Calder\'on preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles

TL;DR

This paper investigates Calderón preconditioning for boundary integral equations in electromagnetic scattering, finding it effective mainly for multi-particle cases and offering computational savings with block-diagonal approaches.

Contribution

The study provides an experimental comparison of Calderón preconditioning versus mass-matrix preconditioning for dielectric particle scattering, highlighting when each method is most effective.

Findings

Calderón preconditioning often outperforms mass-matrix preconditioning for multi-particle scattering.

Block-diagonal Calderón preconditioning reduces computational cost significantly.

Performance varies depending on particle geometry, wavenumber, and refractive index.

Abstract

We consider the simulation of electromagnetic scattering by single and multiple isotropic homogeneous dielectric particles using boundary integral equations. Galerkin discretizations of the classical Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) boundary integral equation formulation provide accurate solutions for complex particle geometries, but are well-known to lead to ill-conditioned linear systems. In this paper we carry out an experimental investigation into the performance of Calder\'on preconditioning techniques for single and multiple absorbing obstacles, which involve a squaring of the PMCHWT operator to produce a well-conditioned second-kind formulation. For single-particle scattering configurations we find that Calder\'on preconditioning is actually often outperformed by simple "mass-matrix" preconditioning, i.e. working with the strong form of the discretized PMCHWT operator. In the case of scattering by multiple particles we find that a significant saving in computational cost can be obtained by performing block-diagonal Calder\'on preconditioning in which only the self-interaction blocks are preconditioned. Using the boundary element software library Bempp (www.bempp.com) the numerical performance of the different methods is compared for a range of wavenumbers, particle geometries and complex refractive indices relevant to the scattering of light by atmospheric ice crystals.