M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-140, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a decaying property (refer to (5) Let S denote the set of all matrices that have the sparsity pattern of a band matrix B (A 11 ) The formation of a sparse approximate inverse (SPAI) is to nd a band matrix B 11 2 S such that min B2S 11 B Ik F = kA 11 B 11 Ik F : 11 24 Refer to [8, 24, 16]. Brie y as with most SPAI methods, the use of F norm decouples the minimisation into least squares (LS) problems for individual columns c j of B 11 . More precisely, owning 11 B 11 Ik F = k 11 c j e j k 2 ; the j th LS problem is to solve A 11 c j = e j which is not expensive ....

M. W. Benson and P. O. Frederickson P. O. (1982), Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22, pp.127-140.

....to solve the problems without known contact information. The global preconditioner that we implement is the sparse approximate inverse preconditioning technique. The sparse approximate inverse preconditioning technique that we discuss here is based on the idea of least squares approximation [1, 2]. A particular class of sparse approximate inverse preconditioners is constructed based on the Frobenius norm minimization idea. Since we want M to be a good approximation to A , it is ideal if MA I. This approach is to approximate A from the left, and M is called a left preconditioner. It ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-- 140, 1982.

....are based on various Gauss eliminations [18] they are inherently sequential in both the construction and the application phases and are difficult to implement on parallel platforms. The sparse approximate inverse (SAI) preconditioner is an interesting alternative of the ILU type preconditioner [3, 4]. Instead of computing an approximation of A, the SAItype preconditioning techniques compute a sparse approximation of A directly. In this case, the preconditioned system is MAx = M b; 4) where M A . Unlike the ILU type preconditioners, which need triangular solution procedures, the SAI ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-- 140, 1982.

....size for this sphere and because of the rapid decay of the Green s functions, we hope to include the most relevant contributions from A in the approximate matrix A. 2. 1 Frobenius norm minimization methods A natural way to compute an explicit preconditioner is based on Frobenius norm minimization [3, 4, 5, 16]. The idea is to compute the sparse approximate inverse as the matrix M which minimizes #I MA#F (or #I AM#F for right preconditioning) subject to certain sparsity constraints. The Frobenius norm is usually chosen since it allows the decoupling of the constrained minimization problem into n ....

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathematica, 22:127--140, 1982.

....di erent norms like k k 1 ; however, to preserve sparsity of M it is necessary to impose some constraint a priori on its structure, and the minimization is performed relative to a class of matrices having a predetermined sparsity pattern. First references to this method can be found in [5] [6], 7] and [21] Several e orts have been devoted to nding a good pattern that can retain the entries of A 1 having large modulus. For instance, when A is a banded matrix with a good degree of diagonal dominance, a banded approximation to A 1 is justi ed . In particular, when A is a banded ....

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathematica, 22:127-140, 1982.

....for M , the cost of computing M can be greatly reduced. Possible choices include powers of A or A A, as suggested 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 ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Mathematica 22, 1982, pp. 127{ 140.

....directly a sparse approximate inverse of A [5,9,17] For a comparative study of various approximate inverse preconditioners 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 ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Mathematica 22, 1982, pp. 127-140.

....directly a sparse approximate inverse of A [5,9,17] For a comparative study of various approximate inverse preconditioners 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 ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Mathematica 22, 1982, pp. 127-140.

....stems from approximating the Schur complement matrix S using one of several approximate inverse techniques described next. Given an arbitrary matrix A, approximate inverse preconditioners consist of finding an approximation Q to its inverse, by solving approximately the optimization problem [3]: min Q2S kI Gamma AQk 2 F ; in which S is a certain set of n Theta n sparse matrices. This minimization problem can be decoupled into n minimization problems of the form min m j ke j Gamma Am j k 2 2 ; j = 1; 2; n; where e j and m j are the jth columns of the identity matrix ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....This phenomenon of instability of the LU factors was analyzed in particular by Elman [60] This weakness of ILU factorizations, coupled with their sequential nature, spurred researchers to consider radical alternatives. The approximate inverse methods which were first proposed in the late 1970s [15] were in this category. It is only with the advent of massive parallel processing that such methods were considered as serious contenders of the now standard ILU methods [86] A flurry of publications followed this work and the work by Kolotilina and Yeremin [105, 106] To cite just a few, 46, ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....Consider solving the right preconditioned linear system, AMy = b; x = My; 1) where A is large, sparse matrix and M is a right preconditioner. We want to find a sparse matrix M so that kAM Gamma Ik is small in some norm. This approach was first studied by Benson [2] and Benson and Frederickson [3]. More precisely, they minimize the residual matrix, min M kAM Gamma Ik 2 F ; 2) subject to some constraint on the number and position of the nonzero entries of M . The Frobenius norm is particularly useful for parallel implementation. Notice that kAM Gamma Ik 2 F = n X j=1 kAm j ....

