Inside-Outside Duality and the Determination of Electromagnetic Interior Transmission Eigenvalues

Abstract We introduce an inside-outside duality approach for the determination of interior transmission eigenvalues of a possibly anisotropic dielectric electromagnetic scattering object using timeharmonic electromagnetic far field data. To this end, we exploit a self-adjoint factorization of the far field operator to link the electromagnetic interior transmission eigenvalues to the maximal or minimal phase of the eigenvalues of the corresponding far field operator, depending whether the sign of the contrast function is positive or negative

Recursive integral method for transmission eigenvalues

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.Comment: 18 pages, 8 figure

The Inside-Outside Duality in Inverse Scattering Theory

In this thesis we investigate a connection between far field data that arises from time-harmonic scattering problems and interior eigenvalues of corresponding scattering objects. This connection has been used to develop the so-called ``inside-outside duality`` method, which can be used to detect the interior eigenvalues from far field data. In this method a particular focus lies on the behavior of certain eigenvalues of the far field operator, which characterizes the interior eigenvalues. This thesis is separated into two parts. In the first part, we consider acoustic, time-harmonic scattering from impenetrable and penetrable scattering objects. We start by considering acoustic scattering from impenetrable objects and subsequently outline the principle arguments for the derivation of the inside-outside duality. In this context we also show how to work with near field data instead of far field data. In the remainder of the first part, the arguments are then adapted to scattering from penetrable scattering objects that may contain cavities. For all scattering scenarios under investigation, numerical examples for the verification of the theoretical results are provided. In the second part of this thesis we consider elastic and electromagnetic scattering problems. In the case of elastic scattering, we assume an isotropic background medium in which either a rigid or a penetrable scattering object is embedded. For electromagnetic scattering, we consider penetrable objects that may contain cavities. The main challenge in this part lies in adapting the preceding arguments for the different scattering equations. Therefore we focus on theoretical results, which can potentially be used to detect interior eigenvalues from corresponding far field data