A recently proposed Minimum Discarded Fill #MDF # ordering #or pivoting# technique is e#ective in #nding high quality ILU ### preconditioners# especially for problems arising from unstruc# tured #nite element grids. This algorithm can identify anisotropy in complicated physical structures and orders the unknowns in a #preferred# direction. However# the MDF ordering is costly# when # increases. In this paper# several less expensive variants of the MDF technique are explored to produce cost# e#ective ILU preconditioners. The Incomplete MDF and Threshold MDF orderings combine MDF ideas with drop tolerance techniques to identify the sparsity pattern in the ILU preconditioners. These techniques produce orderings that encourage fast decay of the entries in the ILU factorization. The Minimum Update Matrix #MUM # ordering technique is a simpli#cation of the MDF ordering and is an analogue of the minimum degree algorithm. The MUM ordering is especially e#ective for large matrices arising from Navier#Stokes problems. Key Words. minimum discarded #ll#MDF ## incomplete MDF # threshold MDF # minimum up# dating matrix#MUM ## incomplete factorization# matrix ordering# preconditioned conjugate gradient# high#order ILU factorization. AMS#MOS# subject classi#cation. 65F10# 76S05 1.

We describe the basis for a matrix ordering heuristic for improving incomplete factorization for preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from well-known algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners. Keywords: Preconditioned conjugate gradient, preconditioner, matrix ordering, weighted graph Running Title: Weighted Graph Ordering for PCG Methods. AMS Subject Classification: 65F10 This work was supported by by the Natural Sciences and Engineering Research Council of Canada, and by the Information Technology Research Center, which is funded by the Province of Ontario. y Present address Dep...

y work of this author was performed while he was on a sabbatical visit to NERSC.

Methods for solving sparse linear systems of equations can be categorized under two broad classes- direct and iterative. Direct methods are methods based on gaussian elimination. This report discusses one such direct method namely Nested dissection. Nested Dissection, originally proposed by Alan George, is a technique for solving sparse linear systems efficiently. This report is a survey of some of the work in the area of nested dissection and attempts to put it together using a common framework.

. A short note describes the possible dangers of combining (unperturbed) modified incomplete factorizations with certain ordering strategies of the matrix. Sufficient conditions are given for orderings that lead to zero or vanishing pivots, and numerical tests are given illustrating the latter phenomemon. In that case introducing perturbations of O(h 2 ) seems to alleviate the problem. 1. Introduction. It is part of numerical folklore that the combination of redblack ordering and modified incomplete factorizations leads to zero pivots. This fact has been mentioned in, for instance, [16] and [10], and in a larger context it is part of the existence analysis of incomplete factorizations. Conditions for the existence of incomplete factorizations have been derived in [18] (essentially, strict diagonal dominance) , [8] (lower semistrict diagonal dominance), and [19]. In this last reference a necessary and sufficient condition is given for the case of an irreducible, symmetric, M-matrix....

We review current methods for the direct solution of sparse linear equations. We discuss basic concepts such as fill-in, sparsity orderings, indirect addressing and compare general sparse codes with codes for dense systems. We examine methods for greatly increasing the efficiency when the matrix is symmetric positive definite. We consider frontal and multifrontal methods emphasizing how they can take advantage of vectorization, RISC architectures, and parallelism. Some comparisons are made with other techniques and the availability of software for the direct solution of sparse equations is discussed.

Abstract. Combinatorial algorithms have long played a crucial, albeit under-recognized role in scientific computing. This impact ranges well beyond the familiar applications of graph algorithms in sparse matrices to include mesh generation, optimization, computational biology and chemistry, data analysis and parallelization. Trends in science and in computing suggest strongly that the importance of discrete algorithms in computational science will continue to grow. This paper reviews some of these many past successes and highlights emerging areas of promise and opportunity. 1

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In this paper we demonstrate that tradeoffs can be made between parallelism and fill in nested dissection algorithms for Gaussian elimination, both in theory and in practice. We present a new "less parallel nested dissection" algorithm (LPND), and prove that, unlike the standard nested dissection algorithm, when applied to a chordal graph LPND finds a zero-fill elimination order. We have also implemented the LPND algorithm. On a variety of benchmarks it generates less fill than state-of-the-art implementations of the nested dissection (METIS), minimum-degree (AMD), and hybrid (BEND) algorithms on a large body of test matrices. We have also implemented another nested dissection algorithm that is different from METIS and that uses the same separator algorithm used by our implementation of LPND. This algorithm, as well as LPND, generates less fill than METIS, and on large graphs significantly outperforms AMD. The latter comparison is notable, because although it is known that, for certain...

This paper presents an algorithm for finding parallel elimination orders for Gaussian elimination. Viewing a system of equations as a graph, the algorithm can be applied directly to interval graphs and chordal graphs. For general graphs, the algorithm can be used to parallelize the order produced by some other heuristic such as minimum degree. In this case, the algorithm is applied to the chordal completion that the heuristic generates from the input graph. In general, the input to the algorithm is a chordal graph G with n nodes and m edges. The algorithm produces an order with height at most O(log 3 n) times optimal, fill at most O(m), and work at most O(W (G)), where W (G) is the minimum possible work over all elimination orders for G. Experimental results show that when applied after some other heuristic, the increase in work and fill is usually small. In some instances the algorithm obtains an order that is actually better, in terms of work and fill, than the original on...