A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form

Kharchenko, S. A.; Kolotilina, L. Yu.; Nikishin, A. A.; Yeremin, A. Yu.

doi:10.1002/1099-1506(200104/05)8:3<165::aid-nla235>3.0.co;2-9

Cited by 34 publications

(24 citation statements)

References 10 publications

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“…We mention that breakdown-free variants of AINV have also been developed, independently, by Bridson [14] and by Kharchenko et al [29]. Bridson's scheme is different from ours.…”

Section: Introductionmentioning

confidence: 83%

Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method

Benzi

Cullum

Tůma³

2000

SIAM J. Sci. Comput.

152

162

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Abstract. We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly ill-conditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.Key words. sparse linear systems, finite element matrices, preconditioned conjugate gradients, factorized sparse approximate inverses, incomplete conjugation, stabilized AINV, diagonally compensated reduction AMS subject classifications. Primary, 65F10, 65N22, 65F50; Secondary, 15A06 PII. S10648275993569001. Introduction. We consider the solution of sparse linear systems Ax = b, where A is a symmetric and positive definite (SPD) matrix, by the preconditioned conjugate gradient method. In the last few years there has been considerable interest in explicit preconditioning techniques based on directly approximating A −1 with a sparse matrix M ; see, e.g., [7] Although the main motivation for the development of sparse approximate inverse preconditioners comes from parallel processing, it is becoming clear that these techniques are also of interest because of their robustness. Sparse approximate inverses are often applicable to difficult problems where other preconditioners may break down [4]. For instance, incomplete factorization preconditioners, while widely popular and fairly robust, are not always reliable, in that the incomplete factorization process may

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“…We mention that breakdown-free variants of AINV have also been developed, independently, by Bridson [14] and by Kharchenko et al [29]. Bridson's scheme is different from ours.…”

Section: Introductionmentioning

confidence: 83%

Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method

Benzi

Cullum

Tůma³

2000

SIAM J. Sci. Comput.

152

162

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“…Note that this generalization of the BIF algorithm to the nonsymmetric case is similar to generalizations of other biconjugation-based algorithms to get LDU factorization or its inverse. For instance while AINV ( [3,4]) runs the processes to compute its factors successively, straightforward nonsymmetric generalizations of SAINV [1] and RIF [5] and similar algorithms [10], [28] do interleave the computation.…”

Section: −1mentioning

confidence: 99%

Improved Balanced Incomplete Factorization

Bru¹,

Marı́n²,

Mas³

et al. 2010

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008Comput., 30 ( ), pp. 2302Comput., 30 ( -2318. The improvements are based on a deeper understanding of the Inverse Sherman-Morrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems.

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“…Factorized approximate inverse preconditioners share one drawback with ILU-type techniques: The construction phase is guaranteed to be breakdown-free only for special classes of matrices. Sufficient conditions are that A be symmetric positive definite or an H-matrix; see [29], [3], [28], [2]. For general sparse matrices, instabilities due to very small or zero pivots can occur during the construction of the preconditioner, with disastrous effects.…”

Section: Ilu and Approximate Inverse Preconditionersmentioning

confidence: 99%

Preconditioning Highly Indefinite and Nonsymmetric Matrices

Benzi

Haws²,

Tůma³

2000

SIAM J. Sci. Comput.

116

108

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Abstract. Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131-151] and by Duff and Koster [SIAM J. Matrix Anal. Appl., 20 (1999), pp. 889-901; Tech. report Ral-Tr-99-030, Rutherford Appleton Laboratory, Chilton, UK, 1999]. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing.

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A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form

Cited by 34 publications

References 10 publications

Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method

Robust Approximate Inverse Preconditioning for the Conjugate Gradient Method

Improved Balanced Incomplete Factorization

Preconditioning Highly Indefinite and Nonsymmetric Matrices

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