M. Benzi and M. Tuma. Orderings for factorized sparse approximate inverse preconditioners. SIAM J. Sci. Comput., 21(5):1851--1868, Sept. 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Z i # Z i , drop u i Z j ;# 37 cannot account for sparsity due to the dropping of small elements, but the outer product form automatically does, this version often is more efficient [9] 3.8 Ordering Before using AINV, one more thing must be considered: the ordering of the nodes. In [6, 10] it was made clear that ordering has a significant effect on the construction time of the approximate inverse, and on the convergence of the preconditioned system. For fairly isotropic problems, the heuristic of inverse factor fill reduction has proven to be very effective; ordering algorithms ....

M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners. To appear in SIAM J. of Sci. Comput. (Revised version of Los Alamos National Laboratory Technical Report LA-UR-98-2175, May 1998)

....jj Z j , # . Return W, Z, and D. 37 cannot account for sparsity due to the dropping of small elements, but the outer product form automatically does, this version often is more efficient [9] 3.8 Ordering Before using AINV, one more thing must be considered: the ordering of the nodes. In [6, 10] it was made clear that ordering has a significant effect on the construction time of the approximate inverse, and on the convergence of the preconditioned system. For fairly isotropic problems, the heuristic of inverse factor fill reduction has proven to be very effective; ordering algorithms ....

M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners. To appear in SIAM J. of Sci. Comput. (Revised version of Los Alamos National Laboratory Technical Report LA-UR-98-2175, May 1998)

....it is guaranteed to produce a positive definite preconditioner, though breakdown is possible in the general case, and has generally been shown to be very robust[7] One important issue for factored approximate inverses is the ordering of the rows and columns of the matrix. As demonstrated in [8, 12, 13], performance can be significantly improved by an appropriate reordering e.g. nested dissection (we use the Metis routine[29] On the other hand, one might argue that if the multiresolution bases here are constructed correctly, the transformed A will be well enough conditioned that ordering ....

M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), no. 5, pp. 1851-1868.

....of inner products from the pivot calculation W T i AZ i . Fortunately, many of these inner products can be avoided. We begin by considering those inner products which are zero even without small entries dropped in the algorithm, i.e. when the true inverse factors are computed. Because the 3 In [3, 7] other orderings were considered for AINV, but as nested dissection is generally close to best in convergence, often best in construction time, and most easily parallelized, this article sticks with just nested dissection. Results for other inverse factor fill reducing orderings are similar. ....

M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners. To appear in SIAM J. of Sci. Comput. (Revised version of Los Alamos National Laboratory Technical Report LA-UR-98-2175, May 1998)

....W are unit upper triangular and D 1 is diagonal. However, the purely structural results presented in section 2 apply equally to other factored approximate inverse schemes. Whether the numerical results carry over is still to be determined. For example, conflicting evidence has been presented in [5] and [16] about the e#ect on FSAI [22] which perhaps will be resolved only when the issue of sparsity pattern selection for FSAI has been settled. Some preliminary work in studying the e#ect of ordering on the performance of AINV has shown promising results [3] A more recent work by the same ....

....about the e#ect on FSAI [22] which perhaps will be resolved only when the issue of sparsity pattern selection for FSAI has been settled. Some preliminary work in studying the e#ect of ordering on the performance of AINV has shown promising results [3] A more recent work by the same authors is [5]. We carry this research forward in sections 2 and 3, realizing significant improvements in the speed of preconditioner computation and observing some beneficial e#ects on convergence but noting that structural information alone is not always enough. We then turn our attention to anisotropic ....

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M. Benzi and M. T uma, Orderings for factorized sparse approximate inverse preconditioners, revised version of Los Alamos National Laboratory Technical Report LA-UR-98-2175, 1998; SIAM J. Sci. Comput., to appear.

....orderings, such as Nested Dissection and Minimum Degree. These orderings are beneficial in that they result in smaller time and space requirements for forming and storing the preconditioner, while at the same time improving the quality of the preconditioner in a significant number of cases; see [10], 13] 20] In order to describe the procedure, let a T i denote the ith row of A. Also, let e i denote the ith unit basis vector. The basic A conjugation procedure can be written as follows. Algorithm 2.1. A orthogonalization algorithm, I (1) Let z (0) i = e i (1 i n) 2) For i = 1; 2; ....

M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., to appear.

....nonsymmetric (one sided) and symmetric (fill reducing) permutations. The effect of symmetric reorderings on preconditioned iterative methods has been considered by many authors before, but mostly on problems which are either symmetric [28] or at least structurally symmetric or nearly so [7] [12], 15] On the other hand, the effect of nonsymmetric permutations has received little attention so far, and the same is true for the use of symmetric permutations on matrices that are highly nonsymmetric, structurally as well as numerically. Note that nonsymmetric permutations, unlike symmetric ....

....not an issue here. Like for ILU, and in contrast to SPAI, the performance of approximate inverse preconditioners in factorized form is sensitive to the ordering of the matrix. For structurally symmetric (or nearly so) matrices having a stable AINV preconditioner, it was shown in [15] and [12] that symmetric reorderings that reduce fill in in the inverse factors, like minimum degree or (generalized) nested dissection, are beneficial in that they tend to reduce the time and storage required to compute the preconditioner while at the same time improving the quality of the preconditioner ....

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M. Benzi and M. Tuma, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), to appear.

....in faster convergence on average. Being a factorized preconditioner, AINV can be used with the conjugate gradient (CG) method for solving symmetric positive definite (SPD) problems, unlike SPAI. Also, sparse matrix reorderings can be used to increase the efficiency of the preconditioner; see [7] [5]. In contrast to other factorized sparse approximate inverse preconditioners, like the FSAI method [18] it is not necessary to prescribe the sparsity pattern of the preconditioner factors in advance. Indeed, significant entries in the inverse factors are automatically captured by means of a drop ....

....of AINV is that, as it stands, the (bi)conjugation process is highly sequential. However, it is possible to exploit certain sparse matrix reorderings, such as those induced by graph partitioning, to introduce parallelism in the preconditioner construction phase. This idea was first put forth in [5]; the remaining of the present paper is devoted to investigating and testing such idea. 3 Graph partitioning Graph partitioning has become a universal tool in parallel computing, where it is routinely used for partitioning a problem among processors in a parallel environment. Here we will ....

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Benzi, M. and Tuma, M., Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., to appear.

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M. Benzi and M. Tuma. Orderings for factorized sparse approximate inverse preconditioners. SIAM J. Sci. Comput., 21(5):1851--1868, Sept. 2000.

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