On the Computation of Square Roots and Inverse Square Roots of Gram Matrices for Surface Integral Equations in Electromagnetics

TL;DR

This paper introduces efficient algorithms for computing square roots and inverse square roots of Gram matrices in electromagnetic boundary element methods, aiding spectral analysis of surface integral operators.

Contribution

The paper presents novel algorithms based on function expansions for computing matrix square roots and inverse square roots, improving spectral analysis in electromagnetic boundary element discretizations.

Findings

Algorithms effectively compute matrix square roots and inverse square roots.

Numerical experiments demonstrate the methods' advantages and limitations.

Application to sphere scattering problem confirms spectrum-revealing capabilities.

Abstract

Surface integral equations (SIEs)-based boundary element methods are widely used for analyzing electromagnetic scattering scenarii. However, after discretization of SIEs, the spectrum and eigenvectors of the boundary element matrices are not usually representative of the spectrum and eigenfunctions of the underlying surface integral operators, which can be problematic for methods that rely heavily on spectral properties. To address this issue, we delineate some efficient algorithms that allow for the computation of matrix square roots and inverse square roots of the Gram matrices corresponding to the discretization scheme, which can be used for revealing the spectrum of standard electromagnetic integral operators. The algorithms, which are based on properly chosen expansions of the square root and inverse square root functions, are quite effective when applied to several of the most relevant Gram matrices used for boundary element discretizations in electromagnetics. Tables containing different sets of expansion coefficients are provided along with comparative numerical experiments that evidence advantages and disadvantages of the different approaches. In addition, to demonstrate the spectrum-revealing properties of the proposed techniques, they are applied to the discretization of the problem of scattering by a sphere for which the analytic spectrum is known.