Transverse electric scattering on inhomogeneous objects: spectrum of integral operator and preconditioning

TL;DR

This paper analyzes the spectrum of the integral operator in TE scattering problems and introduces a regularizing preconditioner that significantly improves the convergence of iterative methods.

Contribution

It identifies the essential spectrum's role in slow convergence and proposes a regularization and deflation preconditioning strategy for efficient solutions.

Findings

Regularizing operator transforms the system to 'identity plus compact' form.

Preconditioning with eigenvalue deflation accelerates GMRES convergence.

Method maintains FFT efficiency in matrix-vector products.

Abstract

The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify the nontrivial essential spectrum of the pertaining integral operator as partly responsible for this behavior, and the main reason why a normally efficient deflating preconditioner does not work. We solve this problem by applying an explicit multiplicative regularizing operator, which transforms the system to the form `identity plus compact', yet allows the resulting matrix-vector products to be carried out at the FFT speed. Such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes, which results in a robust acceleration of the restarted GMRES under constraint memory conditions.