A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography

Saibaba, Arvind K.; Bakhos, Tania; Kitanidis, Peter K.

doi:10.1137/120902690

Cited by 39 publications

(81 citation statements)

References 25 publications

(55 reference statements)

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“…We present a combination of inner FOM and outer GMRES in Algorithm 3. 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.…”

Section: Resultsmentioning

confidence: 99%

“…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].…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, only the seed shift τ has to be chosen and in order to invert P(σ) we only need to decompose P. However, the one-time decomposition of P can numerically be very costly and the suitable choice of the seed shift τ is difficult for a large range of shifts σ 1 , .., σ Nσ , as has been pointed out by [20]. In [1], a polynomial preconditioner is suggested that cheaply approximates P −1 .…”

Section: The Single Shift-and-invert Preconditioner For Shifted Systemsmentioning

confidence: 99%

“…Therefore, we assume that the preconditioner P j (σ) is equivalent to a truncated msFOM inner iteration with collinear factor γ (σ) j of the residuals according to (20). From the following calculation…”

Section: Nested Fom-gmres For Shifted Linear Systemsmentioning

confidence: 99%

“…The required collinearity factor γ (σ) j of the j-th outer iteration in (19) is given by (20), where we assume that msFOM is stopped after m inner iterations…”

mentioning

confidence: 99%

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Nested Krylov Methods for Shifted Linear Systems

Baumann¹,

Gijzen²

2015

SIAM J. Sci. Comput.

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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.

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: The Single Shift-and-invert Preconditioner For Shifted Systemsmentioning

confidence: 99%

Section: Nested Fom-gmres For Shifted Linear Systemsmentioning

confidence: 99%

“…The required collinearity factor γ (σ) j of the j-th outer iteration in (19) is given by (20), where we assume that msFOM is stopped after m inner iterations…”

mentioning

confidence: 99%

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Nested Krylov Methods for Shifted Linear Systems

Baumann¹,

Gijzen²

2015

SIAM J. Sci. Comput.

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A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right‐hand sides

Sun

Huang

Jing

et al. 2018

Numerical Linear Algebra App

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Summary The restarted block generalized minimum residual method (BGMRES) with deflated restarting (BGMRES‐DR) was proposed by Morgan to dump the negative effect of small eigenvalues from the convergence of the BGMRES method. More recently, Wu et al. introduced the shifted BGMRES method (BGMRES‐Sh) for solving the sequence of linear systems with multiple shifts and multiple right‐hand sides. In this paper, a new shifted block Krylov subspace algorithm that combines the characteristics of both the BGMRES‐DR and the BGMRES‐Sh methods is proposed. Moreover, our method is enhanced with a seed selection strategy to handle the case of almost linear dependence of the right‐hand sides. Numerical experiments illustrate the potential of the proposed method to solve efficiently the sequence of linear systems with multiple shifts and multiple right‐hand sides, with and without preconditioner, also against other state‐of‐the‐art solvers.

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Shifted LOPBiCG: A locally orthogonal product‐type method for solving nonsymmetric shifted linear systems based on Bi‐CGSTAB

Zhao,

Sogabe,

Kemmochi

et al. 2023

Numerical Linear Algebra App

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When solving shifted linear systems using shifted Krylov subspace methods, selecting a seed system is necessary, and an unsuitable seed may result in many shifted systems being unsolved. To avoid this problem, a seed‐switching technique has been proposed to help switch the seed system to another linear system as a new seed system without losing the dimension of the constructed Krylov subspace. Nevertheless, this technique requires collinear residual vectors when applying Krylov subspace methods to the seed and shifted systems. Since the product‐type shifted Krylov subspace methods cannot provide such collinearity, these methods cannot use this technique. In this article, we propose a variant of the shifted BiCGstab method, which possesses the collinearity of residuals, and apply the seed‐switching technique to it. Some numerical experiments show that the problem of choosing the initial seed system is circumvented.

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A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography

Cited by 39 publications

References 25 publications

Nested Krylov Methods for Shifted Linear Systems

Nested Krylov Methods for Shifted Linear Systems

A block GMRES method with deflated restarting for solving linear systems with multiple shifts and multiple right‐hand sides

Shifted LOPBiCG: A locally orthogonal product‐type method for solving nonsymmetric shifted linear systems based on Bi‐CGSTAB

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