Sparsifying preconditioner for the time-harmonic Maxwell's equations

TL;DR

This paper introduces a sparsifying preconditioner for the time-harmonic Maxwell's equations that reduces computational costs by approximating dense systems with sparse ones, maintaining efficiency across frequencies.

Contribution

It generalizes the sparsifying preconditioner from scalar wave equations to vector Maxwell's equations, enabling efficient iterative solutions.

Findings

Preconditioner keeps iteration count low and frequency-independent.

Enables efficient solution of dense integral systems.

Bridges dense integral equations and sparse PDE discretizations.

Abstract

This paper presents the sparsifying preconditioner for the time-harmonic Maxwell's equations in the integral formulation. Following the work on sparsifying preconditioner for the Lippmann-Schwinger equation, this paper generalizes that approach from the scalar wave case to the vector case. The key idea is to construct a sparse approximation to the dense system by minimizing the non-local interactions in the integral equation, which allows for applying sparse linear solvers to reduce the computational cost. When combined with the standard GMRES solver, the number of preconditioned iterations remains small and essentially independent of the frequency. This suggests that, when the sparsifying preconditioner is adopted, solving the dense integral system can be done as efficiently as solving the sparse system from PDE discretization.