Krylov subspace recycling for sequences of shifted linear systems

Soodhalter, Kirk M.; Szyld, Daniel B.; Xue, Fei

doi:10.1016/j.apnum.2014.02.006

Cited by 54 publications

(48 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, the collinearity factor for the inner Krylov method (multishift FOM) is given by (20) without any further manipulations. When combining multishift IDR(s) and QMRIDR(s) as presented in Algorithm 4, a new variant of IDR(s) has been developed which leads to collinear residuals with collinearity factor given by (27), cf. Algorithm 2.…”

Section: Resultsmentioning

confidence: 99%

“…In contrast to [26,27], we restrict ourselves to such types of preconditioners that preserve the shift-invariance property of the Krylov subspaces in (5). Based on the requirements of a single preconditioner that preserves shiftinvariance, a flexible preconditioner is designed that requires collinear residuals for the inner iteration.…”

Section: Flexible Preconditioning For Multi-shift Krylov Methodsmentioning

confidence: 99%

Section: A Variant Of Multi-shift Idr(s) With Collinear Residualsmentioning

confidence: 99%

“…In Algorithm 2, we present the collinear IDR variant (called collIDR(s)) using (27) for the choice of the free IDR parameter ω (σ) j+1 . Note that in (27) the explicit dependence of the collinearity factor on the shift becomes obvious. Since in Algorithm 2 we loop over N σ distinct shifts, all shifted quantities are labeled with a superscript that depends on this loop index k. Concerning the choice of the s-dimensional shadow space S and the parameter ω j+1 of the unshifted system, we refer to Section 4.1 and 4.2 of [25], respectively.…”

Section: A Variant Of Multi-shift Idr(s) With Collinear Residualsmentioning

confidence: 99%

“…We assume that the factor γ (σ) j is given from (27) by an inner collIDR(s) iteration. Then, the flexible QMRIDR relation applied to a shifted problem yields:…”

Section: In Flexible Qmridr (Fqmridr) Which Has Been Introduced In [3mentioning

confidence: 99%

See 4 more Smart Citations

Nested Krylov methods for shifted linear systems

Baumann¹,

Gijzen²

2015

Preprint

View full text Add to dashboard Cite

SummaryShifted linear systems are of the formwhere A ∈ C N ×N , b ∈ C N and {σ k } Nσ k=1 ∈ C is a sequence of numbers, called shifts. In order to solve (1) for multiple shifts efficiently, shifted Krylov methods make use of the shift-invariance property of their respective Krylov subspaces, i.e.and, therefore, compute one basis of the Krylov subspace (2) for all shifted systems. This leads to a significant speed-up of the numerical solution of the shifted problems because obtaining a basis of (2) is computational expensive. However, in practical applications, preconditioning of (1) is required, which in general destroys the shift-invariance. One of the few known preconditioners that lead to a new, preconditioned shifted problem is the so-called shift-and-invert preconditioner which is of the form (A − τ I) where τ is the seed shift. Most recently, multiple shift-and-invert preconditioners have been applied within a flexible GMRES iteration, cf. [10,20]. Since even the one-time application of a shift-and-invert preconditioner can be computationally costly, polynomial preconditioners have been developed for shifted problems in [1]. They have the advantage of preserving the shift-invariance (2) and being computational feasible at the same time.The presented work is a new approach to the iterative solution of (1). We use nested Krylov methods that use an inner Krylov method as a preconditioner for an outer Krylov iteration, cf. [17,24] for the unshifted case. In order to preserve the shift-invariance property, our algorithm only requires the inner Krylov method to produce collinear residuals of the shifted systems. The collinearity factor is then used in the (generalized) Hessenberg relation of the outer Krylov method. In this article, we present two possible combinations of nested Krylov algorithms, namely a combination of inner multi-shift FOM and outer multi-shift GMRES as well as inner multi-shift IDR(s) and outer multi-shift QMRIDR(s). However, we will point out that in principle every combination is possible as long as the inner Krylov method leads to collinear residuals. Since multi-shift IDR [31] does not lead to collinear residuals by default, the development of a collinear IDR variant that can be applied as an inner method within the nested framework is a second main contribution of this work.The new nested Krylov algorithm has been tested on several shifted problems. In particular, the inhomogeneous and time-harmonic linear elastic wave equation leads to shifts that directly correspond to different frequencies of the waves.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Flexible Preconditioning For Multi-shift Krylov Methodsmentioning

confidence: 99%

Section: A Variant Of Multi-shift Idr(s) With Collinear Residualsmentioning

confidence: 99%

Section: A Variant Of Multi-shift Idr(s) With Collinear Residualsmentioning

confidence: 99%

“…We assume that the factor γ (σ) j is given from (27) by an inner collIDR(s) iteration. Then, the flexible QMRIDR relation applied to a shifted problem yields:…”

Section: In Flexible Qmridr (Fqmridr) Which Has Been Introduced In [3mentioning

confidence: 99%

See 3 more Smart Citations

Nested Krylov methods for shifted linear systems

Baumann¹,

Gijzen²

2015

Preprint

View full text Add to dashboard Cite

show abstract

A survey of subspace recycling iterative methods

2020

Self Cite

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

Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

Gratton¹,

Mercier

Tardieu

et al. 2016

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right‐hand sides. These preconditioners based on limited memory quasi‐Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first‐level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half‐plane, respectively. Third, we focus on theoretical properties of the Ritz‐LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large‐scale symmetric saddle‐point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

Krylov subspace recycling for sequences of shifted linear systems

Cited by 54 publications

References 32 publications

Nested Krylov methods for shifted linear systems

Nested Krylov methods for shifted linear systems

A survey of subspace recycling iterative methods

Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

Contact Info

Product

Resources

About