## On computing inverse entries of a sparse matrix in an out-of-core environment

### Patrick R. Amestoy, Iain S. Duff, Jean-Yves L'Excellent, Yves Robert, François-Henry Rouet, and Bora Uçar

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 out-of-core storage, we show how to compute such a subset efficiently, 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 volume of communication (data transferred) between disk and main memory. This leads to a partitioning problem that we prove is NP-complete. 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 compare the proposed algorithms and illustrate the performance of our algorithms in practice using the \textttMUMPS software package on a set of real-life problems.
Read More: http://epubs.siam.org/doi/abs/10.1137/100799411

Key words. Sparse matrices, direct methods for linear systems and matrix inversion, multifrontal method, graphs and hypergraphs.