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....Advanced Computer Science (RIACS) NASA Ames Research Center. 970 EFFECTIVE SAIP 971 straightforward and can be e#ectively implemented. In addition, several other potential problems with using ILU preconditioners are solved. The idea of using a sparse approximation of A 1 as a preconditioner [3, 4] was proposed only slightly later than ILU was suggested [24] The original crude approach was far less e#ective than the latter, and hence did not gain popularity. To construct a good, but sparse approximation M # A 1 , a key issue is the sparsity pattern of M . Initial approaches failed to ....

M. Benson and P. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

....matrices [7, 19] For these algorithms to be practical, they must provide approximations that are sparse. Block Approximate Inverse Techniques 4 A number of techniques have recently been developed to construct a sparse approximate inverse of a matrix, to be used as a preconditioner [5, 6, 8, 10, 15, 17, 18]. Many of these techniques approximate each row or column independently, focusing on (in the column oriented case) the individual minimizations min x ke j Gamma Axk 2 ; j = 1; 2; n (4) where e j is the j th column of the identity matrix. Such a preconditioner is distinctly easier than ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22 (1982), pp. 127-140.

....for example, various incomplete block factorizations, and have application to improving preconditioners. In this paper, we focus on methods of finding approximate inverses based on minimizing the Frobenius norm of the residual matrix I Gamma AM , first suggested by Benson and Frederickson [5, 6]. We look for a right approximate inverse so that practical variations that use flexible right preconditioning may be used, although left approximate inverses may be better for certain matrices depending on the structure of the inverse. Considering right approximate inverses, we seek to minimize ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22 (1982), pp. 127-140.

....Yet another possible strategy will be mentioned in Section 2.4. 2.2.2 Approximate inverse techniques A second, much cheaper approximation for (1) comes from approximate inverse techniques. Their most common application has been to independently approximate all the rows or columns of an inverse [2, 6, 8, 18, 15] or its factors [1, 19] and use it as a preconditioner. In the column case, for example, this can be done by minimizing the 2 norm of the residual, min x j ke j Gamma Ax j k 2 (5) for each column j of the matrix, where e j is the j th coordinate vector, and where x j is somehow constrained to ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22 (1982), pp. 127-140.

....1 Ax = M 1 b (2) can be more eciently solved by Krylov subspace methods. To this end, there exist many types of sparse preconditioners [1] 11] 15] and [17] see the references therein for details. Of particular interest is the so called sparse approximate inverse (SPAI) preconditioner; see [2], 9] and [18] One diculty associated with SPAI is the determination of a suitable sparse pattern for the preconditioner M 1 that approximates the unknown matrix A 1 . In [5] a wavelet sparse approximate inverse preconditioner (WSPAI) was proposed using the wavelet transform to tackle this ....

....Then an approximate solution will be e y j = e x j ; j ; 0; j : We can see that this approximate solution will be fairly accurate if x represents a smooth function with weak or no singularities at x 1 because for large j, the wavelet coecients will be small. For example, consider x = [8 7 6 5 4 3 2 1] and use the Haar wavelet transform (with 3 levels and bandwidth = 3) We have e x = h 12:7279 5:6569 2 2 0:7071 0:7071 0:7071 0:7071 i ; e y = h 12:7279 5:6569 2 0 0 0 0 0 i ; e c = e x e y = h 0 0 0 2 0:7071 0:7071 0:7071 0:7071 i : In the wavelet basis, e y is an ....

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Benson M. W. and Frederickson P. O. (1982), Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22, pp.127-140.

