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解大型线性方程组的轮换重新开始Krylov子空间方法(英文).docx

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解大型线性方程组的轮换重新开始Krylov子空间方法(英文) Introduction: Linearequationsareencounteredinvariousscientificandengineeringproblems.Often,thenumberoflinearequationsisverylarge,makingthedirectmethodsforsolvingtheminfeasible.Hence,iterativemethodsareusedforsolvinglargelinearequationsystems.Krylovsubspacemethodsareoneofthemoststudiediterativemethodsforsolvinglargelinearequationsystems.Amongthem,therestartedKrylovsubspacemethodshavebeenwidelyusedbecauseoftheircomputationalefficiency.TherestartedKrylovsubspacemethodsforsolvinglargelinearequationsystemsarebasedontheideaofperiodicallyrestartingtheiterativeprocessinawaysuchthatthenewsearchdirectionischosenfromthesubspacepreviouslygenerated. Inthispaper,wediscussthecyclicrestartedKrylovsubspacemethodsforsolvinglargelinearequationsystems.WewillfocusonthecyclicArnoldiandGMRESmethodsandtheirimprovedversionswithcyclicrestarts.ThesecyclicrestartKrylovsubspacemethodshaveadvantagesoverconventionalKrylovsubspacemethods,astheycanexploitthecyclicstructureofmanylinearequationsystems. CyclicArnoldiMethod: TheArnoldimethodisawell-knownKrylovsubspacemethodthatconstructsanorthonormalbasisfortheKrylovsubspacegeneratedbyAandagiveninitialvector,v.TheArnoldidecompositioncanbewrittenas: AV_m=V_{m+1}H_m whereV_misthem-dimensionalorthonormalbasismatrixfortheKrylovsubspace,H_misthe(m+1)xmupperHessenbergmatrixandAV_misthematrixthatrepresentstheactionofAonV_m.OncetheArnoldidecompositioniscomputed,wecansolvethelinearequationsystembysolvingthereducedsystem: H_my_m=c wherey_misanm-dimensionalvector,whichcanbeusedtoconstructanapproximatesolutiontotheoriginalequationsystemasx_m=V_my_m ThecyclicArnoldimethodisarestartedversionoftheArnoldimethod,whichrestartstheArnoldiprocesseverymiterations.TherestartedArnoldimethodcanbewrittenas: AV_m=V_{m+1}H_m Then,wecomputeanapproximationtothesolutionfromx_k=V_ky_k,wherey_kisthesolutionofthereducedsystemH_ky_k=c.TheArnoldidecompositionisthenrestartedwiththeresidual,r_k=c-AV_ky_k,astheinitialvector. CyclicGMRESMethod: TheGMRES(General