specialmatricesandtheirdirectinversion 85
2.6 Orderingalgorithmsforbettersparsity 100
2.7 DiscussionofsoftwareandthesuppliedMfiles 106
3 Iterativemethods 110
3.1 Solutioncomplexityandexpectations 111
3.2 Introductiontoresidualcorrection 112
3.3 Classicaliterativemethods 113
3.4 Theconjugategradientmethod:theSPDcase 119
vii
viii Contents
3.5 Theconjugategradientnormalmethod:theunsymmetriccase 130
3.6 Thegeneralizedminimalresidualmethod:GMRES 133
3.7 TheGMRESalgorithmincomplexarithmetic 141
3.8 Matrixfreeiterativesolvers:thefastmultipolemethods 144
3.9 DiscussionofsoftwareandthesuppliedMfiles 162
4 Matrixsplittingpreconditioners[T1]:directapproximation
of An×n 165
4.1 Bandedpreconditioner 166
4.2 Bandedarrowpreconditioner 167
4.3 BlockarrowpreconditionerfromDDMordering 168
4.4 Triangularpreconditioners 171
4.5 ILUpreconditioners 172
4.6 Fastcirculantpreconditioners 176
4.7 Singularoperatorsplittingpreconditioners 182
4.8 Preconditioningthefastmultipolemethod 185
4.9 Numericalexperiments 186
4.10 DiscussionofsoftwareandthesuppliedMfiles 187
5 Approximateinversepreconditioners[T2]:direct
approximationof A−1 191
n×n
5.1 Howtocharacterize A−1intermsof A 192
5.2 Bandedpreconditioner 195
5.3 Polynomialpreconditioner p (A) 195
k
5.4 Generalandadaptivesparseapproximateinverses 199
5.5 AINVtypepreconditioner 211
5.6 Multi-stagepreconditioners 213
5.7 Thedualtoleranceself-preconditioningmethod 224
5.8 Nearneighboursplittingforsingularintegralequations 227
5.9 Numericalexperiments 237
5.10 DiscussionofsoftwareandthesuppliedMfiles 238
6 Multilevelmethodsandpreconditioners[T3]:coarsegrid
approximation 240
6.1 MultigridmethodforlinearPDEs 241
6.2 MultigridmethodfornonlinearPDEs 259
6.3 Multigridmethodforlinearintegralequations 263
6.4 Algebraicmultigridmethods 270
6.5 Multileveldomaindecompositionpreconditionersfor
GMRES 279
6.6 DiscussionofsoftwareandthesuppliedMfiles 286
Contents ix
7 MultilevelrecursiveSchurcomplements
preconditioners[T4] 289
7.1 Multilevelfunctionalpartition:AMLIapproximatedSchur 290
7.2 Multilevelgeometricalpartition:exactSchur 295
7.3 Multilevelalgebraicpartition:permutation-basedSchur 300
7.4 Appendix:theFEMhierarchicalbasis 305
7.5 DiscussionofsoftwareandthesuppliedMfiles 309
8 Sparsewaveletpreconditioners[T5]:approximation
of A˜n×n and A˜−n×1n 310
8.1 Introductiontomultiresolutionandorthogonalwavelets 311
8.2 Operatorcompressionbywaveletsandsparsitypatterns 320
8.3 BandWSPAIpreconditioner 323
8.4 NewcenteringWSPAIpreconditioner 325
8.5 Optimalimplementationsandwaveletquadratures 335
8.6 Numericalresults 336
8.7 DiscussionofsoftwareandthesuppliedMfiles 338
9 WaveletSchurpreconditioners[T6] 340
9.1 Introduction 341
9.2 Waveletstelescopicsplittingofanoperator 342
9.3 AnexactSchurpreconditionerwithlevel-by-levelwavelets 346
9.4 Anapproximatepreconditionerwithlevel-by-levelwavelets 352
9.5 Someanalysisandnumericalexperiments 357
9.6 DiscussionoftheaccompaniedMfiles 363
10 Implicitwaveletpreconditioners[T7] 364
10.1 Introduction 365
10.2 Wavelet-basedsparseapproximateinverse 368
10.3 Animplicitwaveletsparseapproximateinverse
preconditioner 369
10.4 Implementationdetails 371
10.5 Denseproblems 374
10.6 Sometheoreticalresults 376
10.7 Combinationwithalevel-onepreconditioner 379
10.8 Numericalresults 380
10.9 DiscussionofthesuppliedMfile 381
11 ApplicationI:acousticscatteringmodelling 383
11.1 TheboundaryintegralequationsfortheHelmholtzequationin
R3anditerativesolution 384
11.2 ThelowwavenumbercaseofaHelmholtzequation 397