解大型线性方程组的轮换重新开始Krylov子空间方法(英文)

解大型线性方程组的轮换重新开始Krylov子空间方法(英文)Introduction:Linear equations are encountered in various scientific an

Krylov 解大型线性方程组的轮换重新开始子空间方 () 法英文 Introduction: Linearequationsareencounteredinvariousscientificand engineeringproblems.Often,thenumberoflinearequationsis verylarge,makingthedirectmethodsforsolvingtheminfeasible. Hence,iterativemethodsareusedforsolvinglargelinearequation systems.Krylovsubspacemethodsareoneofthemoststudied iterativemethodsforsolvinglargelinearequationsystems. Amongthem,therestartedKrylovsubspacemethodshavebeen widelyusedbecauseoftheircomputationalefficiency.The restartedKrylovsubspacemethodsforsolvinglargelinear equationsystemsarebasedontheideaofperiodicallyrestarting theiterativeprocessinawaysuchthatthenewsearchdirectionis chosenfromthesubspacepreviouslygenerated. Inthispaper,wediscussthecyclicrestartedKrylovsubspace methodsforsolvinglargelinearequationsystems.Wewillfocus onthecyclicArnoldiandGMRESmethodsandtheirimproved versionswithcyclicrestarts.ThesecyclicrestartKrylovsubspace methodshaveadvantagesoverconventionalKrylovsubspace methods,astheycanexploitthecyclicstructureofmanylinear equationsystems. CyclicArnoldiMethod: TheArnoldimethodisawell-knownKrylovsubspacemethod thatconstructsanorthonormalbasisfortheKrylovsubspace generatedbyAandagiveninitialvector,v.TheArnoldi decompositioncanbewrittenas: AV_m=V_{m+1}H_m whereV_misthem-dimensionalorthonormalbasismatrixfor theKrylovsubspace,H_misthe(m+1)xmupperHessenberg matrixandAV_misthematrixthatrepresentstheactionofAon V_m.OncetheArnoldidecompositioniscomputed,wecansolve thelinearequationsystembysolvingthereducedsystem: H_my_m=c wherey_misanm-dimensionalvector,whichcanbeusedto