TriCG with deflated restarting for symmetric quasi-definite linear systems

TL;DR

This paper introduces a deflation restarting strategy for the TriCG iterative method to enhance convergence when solving symmetric quasi-definite linear systems, especially with challenging off-diagonal blocks.

Contribution

The paper develops a novel deflation strategy and a TriCG with deflated restarting, improving convergence and efficiency for SQD systems with large elliptic singular values.

Findings

Enhanced convergence of TriCG with deflated restarting.

Superior performance demonstrated through numerical experiments.

Effective handling of multiple right-hand sides.

Abstract

TriCG is a short-recurrence iterative method recently introduced by Montoison and Orban [SIAM J. Sci. Comput., 43 (2021), pp. A2502--A2525] for solving symmetric quasi-definite (SQD) linear systems. TriCG takes advantage of the inherent block structure of SQD linear systems and performs substantially better than SYMMLQ. However, numerical experiments have revealed that the convergence of TriCG can be notably slow when the off-diagonal block contains a substantial number of large elliptic singular values. To address this limitation, we introduce a deflation strategy tailored for TriCG to improve its convergence behavior. Specifically, we develop a generalized Saunders--Simon--Yip process with deflated restarting to construct the deflation subspaces. Building upon this process, we propose a novel method termed TriCG with deflated restarting. The deflation subspaces can also be utilized to solve SQD linear systems with multiple right-hand sides. Numerical experiments are provided to illustrate the superior performance of the proposed methods.