Recursive integral method for transmission eigenvalues

TL;DR

This paper introduces a recursive integral method for accurately computing transmission eigenvalues in inverse scattering problems, overcoming challenges of non-selfadjointness and nonlinearity, and demonstrating effectiveness through numerical tests.

Contribution

The paper presents a novel recursive integral method that efficiently computes transmission eigenvalues without prior spectral information, addressing numerical difficulties of existing approaches.

Findings

Method is self-correcting and can separate close eigenvalues.

The approach is robust and effective in numerical experiments.

It overcomes limitations of existing methods for this eigenvalue problem.

Abstract

Recently, a new eigenvalue problem, called the transmission eigenvalue problem, has attracted many researchers. The problem arose in inverse scattering theory for inhomogeneous media and has important applications in a variety of inverse problems for target identification and nondestructive testing. The problem is numerically challenging because it is non-selfadjoint and nonlinear. In this paper, we propose a recursive integral method for computing transmission eigenvalues from a finite element discretization of the continuous problem. The method, which overcomes some difficulties of existing methods, is based on eigenprojectors of compact operators. It is self-correcting, can separate nearby eigenvalues, and does not require an initial approximation based on some a priori spectral information. These features make the method well suited for the transmission eigenvalue problem whose spectrum is complicated. Numerical examples show that the method is effective and robust.