Inverse Scattering Using Asymptotic Techniques

Abstract

This dissertation is dedicated to establish and debate the performance of asymptotic frameworks for inverse scattering in elastic and electromagnetic media. The crux of the study is on the establishment of strong mathematical foundations for the topological derivative based optimization techniques, and on the design and e ective use of elastic scattering coe cients for inverse scattering and enhancement of nearly elastic cloaking e ect. Speci cally, in the rst part of the dissertation, topological derivative based inverse scattering techniques are proposed for detecting characteristically small inclusions. Moreover, the key ingredients to ascertain the performance and robustness of the algorithms; such as resolution, stability with respect to medium and measurement noises, and signal-to-noise ratio, are evaluated for the proposed algorithms to demonstrate their appositeness. In the second half of the dissertation, the notion of elastic scattering coe cients in two-dimensions is introduced and its link with the scattering signature of inclusions is established. An optimization-based procedure is proposed to recover these mathematical objects for an inclusion from its scattering signature. The decay rate of the elastic scattering coe cients, stability and error analyses of their reconstruction procedure, and the estimate of maximal resolving order of the recovery algorithm in terms of the signal-to-noise ratio are discussed. Moreover, the utility of the elastic scattering coe cients in designing the so-called scattering-coe cients-vanishing-structures in elastic media and then to the enhancement of nearly elastic cloaking devices is explained. The conducted research is motivated by the numerous potential applications of these direct and inverse scattering frameworks, for instance, in biomedical imaging, non-destructive testing, geophysical sensing and exploration, and homeland security.