Factorized Sparse Approximate Inverse Preconditionings I. Theory

Abstract

This paper considers construction and properties of factorized sparse approximate inverse preconditionings well suited for implementation on modern parallel computers. In the symmetric case such preconditionings have the form $A \to G_L AG_L^T $, where $G_L $ is a sparse approximation based on minimizing the Frobenius form $\| I - G_L L_A \|_F $ to the inverse of the lower triangular Cholesky factor $L_A $ of A, which is not assumed to be known explicitly. These preconditionings preserve symmetry and/or positive definiteness of the original matrix and, in the case of M-, H-, or block H-matrices, lead to convergent splittings.