Sparse approximate inverses and target matrices

Abstract

If P has a prescribed sparsity and minimizes the Frobenius norm ||I-PA||F it is called a sparse approximate inverse of A. It is well known that the computation of such a matrix P is via the solution of independent linear least squares problems for the rows separately (and therefore in parallel). In this paper we consider the choice of other norms, and introduce the idea of `target' matrices. A target matrix, T, is readily inverted and thus forms part of a preconditioner when ||T-PA|| is minimized over some appropriate sparse matrices P. The use of alternatives to the Frobenius norm which maintain parallelizability whilst discussed in early literature does not appear to have been exploited.

This work was supported by the Engineering and Physical Sciences Research council and NAG Ltd.