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Keywords: parallel computing, iterative methods, preconditioning.

Summary

The efficient solution of sparse linear systems of equations is a central issue in many numerical computations in science and engineering. Typically, the system arises from the discretization of partial differential equations with the number of unknowns frequently growing up to a few millions, thus making the use of iterative solvers and parallel architectures virtually mandatory. Iterative methods based on Krylov subspaces can be, at least in principle, ideally implemented on parallel computers. However, the computation and application of an effective preconditioner are not, and perhaps this is the most important effort for the efficient solution to a large size sparse linear system.

In the present paper a novel parallel preconditioner combining a generalized Factored Sparse Approximate Inverse (FSAI) with a block Incomplete LU (ILU) decomposition is presented. The generalized Block FSAI (BFSAI) is derived by requiring the preconditioned matrix to resemble as much as possible a block diagonal matrix in the sense of the minimal Frobenius norm. A second preconditioning is then applied following an incomplete Block Jacobi strategy. The BFSAI-ILU preconditioner turns out to be a parallel hybrid of FSAI and ILU that proves superior to FSAI for any number of processors and is fully scalable for any given number of blocks.

The outcome of realistic fluid-dynamical and geomechnical applications shows that the BFSAI-ILU quality as a preconditioner progressively deteriorates as the number of blocks, hence of processors, increases. In the limiting case with a number of processors equal to the matrix size, however, it coincides with FSAI, whose performance represents an upper bound for BFSAI-ILU. For a small number of processors, say 2 to 8, BFSAI-ILU is significantly superior to FSAI, proving more than twice as fast in the test cases discussed in the present paper. Therefore, BFSAI-ILU appears to be a robust and promising tool to exploit the parallel potential provided by the modern multi-core processor technology.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 94 PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: Paper 3 A Novel Hybrid FSAI-ILU Preconditioner for the Efficient Parallel Solution of Large Size Sparse Linear Systems C. Janna, M. Ferronato and G. Gambolati DMMMSA, University of Padova, Italy doi:10.4203/ccp.94.3 purchase the full-text of this paper Full Bibliographic Reference for this paper C. Janna, M. Ferronato, G. Gambolati, "A Novel Hybrid FSAI-ILU Preconditioner for the Efficient Parallel Solution of Large Size Sparse Linear Systems", in , (Editors), "Proceedings of the Seventh International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 3, 2010. doi:10.4203/ccp.94.3 Keywords: parallel computing, iterative methods, preconditioning. Summary The efficient solution of sparse linear systems of equations is a central issue in many numerical computations in science and engineering. Typically, the system arises from the discretization of partial differential equations with the number of unknowns frequently growing up to a few millions, thus making the use of iterative solvers and parallel architectures virtually mandatory. Iterative methods based on Krylov subspaces can be, at least in principle, ideally implemented on parallel computers. However, the computation and application of an effective preconditioner are not, and perhaps this is the most important effort for the efficient solution to a large size sparse linear system. In the present paper a novel parallel preconditioner combining a generalized Factored Sparse Approximate Inverse (FSAI) with a block Incomplete LU (ILU) decomposition is presented. The generalized Block FSAI (BFSAI) is derived by requiring the preconditioned matrix to resemble as much as possible a block diagonal matrix in the sense of the minimal Frobenius norm. A second preconditioning is then applied following an incomplete Block Jacobi strategy. The BFSAI-ILU preconditioner turns out to be a parallel hybrid of FSAI and ILU that proves superior to FSAI for any number of processors and is fully scalable for any given number of blocks. The outcome of realistic fluid-dynamical and geomechnical applications shows that the BFSAI-ILU quality as a preconditioner progressively deteriorates as the number of blocks, hence of processors, increases. In the limiting case with a number of processors equal to the matrix size, however, it coincides with FSAI, whose performance represents an upper bound for BFSAI-ILU. For a small number of processors, say 2 to 8, BFSAI-ILU is significantly superior to FSAI, proving more than twice as fast in the test cases discussed in the present paper. Therefore, BFSAI-ILU appears to be a robust and promising tool to exploit the parallel potential provided by the modern multi-core processor technology. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £125 +P&P)
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