....For this purpose, two approaches can be followed: an adaptive technique that dynamically tries to identify the best structure for M ; and a static technique, where the pattern of M is prescribed a priori based on some heuristics. Some early references to this latter class can be found in [3, 4, 5, 16] and in [1] for some applications to boundary element matrices in electromagnetism. In addition, when the coecient matrix is dense, the preconditioner should be constructed from a sparse approximation of A in order to reduce the computational cost of the least squares solutions. 3 (a) Example 1 ....

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathematica, 22:127-140, 1982.

....For this purpose, two approaches can be followed: an adaptive technique that dynamically tries to identify the best structure for M ; and a static technique, where the pattern of M is prescribed a priori based on some heuristics. Some early references to this latter class can be found in [4] [5], 6] 15] and in [1] for some applications to boundary element matrices in electromagnetism. In addition, when the coefficient matrix is dense, the preconditioner should be constructed from a sparse approximation of A in order to reduce the computational cost of the least squares solutions. 3 ....

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathematica, 22:127--140, 1982.

....inverse M are calculated by solving the problem minimize M2G S kAM Gamma Ik 2 F j n X j=1 minimize m j 2G j S kAm j Gamma e j k 2 2 ; 2. 1) where m j is the j th column of M and where k Delta k F and k Delta k 2 are, respectively, the Frobenius and two norms (see also Benson and Frederickson, 1982). Thus each of the columns of M may be calculated independently, and, if required, in parallel, by solving the least squares problem minimize m j 2G j S kAm j Gamma e j k 2 2 : 2.2) The differences between the proposals of Cosgrove et al. 1992) and Huckle and Grote (1994) are primarily ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathemtica, 22, 127--140, 1982.

....AM , in particular, the entries specified by S, and ignoring the other entries (see Equation (4) In the least squares method, each equation is from the normal equations for a minimization based on all the entries in AM . Benson s work was not published more widely until Benson and Frederickson [6] generalized the sparsity pattern S to q local operators. In a grid consisting of nodes connected by edges, column i of the matrix of a q local operator consists of nonzeros corresponding to node i and its q th level nearest neighbors. The matrix of a 0 local operator is a diagonal matrix. The ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

....are sparse approximate inverses (SAI) Most sparse approximate inverse (SAI) approaches seek a sparse matrix M h so that the error of the residual is minimized in some measure. One of the earliest approaches was the Frobenius norm approach proposed by Benson [10] and Benson and Frederickson [13]: min M h kA h M h Gamma Ik 2 F ; subject to some constraints on the number and position of the nonzero entries of M h . The minimization problem is equivalent to n independent least squares problems: min m j kA h m j Gamma e j k2 ; j = 1; n; 36) where m j and e j are ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....for general nonsymmetric matrices with a symmetric structure. We also discuss how to tactically add nonzeros to the approximate L and U factors to improve the quality of the ILU(0) preconditioners. Another class of purely algebraic methods is the sparse approximate inverse preconditioners [7, 8, 89]. They construct a sparse approximation to the inverse of a matrix, instead of a sparse approximation to the matrix itself as the ILU factorizations do. They have not received much attention until recently, where their intrinsic parallelism is recognized. Robust algorithms and theory are still ....

....and performance. Wavelet Sparse Approximate Inverses Although scalable ILU is possible, there is still a trade off between parallelism (number of colors) and convergence rate. This motivates the recent interest in sparse approximate inverse preconditioning. Benson [7] and Frederickson [8] obtained a preconditioner M by considering a minimization problem: min M2S kAM Gamma Ik 2 F ; where S is a class of sparse matrices, for instance, matrices with fixed sparsity pattern. The above minimization problem is equivalent to the following system of minimization problems: min m j 2S ....

[Article contains additional citation context not shown here]

M.W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....directed paths are more important, and should be retained in an approximate inverse sparsity pattern. This idea is supported by the decay in the elements observed by Tang [30] in the discrete Green s function for many problems. These sparsity patterns were rst used by Benson and Frederickson [4] in the symmetric case, who also de ned matrices with these patterns to be q local matrices. Given a graph G(A) of a structurally symmetric matrix A with a full diagonal, the structure of the jth column of a q local matrix consists of vertex j and its qth level nearest neighbors in G(A) A ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127-140.

