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TriCG with deflated restarting for symmetric quasi-definite linear systems

Kui Du, Jia-Jun Fan

TL;DR

This work addresses solving symmetric quasi-definite (SQD) linear systems by enhancing TriCG with deflation to alleviate the slow convergence caused by large elliptic singular values in the off-diagonal block. It introduces gSSY-DR, a deflated restarting variant of the generalized Saunders--Simon--Yip tridiagonalization, to compute approximate elliptic triplets, and integrates these into two new methods: TriCG-DR for single RHS and D-TriCG for multiple RHS. The authors provide a recovery formula to obtain the true solution from the deflated system and establish residual bounds when the deflation is inexact, supported by extensive numerical experiments showing substantial acceleration on synthetic and real problems, including Stokes discretizations. The combined TriCG-DR+D-TriCG framework demonstrates significant iteration- and runtime-reductions, and the work outlines future directions for deflation in TriMR and recycling strategies for broader applicability.

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.

TriCG with deflated restarting for symmetric quasi-definite linear systems

TL;DR

This work addresses solving symmetric quasi-definite (SQD) linear systems by enhancing TriCG with deflation to alleviate the slow convergence caused by large elliptic singular values in the off-diagonal block. It introduces gSSY-DR, a deflated restarting variant of the generalized Saunders--Simon--Yip tridiagonalization, to compute approximate elliptic triplets, and integrates these into two new methods: TriCG-DR for single RHS and D-TriCG for multiple RHS. The authors provide a recovery formula to obtain the true solution from the deflated system and establish residual bounds when the deflation is inexact, supported by extensive numerical experiments showing substantial acceleration on synthetic and real problems, including Stokes discretizations. The combined TriCG-DR+D-TriCG framework demonstrates significant iteration- and runtime-reductions, and the work outlines future directions for deflation in TriMR and recycling strategies for broader applicability.

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.
Paper Structure (8 sections, 4 theorems, 107 equations, 4 figures, 2 tables, 3 algorithms)

This paper contains 8 sections, 4 theorems, 107 equations, 4 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The gSSY tridiagonalization process and TriCG
  3. Deflation of elliptic singular values
  4. A gSSY process with deflated restarting
  5. TriCG with deflated restarting
  6. Multiple right-hand sides
  7. Numerical experiments
  8. Concluding remarks and future work

Key Result

theorem 1

Let $\widetilde{\mathbf{u}}$ be a solution of the deflated system eq:deflated. Then, the solution of the system eq:problem is given by

Figures (4)

  • Figure 1: The convergence histories of TriCG and TriCG-DR under varying dimensionality of deflation subspaces.
  • Figure 2: The convergence histories of TriCG and TriCG-DR on the problems gupta3, g7jac060sc, rajat27, and TSOPF_RS_b300_c2.
  • Figure 3: The convergence histories of TriCG and D-TriCG with deflation subspaces of different dimensions.
  • Figure 4: The convergence histories of TriCG and TriCG-DR+D-TriCG for 10 right-hand sides on the problem channel_domain.

Theorems & Definitions (9)

  • theorem 1
  • proof
  • proposition 2
  • proof
  • theorem 3
  • proof
  • remark 4
  • theorem 5
  • proof