Factorized sparse approximate inverse preconditioning for singular M-matrices

TL;DR

This paper studies the application of the factorized sparse approximate inverse (FSAI) preconditioner to singular M-matrices, revealing conditions for stability and properties of the preconditioned system relevant to Markov chains and graph Laplacians.

Contribution

It establishes restrictions on the nonzero pattern for stable FSAI construction on singular M-matrices and proves key properties of the resulting preconditioner.

Findings

FSAI is well-defined under certain pattern restrictions.

Preconditioned matrix remains a singular M-matrix with positive diagonal entries.

Generated triangular matrices are non-singular and non-negative.

Abstract

Here we consider the factorized sparse approximate inverse (FSAI) preconditioner. We apply the FSAI preconditioner to singular irreducible M-matrices. These matrices arise e.g. in discrete Markov chain modeling or as graph Laplacians. We show, that there are some restrictions on the nonzero pattern needed for a stable construction of the FSAI preconditioner in this case. With these restrictions FSAI is well-defined. Moreover, we proved that the FSAI preconditioner shares some important properties with the original system. The lower triangular matrix $L_G$ and the upper triangular matrix $U_G$, generated by FSAI, are non-singular and non-negative. The diagonal entries of $L_GAU_G$ are positive and $L_GAU_G$, the preconditioned matrix, is a singular M-matrix. Even more, we establish that a (1,2)-inverse is computed for the complete nonzero patter.