On the Fast Direct Solution of a Preconditioned Electromagnetic Integral Equation

TL;DR

This paper introduces a fast direct solver for high-frequency electromagnetic integral equations using a preconditioned combined field formulation, regularization, and low-rank approximations, enabling efficient solutions for multiple right-hand sides.

Contribution

It develops a novel scheme combining preconditioning, regularization, and low-rank techniques to efficiently solve electromagnetic integral equations in high-frequency regimes.

Findings

The method achieves favorable computational complexity.

Numerical results confirm the scheme's accuracy and efficiency.

The approach is practical for multiple right-hand side problems.

Abstract

This work presents a fast direct solver strategy for electromagnetic integral equations in the high-frequency regime. The new scheme relies on a suitably preconditioned combined field formulation and results in a single skeleton form plus identity equation. This is obtained after a regularization of the elliptic spectrum through the extraction of a suitably chosen equivalent circulant problem. The inverse of the system matrix is then obtained by leveraging the Woodbury matrix identity, the low-rank representation of the extracted part of the operator, and fast circulant algebra yielding a scheme with a favorable complexity and suitable for the solution of multiple right-hand sides. Theoretical considerations are accompanied by numerical results both of which are confirming and showing the practical relevance of the newly developed scheme.