Two‐stage spectral preconditioners for iterative eigensolvers

Bergamaschi, Luca; Martínez, Ángeles

doi:10.1002/nla.2084

Cited by 8 publications

(18 citation statements)

References 27 publications

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“…The Broyden tuned preconditioner was used in Reference [26] to accelerate the GMRES for the inner linear systems within the Arnoldi method for eigenvalue computation. The SR1 preconditioner appeared in References [19,25] and also in Reference [27]. This low-rank update has also been employed to accelerate the PCG method in the solution of linear systems involving SPD Jacobian matrices.…”

Section: Digressionmentioning

confidence: 99%

“…A further strategy to improve the preconditioner's efficiency by a low-rank update can be found in References [8,9,27,34,35]. A spectral preconditioner for SPD linear systems is defined as…”

Section: Spectral Preconditionersmentioning

confidence: 99%

“…k is shown to be SPD in the subspace orthogonal to u k [22,49]. Following the works in [27,50], which are based on a two-stage algorithm, the columns of W are orthonormal approximate eigenvectors of A provided by a gradient method, DACG [47], satisfying…”

Section: Fe Discretization Of a Parabolic Equationmentioning

confidence: 99%

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A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Bergamaschi

2020

Algorithms

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The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

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Section: Digressionmentioning

confidence: 99%

Section: Spectral Preconditionersmentioning

confidence: 99%

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A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Bergamaschi

2020

Algorithms

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“…The Broyden tuned preconditioner has been used in [16] to accelerate the GMRES for the inner linear systems within the Arnoldi method for eigenvalue computation. The SR1 preconditioner appeared in the cited works [9,15] and also in [17]. This low-rank update has also been employed to accelerate the PCG method in the solution of linear systems involving SPD Jacobian matrices.…”

Section: Digressionmentioning

confidence: 99%

“…A further strategy to improve the preconditioner's efficiency by a low-rank update can be found in [17,[24][25][26]. A spectral preconditioner for SPD linear systems is defined as…”

Section: Spectral Preconditionersmentioning

confidence: 99%

Low-rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Bergamaschi¹

2020

Preprint

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The aim of this survey is to review some recent developements in devising efficient preconditioners for sequences of linear systems A x = b. Such a problem arise in many scientific applications, such as discretization of transient PDEs, solution of eigenvalue problems, (Inexact) Newton method applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. Full purpose preconditioners such as the Incomplete Cholesky (IC) factorization or approximate inverses are aimed at clustering eigenvalues of the preconditioned matrices around one. In this paper we will analyze a number of techniques of updating a given IC preconditioner (which we denote as P0 in the sequel) by a low-rank matrix with the aim of further improving this clustering. The most popular low-rank strategies are aimed at removing the smallest eigenvalues (deflation) or at shifting them towards the middle of the spectrum. The low-rank correction is based on a (small) number of linearly independent vectors whose choice is crucial for the effectiveness of the approach. In many cases these vectors are approximations of eigenvectors corresponding to the smallest eigenvalues of the preconditioned matrix P0 A. We will also review some techniques to efficiently approximate these vectors when incorporated within a sequence of linear systems all possibly having constant (or slightly changing) coefficient matrices. Numerical results concerning sequences arising from discretization of linear/nonlinear PDEs and iterative solution of eigenvalue problems show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

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Structured Matrices in Numerical Linear Algebra

2019

Springer INdAM Series

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Given an n × n symmetric positive definite (SPD) matrix A and an SPD preconditioner P we propose a new class of generalized block tuned (GBT) preconditioners. These are defined as a p-rank correction of P with the property that arbitrary (positive) parameters γ 1 , . . . , γ p are eigenvalues of the preconditioned matrix. We propose to employ these GBT preconditioners to accelerate the iterative solution of linear systems like (A − θI)s = r in the framework of iterative eigensolvers. We give theoretical evidence that a suitable, and effective, choice of the scalars γ j is able to shift p eigenvalues of P (A − θI) very close to one. Numerical experiments on various matrices of very large size show that the proposed preconditioner is able to yield an almost constant number of iterations, for different eigenpairs, irrespective of the relative separation between consecutive eigenvalues. We also give numerical evidence that the GBT preconditioner is always far superior to the spectral preconditioner [1], on matrices with highly clustered eigenvalues.

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Two‐stage spectral preconditioners for iterative eigensolvers

Cited by 8 publications

References 27 publications

A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Low-rank Updates of Preconditioners for Sequences of Symmetric Linear Systems

Structured Matrices in Numerical Linear Algebra

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