Thi-Phong Nguyen

Fioralba Cakoni, Houssem Haddar, Thi-Phong Nguyen

This paper considers the imaging of local perturbations of an infinite penetrable periodic layer. A cell of this periodic layer consists of several bounded inhomogeneities situated in a known homogeneous media. We use \mfied{a differential linear sampling method} to reconstruct the support of perturbations without using the Green's function of the periodic layer nor reconstruct the periodic background inhomogeneities. The justification of this imaging method relies on the well-posedeness of a nonstandard interior transmission problem, which until now was an open problem except for the special case when the local perturbation didn't intersect the background inhomogeneities. The analysis of this new interior transmission problem is the main focus of this paper. We then complete the justification of our inversion method and present some numerical examples that confirm the theoretical behavior of the differential indicator function determining the reconstructable regions in the periodic layer.

Thi-Phong Nguyen

We discuss the use of differential sampling method to image local perturbations in anisotropic periodic layers, extending earlier works on the isotropic case. We study in particular the new interior transmission problem that is associated with the inverse problem when only a single Floquet-Bloch mode is used. We prove Fredholm properties of this problem under similar assumptions as for classical interior transmission problems. The result of the analysis is then exploited to design an indicator function for the local perturbation. The resulting numerical algorithm is validated for two dimensional numerical experiments with synthetic data.