Ideal Preconditioners for Saddle Point Systems with a Rank-Deficient Leading Block

TL;DR

This paper develops and analyzes preconditioners for symmetric saddle point systems with rank-deficient leading blocks, achieving eigenvalue clustering and robustness under various rank conditions.

Contribution

It introduces two preconditioners that produce a constant eigenvalue count and a third that remains effective without strict rank assumptions.

Findings

Preconditioners achieve eigenvalue clustering with a constant number of eigenvalues.

The third preconditioner is robust even when rank assumptions are relaxed.

The inverse properties of certain saddle point systems are characterized.

Abstract

We consider the iterative solution of symmetric saddle point systems with a rank-deficient leading block. We develop two preconditioners that, under certain assumptions on the rank structure of the system, yield a preconditioned matrix with a constant number of eigenvalues. We then derive some properties of the inverse of a particular class of saddle point system and exploit these to develop a third preconditioner, which remains ideal even when the earlier assumptions on rank structure are relaxed.