Structural properties of the graph of augmented sparse approximate inverses

Graph-theoretical properties of an adaptive fill-in strategy for the construction of sparse approximate inverses of sparse matrices are discussed. The approximate inverse is based on minimization of the Frobenius norm and lends itself naturally to a parallel implementation. General sparsity can be exploited in a straightforward fashion. Sparse-graph algorithms for the location of fill-in within each column of the approximate inverse preconditioning are described using level sets. Examples illustrating the approach are presented. Properties of the fill-in strategy for regularly structured sparsity patterns are established