Fourier analysis of the CGMN method for solving the Helmholtz equation

TL;DR

This paper uses Fourier analysis to study the convergence of the CGMN iterative method for solving the Helmholtz equation, providing insights into optimal parameters and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a Fourier analysis framework for the CGMN method applied to the Helmholtz equation and identifies optimal relaxation parameters for improved convergence.

Findings

Fourier analysis yields an expression for the iteration matrix.

Optimal relaxation parameters are identified for the method.

Numerical experiments confirm the theoretical analysis.

Abstract

The Helmholtz equation arises in many applications, such as seismic and medical imaging. These application are characterized by the need to propagate many wavelengths through an inhomogeneous medium. The typical size of the problems in 3D applications precludes the use of direct factorization to solve the equation and hence iterative methods are used in practice. For higher wavenumbers, the system becomes increasingly indefinite and thus good preconditioners need to be constructed. In this note we consider an accelerated Kazcmarz method (CGMN) and present an expression for the resulting iteration matrix. This iteration matrix can be used to analyze the convergence of the CGMN method. In particular, we present a Fourier analysis for the method applied to the 1D Helmholtz equation. This analysis suggests an optimal choice of the relaxation parameter. Finally, we present some numerical experiments.