We consider the scattering problem of electromagnetic waves by an infinitely long cylinder in three dimensions.The cylinder is dielectric, isotropic and inhomogeneous(with respect to the lateral directions). The incoming wave is time-harmonic and obliquely incident on the scatterer. We examine the well-posedness of the direct problem (uniqueness and existence of solution) using a Lippmann- Schwinger integral equation formulation. We prove uniqueness of the inverse problem to reconstruct the refractive index of an isotropic circular cross-section using the discreteness of the corresponding transmission eigenvalue problem and solutions based on separation of variables. We solve numerically the inverse problem for media with radial symmetric parameters using a Newton - type scheme. The direct problem is also solved numerically to provide us with the necessary far-field patterns of the scattered fields. We present numerical reconstructions justifying the applicability of the proposed method.

In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous media with a conductive boundary. These are a new class of eigenvalue problems that is not elliptic, not self-adjoint, and non-linear, which gives the possibility of complex eigenvalues. We will discuss the existence of the eigenvalues as well as their dependence on the material parameters. Due to the fact that this is a non-standard eigenvalue problem, a discussion of the numerical calculations will also be highlighted. This is joint work with: R.-C. Ayala, O. Bondarenko, A. Kleefeld, and N. Pallikarakis.

The Generalized Sampling Method has been introduced to justify the so-called Linear Sampling Method of Colton and Kirsch (1996). It offers a framework that allow more flexibility than the Factorization Method of Kirsch which made it possible to extended a little the theoretical analysis of sampling methods. In this contribution we will point out the remaining difficulties of the Generalized Linear Sampling methods namely the form of the regularization term, the treatment of noisy measurements and some configuration of the sources and the receivers that break the symmetry of the near field operator. We will propose solution to address some of this challenges. Numerical illustrations will be provided on various type of measurements from Electrical Impedance Tomography, acoustics and elasticity scattering.