Computing interior eigenvalues of domains from far fields

Abstract

Interior eigenvalues of bounded scattering objects can be rigorously characterised from multistatic and multi-frequency far eld data, that is, from the behavior of scattered waves far away from the object. This characterisation, the so-called inside-outside duality, holds for various types of penetrable and impenetrable scatterers and is based on the behavior of a particular eigenvalue of the far eld operator. It naturally leads to a numerical algorithm for computing interior eigenvalues of a scatterer that does not require shape or physical properties of the scatterer as input. Since the non-linear inverse problem to compute such interior eigenvalues from far eld data is ill-posed, we propose a regularising algorithm that is shown to converge as the noise level of the far eld data tends to zero. We illustrate feasibility and accuracy of our algorithm by numerical experiments where we compute interior transmission eigenvalues and Robin eigenvalues of the Laplacian in three-dimensional domains from scattering data of these domains due to plane incident waves.