The adaptive augmented GMRES method for solving ill-posed problems

Kakizawa, Nao; Nodera, Takashi

doi:10.21914/anziamj.v50i0.1444

Cited by 6 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some suggestions for subspaces are provided in [127,137]. For problems with unsuitable augmentation spaces, an adaptive augmented GMRES was considered in [144] to automatically select a suitable subspace from a set of user-specified candidates.…”

Section: 1mentioning

confidence: 99%

Computational methods for large-scale inverse problems: a survey on hybrid projection methods

Chung,

Gazzola

2021

Preprint

View full text Add to dashboard Cite

This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes a broad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems.

show abstract

Section: 1mentioning

confidence: 99%

Computational methods for large-scale inverse problems: a survey on hybrid projection methods

Chung,

Gazzola

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Often, it has been shown beneficial to employ a strategy known as range‐restriction, wherein one uses a power of the operator times the right‐hand side, for example,

𝒱_{j} = 𝒦_{j} false(false(I - Q false) A, false(I - Q false) A r_{0} false)

. We note that based on the work in [15], another adaptive augmented method has also been developed [93]. This body of literature is not necessarily concerned with recycling spectral information as much as it is with an augmentation space that encodes certain known features of the solution.…”

Section: Practical Realizations Of the Recycling Frameworkmentioning

confidence: 99%

A survey of subspace recycling iterative methods

2020

View full text Add to dashboard Cite

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.

show abstract

“…Often, it has been shown beneficial to employ a strategy known as range-restriction, wherein one does not develop the Krylov subspace with respect to the right-hand side but rather uses a power of the operator times the right-hand side, e.g.,  =  ( − ) , ( − ) 0 . 3 We note that based on the work in [84] , another adaptive augmented method has also been developed [88] . This body of literature is not necessarily concerned with recycling spectral information as much as it is with an augmentation space that encodes certain known features of the solution.…”

Section: Augmented Gmres For Ill-posed Problemsmentioning

confidence: 99%

A survey of subspace recycling iterative methods

Soodhalter¹,

Sturler²,

Kilmer³

2020

Preprint

View full text Add to dashboard Cite

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 guesses 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.

show abstract

The adaptive augmented GMRES method for solving ill-posed problems

Cited by 6 publications

References 5 publications

Computational methods for large-scale inverse problems: a survey on hybrid projection methods

Computational methods for large-scale inverse problems: a survey on hybrid projection methods

A survey of subspace recycling iterative methods

A survey of subspace recycling iterative methods

Contact Info

Product

Resources

About