Adaptive GMRES(m) for the Electromagnetic Scattering Problem

Gustavo Espínola,Christian Schaerer,Juan C Cabral

doi:10.5540/tema.2020.021.01.191

Gustavo Espínola, Christian Schaerer + Show 1 more

Open Access

https://doi.org/10.5540/tema.2020.021.01.191

Copy DOI

Abstract

In this article, an adaptive version of the restarted GMRES (GMRES(m)) is introduced for the resolution of the ﬁnite difference approximation of the Helmholtz equation. It has been observed that the choice of the restart parameter m strongly affects the convergence of standard GMRES(m). To overcome this problem, the GMRES(m) is formulated as a control problem in order to adaptively combine two strategies: a) the appropriate variation of the restarted parameter m, if a stagnation in the convergence is detected; and b) the augmentation of the search subspace using vectors obtained at previous cycles. The proposal is compared with similar iterative methods of the literature based on standard GMRES(m) with ﬁxed parameters. Numerical results for selected matrices suggest that the switching adaptive proposal method could overcome the stagnation observed in standard methods, and even improve the performance in terms of computational time and memory requirements.

Highlights

This article deals with the numerical resolution of a Helmholtz scattering problem by using an adaptive version of the restarted Generalized Minimal Residual Method (GMRES)
The resolution of the Helmholtz scattering equation using iterative methods is difficult since the problem is ill-posed for a set of frequencies that physically corresponds to the resonance modes of the domain to be solved, the discretization grid has to be refined as a function of the frequency of the operator, and the oscillatory and non-local structure of the solution affects the numerical methods [10]
The standard incomplete LU is ineffective, and in some cases, it does not assure a better rate of convergence [19] and when it is used with GMRES, the performance deteriorates as the wavenumber becomes larger [10]

Summary

INTRODUCTION

In problems with stagnation, according to remark (d) of LGMRES, the error approximation vectors do not help to improve the rate of convergence, i.e., φ j−1 = u j−1 − u j−2 ≈ 0 These errors vector are discarded and only the harmonic Ritz vectors are maintained to enrich the search subspace. As can be seen in the numerical results of the LGMRES-E, keeping constant m and enriching the search subspace, it does not necessarily avoid stagnation To solve this problem a controller to augment the size of the search subspace is proposed for enlarging the Arnoldi basis, since decreasing the restart parameter does not contribute to an improvement in the convergence [4]. The pseudocode for the j-th cycle of the proposed method denoted as A-LGMRES-E(m, d, l) is presented in the Algorithm 1

AN ELECTROMAGNETIC CAVITY PROBLEM

NUMERICAL EXPERIMENTS

Methods

CONCLUSION