Analysis of acceleration strategies for restarted minimal residual methods

Eiermann, Michael; Ernst, Oliver G.; Schneider, O.

doi:10.1016/s0377-0427(00)00398-8

Cited by 76 publications

(82 citation statements)

References 21 publications

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“…Nevertheless to retain the convergence rate a number of techniques have been proposed; they fall in the class of augmented and deflated methods; see e.g. [4,10,11,16,24]. Deflated methods compute spectral information at a restart and use this information to improve the convergence of the subspace method.…”

Section: Introductionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

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Section: Introductionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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show abstract

“…Em particular, através da busca de aproximações de autovalores facilmente calculáveis usando a matriz de Hessenberg do ciclo anterior ( [83], [84], [85]). Outro proposta, apresentada na seção 5.2, é a preservação de subespaços do ciclo anterior mais relevantes para a garantia da ortogonalidade do método ( [42], [45], [113] [99], [111], [112], [114], [132]). Um avanço mais recente que será apresentado na seção 5.4, está relacionado a se permitir que o produto matriz-vetor seja calculado de forma inexata, necessidade sentida em áreas aonde a matriz não é disponível ou é muito cara para ser calculada exatamente ( [19], [20], [52], [112], [114], [129]).…”

Section: Novos Desenvolvimentos Dos Métodos De Krylovunclassified

“…Para outros métodos de Krylov, para solução de sistemas lineares, também se pode utilizar técnica equivalente, ver [35,104]. Essas alternativas também são conhecidas como métodos de aumento por deflação [45].…”

Section: Novos Desenvolvimentos Dos Métodos De Krylovunclassified

Avanços em Métodos de Krylov para Solução de Sistemas Lineares de Grande Porte

2009

Notas Em Matemática Aplicada

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“…Two such recently introduced methods are known as GCROT and GCRO-DR [8,11,31]. While GCROT selects subspaces based on canonical angles between the search space and its image under the system matrix, GCRO-DR employs subspaces spanned by harmonic Ritz vectors of the system matrix with respect to the search space, i.e., approximate eigenvectors.…”

Section: 2mentioning

confidence: 99%