W. P. Tang and W. L. Wan. Sparse approximate inverse smoother for multi-grid. To appear in SIAM J. Matrix Anal. Appl., 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....matrix norm. Several algorithms have been proposed for constructing approximate inverses. These can be loosely categorized first by their result: some algorithms produce a single sparse matrix approximating A e.g. SPAI[22] Chow and Saad s MR method[15, 16] Tang and Wan s local inverse[35] and others produce factored approximations (approximate inverses of the triangular factors) e.g. FSAI[25] AINV[4] The factored form has the advantages of guaranteed non singularity, extra sparsity from good orderings, and apparently more effect per nonzero thanks to its more implicit ....

....the matrix is known only as a linear operator. Actually, a little more is known: the adjoint of the operator may be used in the algorithm as well. This rules out the Frobenius norm minimization algorithms such as SPAI[22] and FSAI[25] as well as Tang and Wan s local inverse method[35], since they all require the ability to access submatrices of M # . Chow and Saad s MR method[15, 16] is a possibility as it only uses the matrix as an operator. However, the impressive performance[5] of the incomplete inverse 33 factorization algorithms makes them the most attractive ....

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W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multigrid, technical report CAM 98-18, Dept. of Math., University of California at Los Angeles, 1998. 118

....by Huckle [16] and Chow [11] Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3, 4] 2 they were shown to be e ective on various dicult elliptic problems on unstructured grids by Tang and Wan [23]. Advantages of sparse approximate inverse smoothers over classical smoothers, such as damped Jacobi, GaussSeidel or ILU, are inherent parallelism, possible local adaptivity and improved robustness. Here we shall consider sparse approximate inverse (SPAI) smoothers based on the SPAI Algorithm by ....

....either if A is weakly diagonally dominant, or if A has at most seven nonzero o diagonal entries per row. To our knowledge this is the rst fairly general theoreti3 cal result on the smoothing property of iterative methods that are based on sparse approximate inverses. Previously Tang and Wan [23] analyzed the smoothing property of sparse approximate inverse smoothers for boundary value problems with constant coecients on a two dimensional regular grid. From a comparison of the SPAI 0 and damped Jacobi smoothers via numerical experiments, we conclude that the parameter free SPAI 0 smoother ....

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W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multi-grid, SIAM J. Matrix Anal. Appl. 21, 2000, pp. 1236-1252.

....we refer to Benzi and Tuma [6] Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3,4] they were shown to be e ective on various dicult elliptic problems on unstructured grids by Tang and Wan [25]. Advantages of sparse approximate inverse smoothers over classical smoothers, such as damped Jacobi, Gauss Seidel, or ILU, are inherent parallelism, possible local adaptivity, and improved robustness. Here we shall consider sparse approximate inverse (SPAI) smoothers based on the SPAI Algorithm ....

....thus minimize kI MAk in the Frobenius norm for the sparsity pattern chosen a priori. This eliminates the search for an e ective sparsity pattern of M , and thus greatly reduces the cost of computing the approximate inverse. The SPAI 1 smoother coincides with the SAI(0,1) smoother of Tang and Wan [25]. For SPAI 0, M = diag(m kk ) is diagonal and can be calculated directly. It is simply given by m kk = a kk ka k k 2 2 ; 1 k n; 14) where a k is the k th row of A note that M is always well de ned if A is nonsingular. In contrast to damped Jacobi, SPAI 0 is parameter free. To ....

W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multigrid, SIAM J. Matrix Anal. Appl. 21, 2000, pp. 1236-1252.

....matrix norm. 6 Several algorithms have been proposed for constructing approximate inverses. These can be loosely categorized first by their result: some algorithms produce a single sparse matrix approximating A 1 e.g. SPAI[22] Chow and Saad s MR method[15, 16] Tang and Wan s local inverse[35] and others produce factored approximations (approximate inverses of the triangular factors) e.g. FSAI[25] AINV[4] The factored form has the advantages of guaranteed non singularity, extra sparsity from good orderings, and apparently more effect per nonzero thanks to its more implicit ....

....linear operator. Actually, a little more is known: the adjoint of the operator (M T # AM 1 # ) T = M T # A T M 1 # may be used in the algorithm as well. This rules out the Frobenius norm minimization algorithms such as SPAI[22] and FSAI[25] as well as Tang and Wan s local inverse method[35], since they all require the ability to access submatrices of M T # AM 1 # . Chow and Saad s MR method[15, 16] is a possibility as it only uses the matrix as an operator. However, the impressive performance[5] of the incomplete inverse 33 factorization algorithms makes them the most ....

[Article contains additional citation context not shown here]

W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multigrid, technical report CAM 98-18, Dept. of Math., University of California at Los Angeles, 1998. 117

....we refer to Benzi and Tuma [6] Approximate inverse techniques are also gaining in importance as smoothers for multigrid methods. First introduced by Benson and Frederickson [3,4] they were shown to be e ective on various dicult elliptic problems on unstructured grids by Tang and Wan [25]. Advantages of sparse approximate inverse smoothers over classical smoothers, such as damped Jacobi, Gauss Seidel, or ILU, are inherent parallelism, possible local adaptivity, and improved robustness. Here we shall consider sparse approximate inverse (SPAI) smoothers based on the SPAI Algorithm ....

