Augmented GMRES‐type methods

Baglama, James; Reichel, Lothar

doi:10.1002/nla.518

Cited by 33 publications

(46 citation statements)

References 19 publications

(43 reference statements)

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“…We conclude this section with some comments on how the methods of the present paper relate to those described in [2]. In the latter methods a user prescribes a subspace W that is to be part of the solution subspace.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…For instance, W may be chosen as the span of the vectors (1.9). The kth approximate solution of (1.1), denoted by x k , determined by the augmented RRGMRES described in [2] satisfies…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Other methods to extend Krylov subspaces to include certain desired vectors are described in [1,2]. The FGMRES methods of the present paper are related to but different from the methods described in [2]. We find the use of FGMRES attractive, because this method provides a unified approach to include user-specified vectors in the solution subspace.…”

Section: Introductionmentioning

confidence: 98%

“…Our interest in FGMRES stems from the fact that the method provides a convenient framework for explicitly modifying the Krylov subspace (1.7) used by standard GMRES. Other methods to extend Krylov subspaces to include certain desired vectors are described in [1,2]. The FGMRES methods of the present paper are related to but different from the methods described in [2].…”

Section: Introductionmentioning

confidence: 99%

“…We find the use of FGMRES attractive, because this method provides a unified approach to include user-specified vectors in the solution subspace. Further comments on how the methods of this paper relate to those of [2] are provided at the end of Section 3. We also note that the application of regularization matrices provides an implicit way to change the solution subspace; see, e.g., [8,12,16,25] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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FGMRES for linear discrete ill-posed problems

Morikuni

Reichel

Hayami

2014

Applied Numerical Mathematics

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GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear discrete ill-posed problems. The FGMRES method, proposed by Saad, is a generalization of GMRES that allows larger flexibility in the choice of solution subspace than GMRES. This paper explores application of FGMRES to the solution of linear discrete ill-posed problems. Numerical examples illustrate that FGMRES with a suitably chosen solution subspace may determine approximate solutions of higher quality than commonly applied iterative methods.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…For instance, W may be chosen as the span of the vectors (1.9). The kth approximate solution of (1.1), denoted by x k , determined by the augmented RRGMRES described in [2] satisfies…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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FGMRES for linear discrete ill-posed problems

Morikuni

Reichel

Hayami

2014

Applied Numerical Mathematics

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A survey of subspace recycling iterative methods

2020

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This survey concerns subspace recycling methods, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial vectors over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.

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Hybrid enriched bidiagonalization for discrete ill‐posed problems

Hansen

Dong

Abe

2019

Numerical Linear Algebra App

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Summary The regularizing properties of the Golub–Kahan bidiagonalization algorithm are powerful when the associated Krylov subspace captures the dominating components of the solution. In some applications the regularized solution can be further improved by enrichment, that is, by augmenting the Krylov subspace with a low‐dimensional subspace that represents specific prior information. Inspired by earlier work on GMRES, we demonstrate how to carry these ideas over to the bidiagonalization algorithm, and we describe how to incorporate Tikhonov regularization. This leads to a hybrid iterative method where the choice of regularization parameter in each iteration also provides a stopping rule.

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Augmented GMRES‐type methods

Cited by 33 publications

References 19 publications

FGMRES for linear discrete ill-posed problems

FGMRES for linear discrete ill-posed problems

A survey of subspace recycling iterative methods

Hybrid enriched bidiagonalization for discrete ill‐posed problems

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