Deflated and Augmented Krylov Subspace Techniques

Chapman, Andrew; Saad, Yousef

doi:10.1002/(sici)1099-1506(199701/02)4:1<43::aid-nla99>3.3.co;2-q

Cited by 71 publications

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“…Deflation is also used in iterative methods for non-symmetric systems of equations [5,9,10,13,24,27,33]. In these papers the smallest eigenvalues have been shifted away from the origin.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

Vuik

Segal

Meijerink

1999

Journal of Computational Physics

129

146

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Knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs. A mathematical model for the prediction of fluid pressures is given by a time-dependent diffusion equation. Application of the finite element method leads to a system of linear equations. A complication is that the underground consists of layers with very large differences in permeability. This implies that the symmetric and positive definite coefficient matrix has a very large condition number. Bad convergence behavior of the CG method has been observed; moreover, a classical termination criterion is not valid in this problem. After diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability. For the IC preconditioner the same behavior is observed. To annihilate the effect of the extreme eigenvalues a deflated CG method is used. The convergence rate improves considerably and the termination criterion becomes again reliable. Finally a cheap approximation of the eigenvectors is proposed.

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“…Deflation is also used in iterative methods for non-symmetric systems of equations [5,9,10,13,24,27,33]. In these papers the smallest eigenvalues have been shifted away from the origin.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

Vuik

Segal

Meijerink

1999

Journal of Computational Physics

129

146

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“…Algorithm 2.1 is used for the numerical examples of Section 4; however, we remark that a black-box implementation of our method has to allow the k-loop (lines [8][9][10][11][12][13][14][15][16][17][18][19] to be exited as soon as the range of [W V p+1:p+k ] contains a vector x, such that the associated residual error Ax − b is sufficiently small. The norm of this residual error can be evaluated inexpensively just as in standard GMRES and RRGMRES.…”

Section: Compute the Solution Y J Of The Least-squares Problemmentioning

confidence: 99%

“…We remark that Chapman and Saad [8] and Morgan [14,15] proposed augmentation of Krylov subspaces generated by restarted GMRES by spaces spanned by certain eigenvectors or Ritz vectors. Morgan [16] recently presented an elegant implementation of the latter approach.…”

Section: Introductionmentioning

confidence: 99%

Augmented GMRES‐type methods

Baglama

Reichel

2007

Numerical Linear Algebra App

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SUMMARYGMRES is a popular iterative method for the solution of large linear systems of equations with a square nonsymmetric matrix. The method generates a Krylov subspace in which an approximate solution is determined. We present modifications of the GMRES and the closely related RRGMRES methods that allow augmentation of the Krylov subspaces generated by these methods by a user-supplied subspace. We choose this subspace to enable the representation of certain known nonsmooth features of the desired solution, such as jumps, or to make it possible to represent certain smooth functions, such as constants or linear functions. The latter choice of augmenting subspace appears to be new. Applications to the solution of both well-posed and ill-posed problems are presented.

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“…Segundo [1], uma motivação para propor o LGM-RES se deu pelo fato de que o GMRES(m) não mantém a ortogonalidade entre espaços aproximados generalizados em sucessivas reinicializações, tornando assim sua convergência vagarosa ou até mesmo levando a divergência. Tambémé sabido, conforme [2], que métodos aumentados formam uma classe de técnicas de aceleração; esses métodos buscam evitar a divergência melhorando assim as informações obtidas com o GMRES em cada reinicialização. Tipicamente, um subespaço A-invarianteé agrupado ao subespaço aproximado de Krylov, resultando assim em um subespaço de Krylov aumentado.…”

Section: Descrição Dos Métodosunclassified

Seleção Dinâmica da Dimensão do Subespaço de Krylov no Método GMRES(m) e suas Variantes

Gonçalez¹,

Cunha²

2006

TEMA - Tendências Em Matemática Aplicada E Computacional

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Resumo. Nesse trabalho apresentamos alguns algoritmos adaptativos do Método do Resíduo Mínimo Generalizado (GMRES) [10], um método iterativo para resolver sistemas de equações lineares com matrizes não simétricas e esparsas, o qual baseiase nos métodos de projeção ortogonal sobre um subespaço de Krylov.O GMRES apresenta uma versão reinicializada, denotada por GMRES(m), também proposta em [10], com o intuito de permitir a utilização do método para resolver sistemas de equações lineares cuja matriz dos coeficientes apresenta uma grande dimensão. No entanto, escolher um valor apropriado, m, para a dimensão da base do subsespaço de Krylové bastante difícil.Dessa forma, nesse trabalho, acrescentamos ao GMRES(m) e algumas de suas variantes um critério que tem por objetivo escolher, ao longo das iterações, um m tal que se obtenha a convergência do método, possivelmente de forma mais rápida.Aproximadamente duas centenas de testes foram realizados utilizando as matrizes da coleção Harwell-Boeing, que foram utilizados para mostrar o comportamento dos algoritmos adaptativos. Foram obtidos resultados muito bons conforme apresentaremos nesse trabalho. IntroduçãoNesse trabalho são estudados aprimoramentos ao método iterativo "Generalized Minimum Residual" (GMRES), proposto por Saad e Schultz em 1986 [10], e algumas de suas variantes, utilizados na solução de sistemas de equações lineares da formaonde A ∈ R n×né não-simétrica e esparsa e x, b ∈ R n . No GMRES original, conforme proposto por Saad e Schultz, o resíduo inicial r 0 = b − Ax 0é utilizado para gerar uma base ortonormal para um subespaço de Krylov, K k (A, r 0 ) = span(r 0 , Ar 0 , . . . , A k−1 r 0 ), (1.2) 1

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Deflated and Augmented Krylov Subspace Techniques

Cited by 71 publications

References 0 publications

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

Augmented GMRES‐type methods

Seleção Dinâmica da Dimensão do Subespaço de Krylov no Método GMRES(m) e suas Variantes

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