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design and implementation of a scalable parallel direct solver for sparse symmetric positiv

design and implementation of a scalable parallel direct solver for sparse symmetric positiv

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时间：2019-02-28

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1、DesignandImplementationofaScalableParallelDirectSolverforSparseSymmetricPositiveDe niteSystems:PreliminaryResultsyyzzAnshulGuptaFredGustavsonMaheshJoshiGeorgeKarypiszVipinKumarAbstractSolvinglargesparsesystemsoflinearequationsisatthecoreofmanyproblem

2、sinengineeringandscienti ccomputing.Ithaslongbeenachallengetodevelopparallelformulationsofsparsedirectsolversduetoseveraldi erentcomplexstepsinvolvedintheprocess.Inthispaper,wedescribeoneofthe rstecient,practical,androbustparallelsolversforsparsesymm

3、etricpositivede nitelinearsystemsthatwehavedevelopedanddiscussthealgorithmicandimplementationissuesinvolvedinitsdevelopment.1IntroductionSolvinglargesparsesystemsoflinearequationsisattheheartofmanyengineeringandscienti ccomputingapplications.Therearet

4、womethodstosolvethesesystems-directanditerative.Directmethodsarepreferredformanyapplicationsbecauseofvariouspropertiesofthemethodandthenatureoftheapplication.Awideclassofsparselinearsystemsarisinginpracticehaveasymmetricpositivede nite(SPD)coecientma

5、trix.TheproblemistocomputethesolutiontothesystemAx=b;whereAisasparseandSPDmatrix.SuchasystemiscommonlysolvedusingCholeskyfactorization.Adirectmethodofsolutionconsistsoffourconsecutivephasesviz.ordering,symbolicfactorization,numericalfactorizationandso

6、lutionoftriangularsystems.Duringtheorderingphase,TapermutationmatrixPiscomputedsothatthematrixPAPwillincuraminimal llduringthefactorizationphase.Duringthesymbolicfactorizationphase,thenon-zerostructureofthetriangularCholeskyfactorLisdetermined.Thesymb

7、olicfactorizationphaseexistsinordertoincreasetheperformanceofthenumericalfactorizationphase.ThenecessaryoperationstocomputethevaluesTTinLthatsatisfyPAP=LL,areperformedduringthephaseofnumericalfactorization.Finally,0thesolutiontoAx=biscomputedbysolving

8、twotriangularsystemsviz.Ly=bfollowedbyT000Lx=y,whereb=Pbandx=Px.Solvingtheformersystemiscalledforwardeliminationandthelatterprocessofsolutioniscalledbackwardsubstitution.The nalsolution,x,isobtainedT0usingx=Px.Inthispaper,wedescribeoneofthe rs

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