Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

Martínez, Ángeles

doi:10.1002/nla.2032

Cited by 17 publications

(23 citation statements)

References 24 publications

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“…Finally, the SR1 update provides clearly symmetric matrices, upon symmetry of P 0 . If in addition P 0 is SPD the new update P is also likely to be SPD as well, depending on the quality of the initial preconditioner P 0 , as stated in the following result [25] (Section 3.2).…”

Section: Digressionmentioning

confidence: 99%

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%

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

Bergamaschi

2020

Algorithms

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“…We call (15) the Multiresolution Matrix Decomposition (MMD) of A −1 . We remark that as k increases, the compressed dimension N (k) decreases, and the scale of the subspace spanned by Ψ (k) becomes coarser.…”

Section: Multiresolution Matrix Decompositionmentioning

confidence: 99%

“…These include the Jacobi-Davidson (JD) method [25], implicit restarted Arnoldi/Lanczos method [5,27,13], and the Deflation-accelerated Newton method (DACG) [2]. All these methods give promising results [1,15], especially for finding a small amount of leftmost eigenpairs. However, as reported in [15], the Implicit Restarted Lanczos Method (IRLM) is still the most performing algorithm when a large amount of smallest eigenpairs are required.…”

mentioning

confidence: 99%

“…All these methods give promising results [1,15], especially for finding a small amount of leftmost eigenpairs. However, as reported in [15], the Implicit Restarted Lanczos Method (IRLM) is still the most performing algorithm when a large amount of smallest eigenpairs are required. Therefore, it is highly desirable to develop a new algorithm, based on the architecture of the IRLM, that can further optimize the performance.…”

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confidence: 99%

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A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

Hou¹,

Huang²,

Lam³

et al. 2019

Multiscale Model. Simul.

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In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition components by integrating the multiresolution framework into the Implicitly Restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm. (K) ex and D (K) ex to obtain V (K−1) ini and D (K−1) ini, which will then be the initial spectrum in the consecutive finer level. The efficiency of the cross-level refinement is achieved by a modified version of the orthogonal iteration with the Ritz Acceleration, where we exploit the proximity of the eigenspace across levels to accelerate the Conjugate gradient (CG) method within the refinement step. Using this refined initial spectrum, our second stage is to extend spectrum up to some prescribed control of the well-posedness using the Implicit Restarted Lanczos architecture. Recall that a shifting approach is introduced to reduce 2.2. Operator Compression. The procedures of compressing the solver A −1 with broad-banded spectrum are: (i) construct a partition of the computational basis using local information of A; (ii) construct the

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“…Note that we selected test examples of comparably small sizes in order to be able to compute the eigendecompositions needed for the weight vectors w. The effects and performance gains resulting from the application of tuned preconditioners have been demonstrated with large matrices, e.g. in [27,17].…”

Section: Numerical Experimentsmentioning

confidence: 99%

GMRES Convergence Bounds for Eigenvalue Problems

Freitag

Kürschner

Pestana

2017

Computational Methods in Applied Mathematics

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The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g. tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

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Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems

Cited by 17 publications

References 24 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

A Fast Hierarchically Preconditioned Eigensolver Based on Multiresolution Matrix Decomposition

GMRES Convergence Bounds for Eigenvalue Problems

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