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1、Numer.Math.(1997)78:305–328NumerischeMathematikcSpringer-Verlag1997ElectronicEditionAdaptivefilteringChristianWagner,GabrielWittumInstitutfurComputeranwendungen,Universit¨atStuttgart,Pfaffenwaldring27,D-70569Stuttgart,¨Germany,e-mail:chris@ica3.uni-stuttgart.deReceivedJune26,1996/Revisedversio
2、nreceivedOctober7,1996Summary.Inthispaper,theadaptivefilteringmethodisintroducedandana-lysed.Thismethodleadstorobustalgorithmsforthesolutionofsystemsoflinearequationswhicharisefromthediscretisationofpartialdifferentialequa-tionswithstronglyvaryingcoefficients.Theseiterativealgorithmsarebasedont
3、hetangentialfrequencyfilteringdecompositions(TFFD).Duringtheiterationwithapreliminarypreconditioner,theadaptivetestvectormethodcalculatesnewtestvectorsfortheTFFD.Theadaptivetestvectoriterativemethodallowsthecombinationofthetangentialfrequencydecompositionandotheriterativemethodssuchasmulti-gri
4、d.TheconnectionwiththeTFFDimprovesthero-bustnessoftheseiterativemethodswithrespecttovaryingcoefficients.Interfaceproblemsaswellasproblemswithstochasticallydistributedpropertiesarecon-sidered.Realisticnumericalexperimentsconfirmtheefficiencyofthepresentedalgorithms.MathematicsSubjectClassification
5、(1991):65F10,65N22,65N551.IntroductionIterativesolversforlargesystemsoflinearequationsshowatypicalslowdownafterafewfaststepsinthebeginning.Thisreflectsthattheiterationbehavesdifferentlyondifferenterrorcomponents.Duringthefirststepsmanycomponentsaredampedsubstantially,thereafterahardcoreof”bad”e
6、igenmodesremains,whichisdampedveryslowly.Thisobservationiswellknownandgaverisetothedevelopmentofconjugategradientmethods,seee.g.thestillveryinterestingarticle[11],andmulti-gridmethodsaswell,see[1],[3],[4],[5],[6],[7].Forlinearsystemsarisingfromellipticpartialdifferentialequationsandcomparable
7、ones,multi-gridmethodsimprovedthissubstantiallackbyaddingNumerischeMathematikElectronicEditionpage305ofNumer.Math.(1997)78:305–328306C.Wagner,G.Wittumacoarse-gridcorrection,whichtreatsthesmootherrorcomponents.Fornon-smoothproblems,however,the
