Anderson Accelerated PMHSS for Complex-Symmetric Linear Systems

TL;DR

This paper introduces an Anderson Accelerated PMHSS method tailored for complex-symmetric linear systems, demonstrating improved performance and robustness over existing methods, especially in electromagnetics applications.

Contribution

The paper develops a novel Anderson Accelerated PMHSS method that outperforms traditional preconditioned GMRES in solving complex-symmetric systems, with convergence independent of discretization size.

Findings

AA-PMHSS shows better robustness than existing methods.

The method has faster preconditioner evaluation.

Convergence rate is independent of discretization size.

Abstract

This paper presents the design and development of an Anderson Accelerated Preconditioned Modified Hermitian and Skew-Hermitian Splitting (AA-PMHSS) method for solving complex-symmetric linear systems with application to electromagnetics problems, such as wave scattering and eddy currents. While it has been shown that the Anderson Acceleration of real linear systems is essentially equivalent to GMRES, we show here that the formulation using Anderson acceleration leads to a more performant method. We show relatively good robustness compared to existing preconditioned GMRES methods and significantly better performance due to the faster evaluation of the preconditioner. In particular, AA-PMHSS can be applied to solve problems and equations arising from electromagnetics, such as time-harmonic eddy current simulations discretized with the Finite Element Method. We also evaluate three test systems present in previous literature. We show that the method is competitive with two types of preconditioned GMRES. One of the significant advantages of these methods is that the convergence rate is independent of the discretization size.