Approximate sparsity patterns for the inverse of a matrix and preconditioning

Abstract

We consider a general sparse matrix A. Computing a sparse approximate inverse matrix M by minimizing ∥ AM− E∥ in the Frobenius norm is very useful for deriving preconditioners in iterative solvers, especially in a parallel environment. The problems, that appear in this connection in a distributed memory setting, are the distribution of the data—mainly submatrices of A—on the different processors. An a-priori knowledge of the data that has to be sent to a processor would be very helpful in this connection in order to reduce the communication time. In this paper, we compare different strategies for choosing a-priori an approximate sparsity structure of A −1. Such a sparsity pattern can be used as a maximum pattern of allowed entries for the sparse approximate inverse M. Then, with this maximum pattern we are able to distribute the claimed data in one step to each processor. Therefore, communication is necessary only at the beginning and at the end of a subprocess. Using the characteristic polynomials and the Neumann series associated with A and A T A, we develop heuristic methods to find good and sparse approximate patterns for A −1. Furthermore, we exactly determine the submatrices that are used in the SPAI algorithm to compute one new column of the sparse approximate inverse. Hence, it is possible to predict in advance an upper bound for the data that each processor will need. Based on numerical examples we compare the different methods with regard to the quality of the resulting approximation M.