J. A. George has discovered a method, called nested dissection, for solving a system of linear equations defined on an n = k $\times$ k square grid in O(n log n) space and O($n{3/2}$) time. We generalize this method without degrading the time and space bounds so that it applies to any system of equations defined on a planar or almost-planar graph. Such systems arise in the solution of two-dimensional finite element problems. Our method uses the fact that planar graphs have good separators. More generally, we show that sparse Gaussian elimination is efficient for any class of graphs which have good separators, and conversely that graphs without good separators (including almost all sparse graphs) are not amenable to sparse Gaussian elimination.

Generalized nested dissection

In this paper, the role of elimination trees in the direct solution of large sparse linear systems is examined. The notion of elimination trees is described and its relation to sparse Cholesky factorization is discussed. The use of elimination trees in the various phases of direct factorization are surveyed: in reordering, sparse storage schemes, symbolic factorization, numeric factorization, and different computing environments.

The role of elimination trees in sparse factorization

This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementatjon techniques. The use of the method in different computational environments is also described.

The multifrontal method for sparse matrix solution: theory and practice

Over the past fifteen years, the implementation of the minimum degree algorithm has received much study, and many important enhancements have been made to it. This paper describes these various enhancements, their historical development, and some experiments showing how very effective they are in improving the execution time of the algorithm. A shortcoming is also presented that exists in all of the widely used implementations of the algorithm, namely, that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering. For example, changing the input ordering can lead to an increase (or decrease) of as much as a factor of three in the cost of the subsequent numerical factorization. This sensitivity is caused by the lack of an effective tie-breaking strategy, and the authors’ experiments illustrate the importance of developing such a strategy

The evolution of the minimum degree ordering algorithm

The most widely used ordering scheme to reduce fills and operations in sparse matrix computation is the minimum-degree algorithm. The notion of multiple elimination is introduced here as a modification to the conventional scheme. The motivation is discussed using the k-by-k grid model problem. Experimental results indicate that the modified version retains the fill-reducing property of (and is often better than) the original ordering algorithm and yet requires less computer time. The reduction in ordering time is problem dependent, and for some problems the modified algorithm can run a few times faster than existing implementations of the minimum-degree algorithm. The use of external degree in the algorithm is also introduced.

Modification of the minimum-degree algorithm by multiple elimination

Publisher Summary This chapter discusses the applications of an element model for Gaussian elimination. The system of linear equations is considered A x = b, where A is an N × N sparse symmetric positive definite matrix such as those that arise in finite difference and finite element approximations to elliptic boundary value problems in two and three dimensions. A classic method for solving such systems is Gaussian elimination. The chapter presents a method by which kth equation is used to eliminate the kth variable from the remaining N − k equations for k = 1, 2, …, N − 1 and then back-solve the resulting upper triangular system for the unknown vector x. The chapter reviews the graph-theoretic elimination model and highlights a new element model of the elimination process. The model is used to give simple proofs of inherent lower bounds for the work and storage associated with Gaussian elimination. It also explains the way by which element model, combined with the unusual idea of recomputing rather than saving the factorization, leads to a minimal storage sparse elimination algorithm that requires significantly less storage than regular sparse elimination for the five- or nine-point.

Applications of an Element Model for Gaussian Elimination

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Applications of an Element Model for Gaussian Elimination