....This phenomenon of instability of the LU factors was analyzed in particular by Elman [56] This weakness of ILU factorizations, coupled with their sequential nature, spurred researchers to consider radical alternatives. The approximate inverse methods which were first proposed in the late 1970s [15] were in this category. It is only with the advent of massive parallel processing that such methods were considered as serious contenders of the now standard ILU methods [81] A flury of publications followed this work and the work by Kolotlina and Yeremin [102, 103] To cite just a few, 41, 91] ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....case for block preconditioners for block tridiagonal matrices [7, 19] For these algorithms to be practical, they must provide approximations that are sparse. A number of techniques have recently been developed to construct a sparse approximate inverse of a matrix, to be used as a preconditioner [5, 6, 8, 10, 15, 17, 18]. Many of these techniques approximate each row or column independently, focusing on (in the column oriented case) the individual minimizations min x ke j Gamma Axk 2 ; j = 1; 2; n (4) where e j is the j th column of the identity matrix. Such a preconditioner is distinctly easier than ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22 (1982), pp. 127-140.

....to its purely algebraic nature. For example, the studies of Grote and Huckle [20] and Chow and Saad [10] show that it is robust for a wide range of matrices in the Harwell Boeing collection. An approach of computing sparse approximate inverse is described by Benson [2] and Benson and Frederickson [3]. Consider the right preconditioned linear system: AMy = b; x = My; 1.2) where M is a right preconditioner. A sparse approximate inverse preconditioner M is defined as a solution to the following minimization problem: min M kAM Gamma Ik 2 F ; 1.3) subject to some constraint on the number ....

M. Benson and P. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

....2 F = n X j=1 ke j Gamma Am j k 2 2 ; where e j denotes the jth column of the identity matrix, the computation of M reduces to solving n independent linear least squares problems, subject to sparsity constraints. This approach was first proposed by Benson [10] Other early papers include [11], 12] and [46] Notice that the above approach produces a right approximate inverse. A left approximate inverse can be computed by solving a constrained minimization problem for Sparse Approximate Inverse Preconditioners 5 kI Gamma MAk F = kI Gamma A T M T k F . This amounts to computing ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Mathematica, 22:127--140, 1982.

....Yet another possible strategy will be mentioned in Section 2.4. 2.2.2 Approximate inverse techniques A second, much cheaper approximation for (1) comes from approximate inverse techniques. Their most common application has been to independently approximate all the rows or columns of an inverse [2, 5, 7, 15, 12] or its factors [1, 16] and use it as a preconditioner. In the column case, for example, this can be done by minimizing the 2 norm of the residual, min x j ke j Gamma Ax j k 2 (5) for each column j of the matrix, where e j is the j th coordinate vector, and where x j is somehow constrained to ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22 (1982), pp. 127-140.

....cost, application cost, robustness, efficiency, etc. A comprehensive survey and comparison of several existing sparse approximate inverse techniques can be found in [7] The most popular approach to constructing sparse approximate inverse is based on the idea of Frobenius norm minimization [5, 15, 22]. The sparse approximate inverse is computed as the matrix M which minimizes kI Gamma AMkF , subject to certain sparsity constraint. The Frobenius norm is usually chosen as it can be computed conveniently and there are analyses to show that the approximate inverses resulting from the Frobenius ....

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

....for incomplete block factorizations with large sparse blocks, as well as several other applications, also described later. We focus on methods of finding approximate inverses based on minimizing the Frobenius norm of the residual matrix I Gamma AM , first suggested by Benson and Frederickson [5, 6]. Consider the minimization of F (M) kI Gamma AMk 2 F : 1.3) to seek a right approximate inverse. An important feature of this objective function is that it can be decoupled as the sum of the squares of the 2 norms of the individual columns of the residual matrix I Gamma AM F (M) kI ....

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

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M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-140, 1982.

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M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

No context found.

M. W. Benson and P. O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems, Utilitas Math., 22 (1982), pp. 127--140.

No context found.

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

No context found.

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

No context found.

M. W. Benson and P.O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127-140, 1982.

No context found.

M. W. Benson and P. O. Frederickson. Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems. Utilitas Math., 22:127--140, 1982.

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