Convergence Acceleration of Preconditioned CG Solver Based on Error Vector Sampling for a Sequence of Linear Systems

TL;DR

This paper introduces a novel algebraic auxiliary matrix construction method based on error vector sampling to accelerate the convergence of preconditioned conjugate gradient solvers for sequences of similar linear systems.

Contribution

It presents a new technique for efficiently identifying eigenvectors with small eigenvalues to enhance convergence acceleration methods.

Findings

Auxiliary matrix improves convergence speed in numerical tests.

Method effectively identifies small eigenvalue eigenspaces.

Application to condition number estimation is feasible.

Abstract

In this paper, we focus on solving a sequence of linear systems with an identical (or similar) coefficient matrix. For this type of problems, we investigate the subspace correction and deflation methods, which use an auxiliary matrix (subspace) to accelerate the convergence of the iterative method. In practical simulations, these acceleration methods typically work well when the range of the auxiliary matrix contains eigenspaces corresponding to small eigenvalues of the coefficient matrix. We have developed a new algebraic auxiliary matrix construction method based on error vector sampling, in which eigenvectors with small eigenvalues are efficiently identified in a solution process. The generated auxiliary matrix is used for the convergence acceleration in the following solution step. Numerical tests confirm that both subspace correction and deflation methods with the auxiliary matrix can accelerate the solution process of the iterative solver. Furthermore, we examine the applicability of our technique to the estimation of the condition number of the coefficient matrix. The algorithm of preconditioned conjugate gradient (PCG) method with the condition number estimation is also shown.