Development and use of higher-order asymptotics to solve inverse scattering problems

Abstract

The purpose of this work was to develop new methods to address inverse problems in elasticity, taking advantage of the presence of a small parameter in the considered problems by means of higher-order asymptotic expansions. The first part is dedicated to the localization and size identification of a buried inhomogeneity B true in a 3D elastic domain. In this goal, we focus on the study of functionals J(Ba ) quantifying the misfit between Btrue and a trial homogeneity Ba . Such functionals are to be minimized w.r.t. some or all the characteristics of the trial inclusion Ba (location, size, mechanical properties ...) to find the best agreement with Btrue. To this end, we produce an expansion of J with respect to the size a of Ba , providing a polynomial approximation easier to minimize. This expansion is established in a volume integral equations framework up to the sixth order, and justified by an estimate of the residual. A suited identification procedure is then given and supported by numerical illustrations for simple obstacles in full-space R3. The main purpose of this second part is to characterize a microstructured two- phases layered 1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigenvalues of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we rely on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method to recover the macroscopic parameters (L and material contrast) of such inclusion. To access the key features of the microstructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.