....thus minimize kI MAk in the Frobenius norm for the sparsity pattern chosen a priori. This eliminates the search for an e ective sparsity pattern of M , and thus greatly reduces the cost of computing the approximate inverse. The SPAI 1 smoother coincides with the SAI(0,1) smoother of Tang and Wan [25]. For SPAI 0, M = diag(m kk ) is diagonal and can be calculated directly. It is simply given by m kk = a kk ka k k 2 2 ; 1 k n; 14) where a k is the k th row of A note that M is always well de ned if A is nonsingular. In contrast to damped Jacobi, SPAI 0 is parameter free. To ....

W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multigrid, SIAM J. Matrix Anal. Appl. 21, 2000, pp. 1236-1252. 23

.... Huckle [1] 2] Benzi and Tuma [3] or Chow and Saad [4] For a comparative study of these different approaches we refer to [5] Approximate inverse techniques are also gaining in importance as robust and parallel smoothers for Multi grid methods see Benson [6] and more recently Tang and Wan [7]. 2 Description of the Block SPAI Algorithm The original SPAI algorithm computes the preconditioner M explicitly by minimizing kAM Gamma Ik in the Frobenius norm. A reduction in the Frobenius norm results in clustered singular values and eigenvalues, a reduction in the departure from normality ....

W.-P. Tang and W.L. Wan, Sparse Approximate Inverse Smoother for Multi-Grid, UCLA Computational and Applied Mathematics, Tech. Report CAM-98-18, March 1998.

....combining sparsi cation with the use of higher level neighbors was used by All eon et al. 1] who attributes the technique to Cosnuau [16] For approximating the inverse of dense matrices in electromagnetics, however, their tests showed that higher levels were not warranted. Tang and Wan [31] also used a sparsi cation before applying a q local matrix pattern, for q 1, for approximate inverses used as multigrid smoothers. They showed that the sparsi cation does not cause a deterioration in convergence rate for their problems. Both the work by All eon et al. and Tang and Wan ....

....in convergence rate for their problems. Both the work by All eon et al. and Tang and Wan represent the rst uses of PSM patterns. Instead of applying the sparsi cation to A, it is also appropriate in some cases to apply the sparsi cation to the sparse approximate inverse after it has been computed [27, 31]. This is useful to reduce the cost of using the approximate inverse when it is relatively full. 2.2. Insights from adaptive schemes. Adaptive schemes can generate patterns that are very di erent from the pattern of A, for example, the generated patterns can be much sparser than A. Nevertheless, ....

W.-P. Tang and W. L. Wan, Sparse approximate inverse smoother for multi-grid, Tech. Report CAM 98-18, Department of Mathematics, University of California, Los Angeles, CA, 1998.

....is also possible to exploit sparsity pattern of the original matrix to avoid unnecessary computations, see Section 3.2. In addition to be used as global preconditioners, sparse approximate inverses have been used as local components in some other types of global preconditioners, e.g. in multigrid [3, 42] and multi level preconditioners [44] and in block preconditioning methods in general [14, 16, 21] These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41] 3 New Factored Sparse Approximate Inverse In a series of papers [28, ....

W.-P. Tang and W. L. Wan. Sparse approximate inverse smoother for multi-grid. Technical Report CAM 98-18, Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, 1998.

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W. P. Tang and W. L. Wan. Sparse approximate inverse smoother for multi-grid. To appear in SIAM J. Matrix Anal. Appl., 1999.

....on high performance hardware; they are also a valuable general purpose alternative to ILU for tough problems where ILU breaks down from instabilities. Several algorithms for computing sparse approximations to A 1 , or to its inverse triangular factors L 1 and U 1 , have been proposed: e.g. [5, 6, 7, 19, 27, 30, 36]. Unfortunately, for linear systems arising from elliptic PDE s, there appears to be an inherent problem in the explicit nature of these preconditioners, a fundamental conflict between accuracy and sparsity. As problem sizes increase, their performance (either in terms of convergence rate at a ....

....preconditioner then can be thought of as the multigrid like algorithm in figure 4.1. The key di#erence between this and multigrid is that the smoothing is performed in one step, and only at the coarsest level for each variable, instead of being interleaved with restriction and prolongation. See [36] for an example of approximate inverses used as smoothers in multigrid. This is similar to but not exactly the same as additive multigrid, i.e. BPX[10] The hierarchical basis preconditioners (e.g. 1, 40] are very similar to the new preconditioner. In these, the original system is transformed ....

W.-P. Tang and W. L. Wan, A sparse approximate inverse smoother for multigrid, to appear in SIAM J. Matrix Anal. Appl.

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Wei-Pai Tang and W. L. Wan. \Sparse Approximate Inverse Smoother for Multi-Grid." CAM Report 98-18, UCLA, March 1998. 104

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