Algebraic Multilevel Methods and Sparse Approximate Inverses

Bollhöfer, Matthias; Mehrmann, Volker

doi:10.1137/s0895479899364441

Cited by 14 publications

(12 citation statements)

References 29 publications

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“…There is at present a wide variety of such algorithms; an incomplete list of recent papers includes [20-22, 58, 113, 203, 228, 235, 242, 250, 254, 256, 257, 276, 295] for multilevel variants of ILU, and [53,218,219,227] for multilevel sparse approximate inverse techniques. Combining approximate inverses and wavelet transforms is another way to improve scalability (see [67,80,99]).…”

Section: Algebraic Multilevel Variantsmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,011

659

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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper. c 2002 Elsevier Science (USA)

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Section: Algebraic Multilevel Variantsmentioning

confidence: 99%

Preconditioning Techniques for Large Linear Systems: A Survey

Benzi¹

2002

Journal of Computational Physics

1,011

659

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“…There are two kinds of approaches to constructing M. One kind is the factorized SAI preconditioners proposed initially in [11], and has been developed in, e.g. [12][13][14][15][16][17][18][19][20]. The other kind is based on Frobenius norm minimization and is also among the best known SAI preconditioners.…”

Section: Introductionmentioning

confidence: 99%

A power sparse approximate inverse preconditioning procedure for large sparse linear systems

Jia

Zhu

2008

Numerical Linear Algebra App

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Motivated by the Cayley-Hamilton theorem, a novel adaptive procedure, called a Power Sparse Approximate Inverse (PSAI) procedure, is proposed that uses a different adaptive sparsity pattern selection approach to constructing a right preconditioner M for the large sparse linear system Ax = b. It determines the sparsity pattern of M dynamically and computes the n independent columns of M that is optimal in the Frobenius norm minimization, subject to the sparsity pattern of M. The PSAI procedure needs a matrix-vector product at each step and updates the solution of a small least squares problem cheaply. To control the sparsity of M and develop a practical PSAI algorithm, two dropping strategies are proposed. The PSAI algorithm can capture an effective approximate sparsity pattern of A −1 and compute a good sparse approximate inverse M efficiently. Numerical experiments are reported to verify the effectiveness of the PSAI algorithm. Numerical comparisons are made for the PSAI algorithm and the adaptive SPAI algorithm proposed by Grote and Huckle as well as for the PSAI algorithm and three static Sparse Approximate Inverse (SAI) algorithms. The results indicate that the PSAI algorithm is at least comparable to and can be much more effective than the adaptive SPAI algorithm and it often outperforms the static SAI algorithms very considerably and is more robust and practical than the static ones for general problems. INVERSE PRECONDITIONING PROCEDURE 261 forms do, because they can express denser matrices than the total number of nonzero entries in their factors. However, factorized forms have their own drawbacks, and like ILU [7] they can fail due to breakdown during an incomplete factorization process. A comprehensive survey of sparse approximate inverse preconditioners, together with extensive numerical comparisons aimed at assessing the overall performance of the various methods, can be found in [46]. POWER SPARSE APPROXIMATEWe concentrate on the second kind of SAI preconditioners in this paper. A key issue is to determine the sparsity pattern of M effectively. The initial work is to prescribe it in advance [21][22][23].Once that is given, the computation of M is straightforward and one can solve n independent small least squares problems to get all the columns of M. This is called a static SAI procedure. M is required to be sparse as well as to approximate A −1 . Much work has been done in prescribing the sparsity pattern of M; e.g. [11,19,20,30,39,40,43]. The most common a priori pattern of M is simply that of A, and this is called the structure of 1-local matrix [22]. This choice can give good results for many problems but can also fail for many problems. One improvement is to use the sparsity structure of q-local matrix for q>1. However, M becomes denser quickly when q increases, even though q = 2 is often impractical [30].SAI techniques are based on the implicit assumption that the majority of the entries in A −1 are small, so that it is possible to find a sparse matrix M that is a good approximation to A −1 ....

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“…1 can be found in previous work on SAI preconditioners specifically [6], [7], [9], [11], [21]). Factorized sparse approximate inverse preconditioners are another class of SAI preconditioning techniques developed in [25], [26], [27], [28], [29], [30], [31]. This class of preconditioners are less popular than the kind based on Frobenius norm minimization (1) [4] and can fail due to breakdowns during an incomplete factorization process.…”

Section: Sai Preconditioningmentioning

confidence: 99%

Parallel Sparse Approximate Inverse Preconditioning on Graphic Processing Units

Dehnavi

Fernández

Gaudiot

et al. 2013

IEEE Trans. Parallel Distrib. Syst.

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Abstract-Accelerating numerical algorithms for solving sparse linear systems on parallel architectures has attracted the attention of many researchers due to their applicability to many engineering and scientific problems. The solution of sparse systems often dominates the overall execution time of such problems and is mainly solved by iterative methods. Preconditioners are used to accelerate the convergence rate of these solvers and reduce the total execution time. Sparse approximate inverse (SAI) preconditioners are a popular class of preconditioners designed to improve the condition number of large sparse matrices. We propose a GPU accelerated SAI preconditioning technique called GSAI, which parallelizes the computation of this preconditioner on NVIDIA graphic cards. The preconditioner is then used to enhance the convergence rate of the BiConjugate Gradient Stabilized (BiCGStab) iterative solver on the GPU. The SAI preconditioner is generated on average 28 and 23 times faster on the NVIDIA GTX480 and TESLA M2070 graphic cards, respectively, compared to ParaSails (a popular implementation of SAI preconditioners on CPU) single processor/core results. The proposed GSAI technique computes the SAI preconditioner in approximately the same time as ParaSails generates the same preconditioner on 16 AMD Opteron 252 processors.

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Algebraic Multilevel Methods and Sparse Approximate Inverses

Cited by 14 publications

References 29 publications

Preconditioning Techniques for Large Linear Systems: A Survey

Preconditioning Techniques for Large Linear Systems: A Survey

A power sparse approximate inverse preconditioning procedure for large sparse linear systems

Parallel Sparse Approximate Inverse Preconditioning on Graphic Processing Units

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