Defects localization and applications to inhomogeneous media in acoustic scattering
This thesis is about inverse problems in acoustic scattering for inhomogeneous media with far-field measurements. In the first part of our work, we are interested in the localization of defects, i.e. the areas where the actual index is different from some reference index. We obtain a characterization of the support of the defects from the measurements by an extension of A. Kirsch’s Factorization Method. We propose several numerical methods, one of them allowing us to consider the case of measurements directions which are different from the incidence directions. These algorithms are numerically validated. In a second part, we consider the reconstruction of the values of some unknown index. Using the previous defects localization, we propose two strategies to determine regions of interest on which the reconstruction is focused. Finally, we introduce a new cost function type, for the reconstruction, which capitalizes on the properties demonstrated in the first part.