Solution of Sparse Linear Systems

We consider the solution of a linear system Ax = b on a distributed memory machine when the matrix A is large, sparse and symmetric positive deenite. In a previous paper we developed an algorithm to compute a ll-reducing nested dissection ordering of A on a distributed memory machine. We n o w develop algorithms for the remaining steps of the solution process. The large-grain task parallelism resulting from sparsity is identiied by a tree of separators available from nested dissection. Our parallel algorithms use this separator tree to estimate the structure of the Cholesky factor L and to organize numeric computations as a sequence of dense matrix operations. We present results of an implementation o n a n I n tel iPSCC860 parallel computer. An an alternative to estimating the structure of L using the separator tree, we develop an algorithm to compute the elimination tree on a distributed memory machine. Our algorithm uses the separator tree to achieve better time and space complexity than earlier work. 1. Introduction and Overview. Consider the solution of a system of linear equations Ax = b, where A is an N N, symmetric positive deenite matrix. Direct solution requires a Cholesky decomposition A = LL T , where L is lower triangular. This step is followed by solution of the triangular systems Ly = b and L T x = y. When the matrix A is sparse, the numeric steps are preceded by a symbolic phase in which a symmetric permutation is applied to the rows and columns of A. The purpose of reordering the system is to ensure that the factor L suuers low ll-in, i.e., only a small number of zero values in A become nonzero during factorization. Various serial algorithms such as the minimum degree heuristic and automatic nested dissection 5 produce appropriate orderings of the matrix A by manipulating its graph GA. As a consequence of sparsity, during numeric factorization a speciic column is updated only by columns in a subset of lower numbered columns instead of all lower numbered columns. Such column dependencies, implicit from the ordering, are represented using a tree structure known as an elimination tree. The elimination tree is computed from the structure of A and the ordering, and is used in turn to compute the exact structure of the factor L and to allocate storage to complete the symbolic phase. In the numeric …