A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method

Benzi, Michele; Meyer, Carl D.; Tůma, Miroslav

doi:10.1137/s1064827594271421

Cited by 356 publications

(307 citation statements)

References 27 publications

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“…A first class of methods, described in this subsection, does not require any information about the triangular factors of A: the factorized approximate inverse preconditioner is constructed directly from A. Methods in this class include the FSAI preconditioner introduced by Kolotilina and Yeremin [61], a related method due to Kaporin [59], incomplete (bi)conjugation schemes [15,18], and bordering strategies [71]. Another class of methods first compute an incomplete triangular factorization of A using standard techniques, and then obtain a factorized sparse approximate inverse by computing sparse approximations to the inverses of the incomplete triangular factors of A.…”

Section: Factorized Sparse Approximate Inversesmentioning

confidence: 99%

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A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

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A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

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Section: Factorized Sparse Approximate Inversesmentioning

confidence: 99%

“…This approach, hereafter referred to as the AINV method, is described in detail in [15] and [18]. The AINV method does not require that the sparsity pattern be known in advance, and is applicable to matrices with general sparsity patterns.…”

Section: Factorized Sparse Approximate Inversesmentioning

confidence: 99%

A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

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227

200

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“…Gradient-domain techniques are not only used for compositing image regions, however. Various applications can be performed based on gradient domain constraints, ranging from shadow removal [19], intrinsic image recovery,1 high dynamic range (HDR) compression [20] …”

Section: A Gradient-domain Image Processingmentioning

confidence: 99%

“…Unlike direct solvers, iterative methods (with classical black box preconditioners) are not as robust. This is true even with the most recent advances in creating lower upper (LU)-based preconditioners [20,21]. Approximate inverse preconditioners [22] are known to be more favorable for parallelism.…”

Section: B Sparse Matrix Solvermentioning

confidence: 99%

“…Let us assume that A has been reordered, for example using reverse CutHill-Mckee [17] or spectral [19][20][21] reordering schemes, into the overlapped diagonal blocks below , and (2) where, Mij,Xi and bi for i , j = 1 ,..., 3 are blocks of appropriate size. Let us define the partitions of M, delineated by lines in the illustration in (2), as (3) where, except for the overlap, = for ∂,ω= 0,1.…”

Section: B Sparse Matrix Solvermentioning

confidence: 99%

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Fast Parallel Compositon of Dunhuang Murals based on Matrix Tearing

Chen¹,

Wang²,

Zhao³

et al. 2017

Proceedings of the 3rd International Conference on Wireless Communication and Sensor Networks (WCSN 2016)

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Abstract-In this paper we have developed a robust scalable hybrid scheme for Fast Parallel compositon of Dunhuang murals based on Matrix Tearing. The large cost associated with solution of partitioned linear systems, which is required to perform the matrix-vector multiplication, was avoided by presenting a scheme that requires only the solution of tiny linear systems of the same size as that of the overlap between two partitions. The difficulties associated with the multiple partition approach are resolved by using the reordering.

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An assessment of some preconditioning techniques in shell problems

Benzi

Kouhia²,

Tůma

1998

Commun. Numer. Meth. Engng.

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Preconditioned Krylov subspace methods have proved to be ecient in solving large, sparse linear systems in many areas of scienti®c computing. The success of these methods in many cases is due to the existence of good preconditioning techniques. In problems of structural mechanics, like the analysis of heat transfer and deformation of solid bodies, iterative solution of the linear equation system can result in a signi®cant reduction of computing time. Also many preconditioning techniques can be applied to these problems, thus facilitating the choice of an optimal preconditioning on the particular computer architecture available. However, in the analysis of thin shells the situation is not so transparent. It is well known that the stiness matrices generated by the FE discretization of thin shells are very ill-conditioned. Thus, many preconditioning techniques fail to converge or they converge too slowly to be competitive with direct solvers. In this study, the performance of some general preconditioning techniques on shell problems is examined. PRECONDITIONING TECHNIQUES In this paper, the iterative solution of the discretized equilibrium equations of shells, written as a system of linear equations Ax b 1 is considered. The stiness matrix A is large, sparse and symmetric. In most cases it is also positive de®nite, but can be inde®nite at some stage in non-linear analyses. However, the number of negative eigenvalues is usually small, mostly one, on unstable equilibrium paths which are of practical interest. The preconditioned conjugate gradient (PCG) method is a powerful tool for solving such a system. Even though it is foolproof only for positive de®nite matrices, in practice it usually performs well without breakdowns. To speed up the convergence of the conjugate gradient iteration, preconditioning of the original system (1) is introduced: M À1 Ax M À1 b

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A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method

Cited by 356 publications

References 27 publications

A comparative study of sparse approximate inverse preconditioners

A comparative study of sparse approximate inverse preconditioners

Fast Parallel Compositon of Dunhuang Murals based on Matrix Tearing

An assessment of some preconditioning techniques in shell problems

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