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On computing inverse entries of a sparse matrix in an outofcore environment
, 2010
"... Abstract. The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in outofcore ..."
Abstract

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Abstract. The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in outofcore storage, we show how to compute such a subset e ciently, by accessing only parts of the factors. When there are many inverse entries to compute, we need to guarantee that the overall computation scheme has reasonable memory requirements, while minimizing the cost of loading the factors. This leads to a partitioning problem that we prove is NPcomplete. We also show that we cannot get a close approximation to the optimal solution in polynomial time. We thus need to develop heuristic algorithms, and we propose: (i) a lower bound on the cost of an optimum solution; (ii) an exact algorithm for a particular case; (iii) two other heuristics for a more general case; and (iv) hypergraph partitioning models for the most general setting. We illustrate the performance of our algorithms in practice using the MUMPS software package on a set of reallife problems as well as some standard test matrices. We show that our techniques can improve the execution time by a factor of 50. Key words. Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs. AMS subject classi cations. 05C50, 05C65, 65F05, 65F50 1. Introduction. We
Team HiePACS HighEnd Parallel Algorithms for Challenging Numerical Simulations
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