Low-Rank Factorizations in Data Sparse Hierarchical Algorithms for Preconditioning Symmetric Positive Definite Matrices

Agullo, Emmanuel; Darve, Eric; Giraud, Luc; Harness, Yuval

doi:10.1137/17m1151158

Cited by 5 publications

(7 citation statements)

References 35 publications

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“…It appears that, up to a certain level, we are still able to find a robust coarse space despite having significantly reduced apply. This is left for future work (see [1] for preliminary investigations in this direction) and we do 568 not consider approximation techniques in the high performance implementation we propose below. quired for computing dot products while the matrix-vector product can be performed concurrently on each subdomain and the application of the (one-level) preconditioner only requires neighbor-toneighbor communications.…”

Section: Approximate Casementioning

confidence: 99%

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Agullo¹,

Giraud²,

Poirel³

2019

SIAM J. Matrix Anal. Appl.

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The solution of large sparse linear systems is one of the most time consuming kernels in many numerical simulations. The domain decomposition community has developed many efficient and robust methods in the last decades. While many of these solvers fall into the abstract Schwarz (aS) framework, their robustness has originally been demonstrated on a case-by-case basis. In this paper, we propose a bound for the condition number of all deflated aS methods provided that the coarse grid consists of the assembly of local components that contain the kernel of some local operators. We show that classical results from the literature on particular instances of aS methods can be retrieved from this bound. We then show that such a coarse grid correction can be explicitly obtained algebraically via generalized eigenproblems, leading to a condition number independent of the number of domains. This result can be readily applied to retrieve or improve the bounds previously obtained via generalized eigenproblems in the particular cases of Neumann-Neumann (NN), Additive Schwarz (AS) and optimized Robin but also generalizes them when applied with approximate local solvers. Interestingly, the proposed methodology turns out to be a comparison of the considered particular aS method with generalized versions of both NN and AS for tackling the lower and upper part of the spectrum, respectively. We furthermore show that the application of the considered grid corrections in an additive fashion is robust in the AS case although it is not robust for aS methods in general. In particular, the proposed framework allows for ensuring the robustness of the AS method applied on the Schur complement (AS/S), either with deflation or additively, and with the freedom of relying on an approximate local Schur complement. Numerical experiments illustrate these statements.

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Section: Approximate Casementioning

confidence: 99%

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Agullo¹,

Giraud²,

Poirel³

2019

SIAM J. Matrix Anal. Appl.

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“…Here, we show that both types of schemes are similarly effective and also give a more intuitive explanation of the effectiveness by extending Equation (10). 3is k. For the Cholesky SIF factorL in Equation (6), the Equation (9) holds with the nonzero singular values ofĈ in Equation (10) given by…”

Section: Spectral Analysis For Cholesky and Ulv Sif Preconditioningmentioning

confidence: 99%

“…where the last equality in the second line is due to the Sherman-Morrison-Woodbury formula and the resultŨ T 2Û 2 = 0. The eigenvalues ofL −1 AL −T can then be immediately obtained based on Equation (9).…”

Section: Theorem 1 Suppose the Smaller Of The Row And Column Sizes Omentioning

confidence: 99%

“…Practical numerical tests have shown superior convergence results as compared with standard structured preconditioning based on direct off-diagonal compression. 7 The scaling and compression strategy is later also followed by a series of other work [9][10][11][12] for designing structured preconditioners for both dense and sparse matrices. Related ideas also appear in some work for preconditioning sparse matrices.…”

Section: Introductionmentioning

confidence: 99%

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Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

Xin

Xia

Cauley

et al. 2020

Numerical Linear Algebra App

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In this work, we provide new analysis for a preconditioning technique called structured incomplete factorization (SIF) for symmetric positive definite matrices. In this technique, a scaling and compression strategy is applied to construct SIF preconditioners, where off-diagonal blocks of the original matrix are first scaled and then approximated by low-rank forms. Some spectral behaviors after applying the preconditioner are shown. The effectiveness is confirmed with the aid of a type of two-dimensional and three-dimensional discretized model problems. We further show that previous studies on the robustness are too conservative. In fact, the practical multilevel version of the preconditioner has a robustness enhancement effect, and is unconditionally robust (or breakdown free) for the model problems regardless of the compression accuracy for the scaled off-diagonal blocks. The studies give new insights into the SIF preconditioning technique and confirm that it is an effective and reliable way for designing structured preconditioners. The studies also provide useful tools for analyzing other structured preconditioners. Various spectral analysis results can be used to characterize other structured algorithms and study more general problems. K E Y W O R D Seffectiveness, robustness, scaling and compression strategy, SIF preconditioning, spectral analysis

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“…One type is in [11,12,13,18] based on low-rank strategies for approximating A −1 . Another type is in [1,6,8,9,14,19,21,22] where approximate Cholesky factorizations are computed using low-rank approximations of relevant off-diagonal blocks. Both types of methods have been shown useful for many applications.…”

mentioning

confidence: 99%

Robust and effective eSIF preconditioning for general SPD matrices

Xia¹

2020

Preprint

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We propose an unconditionally robust and highly effective preconditioner for general symmetric positive definite (SPD) matrices based on structured incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular preprocessing. A practical multilevel eSIF scheme is then designed. We give rigorous analysis for both the enhanced scaling-and-compression strategy and the multilevel eSIF preconditioner. The new eSIF framework has some significant advantages and overcomes some major limitations of the SIF strategy. (i) With the same tolerance for compressing the off-diagonal blocks, the eSIF preconditioner can approximate the original matrix to a much higher accuracy. (ii) The new preconditioner leads to much more significant reductions of condition numbers due to an accelerated magnification effect for the decay in the singular values of the scaled off-diagonal blocks. (iii) With the new preconditioner, the eigenvalues of the preconditioned matrix are much better clustered around 1. (iv) The multilevel eSIF preconditioner is further unconditionally robust or is guaranteed to be positive definite without the need of extra stabilization, while the multilevel SIF preconditioner has a strict requirement in order to preserve positive definiteness. Comprehensive numerical tests are used to show the advantages of the eSIF preconditioner in accelerating the convergence of iterative solutions.

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Low-Rank Factorizations in Data Sparse Hierarchical Algorithms for Preconditioning Symmetric Positive Definite Matrices

Cited by 5 publications

References 35 publications

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Effectiveness and robustness revisited for a preconditioning technique based on structured incomplete factorization

Robust and effective eSIF preconditioning for general SPD matrices

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