Restarted GMRES preconditioned by deflation

Erhel, Jocelyne; Burrage, Kevin; Pohl, B.A.

doi:10.1016/0377-0427(95)00047-x

Cited by 181 publications

(163 citation statements)

References 15 publications

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“…Relations (9), (10), (12), (13) and (14) have been shown in [15,Proposition 2]. From relations (13) and (6) respectively, we deduce…”

Section: Flexible Gmres With Deflated Restartingmentioning

confidence: 74%

“…This class of methods is required when preconditioning with a different (possibly nonlinear) operator at each iteration of a subspace method is considered. This notably occurs when adaptive preconditioners using information obtained from previous iterations [3,14] are used or when inexact solutions of the preconditioning system using e.g. adaptive cycling strategy in multigrid [20] or approximate interior solvers in domain decomposition methods [32,Section 4.3] are considered.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Relations (9), (10), (12), (13) and (14) have been shown in [15,Proposition 2]. From relations (13) and (6) respectively, we deduce…”

Section: Flexible Gmres With Deflated Restartingmentioning

confidence: 74%

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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“…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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“…A disadvantage of this approach is that the convergence behavior in many situations seems to depend quite critically on the value of m. Even in situations in which satisfactory convergence takes place, the convergence is less than optimal, since the history is thrown away so that potential superlinear convergence behavior is inhibited [10]. There are many acceleration techniques that attempt to mimic the convergence of full GMRES more closely, or to accelerate the convergence of the regular GMRES by retaining some historical information at the time of restart [11][12][13][14][15]. Deflation methods are a main class of acceleration techniques for GMRES.…”

Section: Acceleration Techniques For Gmresmentioning

confidence: 99%

“…This spectral preconditioning technique uses approximate eigenvectors generated during only one GMRES cycle. It is improved in [15] with a method, called deflated GMRES (DGMRES) in this paper, that keeps working on the approximate eigenvectors outside of GMRES. DGMRES combines its previous approximate eigenvectors with new ones from each cycle of GMRES.…”

Section: Acceleration Techniques For Gmresmentioning

confidence: 99%

Adaptively Accelerated Gmres Fast Fourier Transform Method for Electromagnetic Scattering

Xin¹,

Rui²

2008

PIER

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Abstract-The problem of electromagnetic scattering by 3D dielectric bodies is formulated in terms of a weak-form volume integral equation. Applying Galerkin's method with rooftop functions as basis and testing functions, the integral equation can be usually solved by Krylov-subspace fast Fourier transform (FFT) iterative methods. In this paper, the generalized minimum residual (GMRES)-FFT method is used to solve this integral equation, and several adaptive acceleration techniques are proposed to improve the convergence rate of the GMRES-FFT method. On several electromagnetic scattering problems, the performance of these adaptively accelerated GMRES-FFT methods are thoroughly analyzed and compared.

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Restarted GMRES preconditioned by deflation

Cited by 181 publications

References 15 publications

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

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

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

Adaptively Accelerated Gmres Fast Fourier Transform Method for Electromagnetic Scattering

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