A new iterative method for solving a class of two-by-two block complex linear systems

TL;DR

This paper introduces the ASSS iterative method and preconditioner for efficiently solving linear systems from discretized optimal control problems, guaranteeing convergence and accelerating GMRES.

Contribution

The paper proposes a novel ASSS iterative method and preconditioner with proven convergence bounds for complex block linear systems from PDE discretizations.

Findings

Spectral radius of the method is always less than 1, ensuring convergence.

The ASSS preconditioner significantly speeds up GMRES convergence.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system of linear equations obtained by using finite element discretization of a distributed optimal control problem together with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solving the system. Numerical results are presented to demonstrate the effectiveness of both the ASSS iteration method and the ASSS preconditioner.