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On the development and use of higher-order asymptotics for solving inverse scattering problems.

Rémi Cornaggia 1, 2 
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
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 asymptoticexpansions.The first part is dedicated to the localization and size identification of a buried inhomogeneity Bᵗʳᵘᵉ in a 3Delastic domain. In this goal, we focused on the study of functionals ��(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 produced an expansion of �� with respect to the size a of Bₐ, providing a polynomial approximation easier to minimize. This expansion, established up to O(a⁶) in a volume integral equations framework, is 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 layered1D inclusion of length L, supposing we already know its low-frequency transmission eigenvalues (TEs). Those are computed as the eigen values of the so-called interior transmission problem (ITP). To provide a convenient invertible model, while accounting for the microstructure effects, we then relied on homogenized approximations of the exact ITP for the periodic inclusion. Focusing on the leading-order homogenized ITP, we first provide a straightforward method tore cover the macroscopic parameters (L and material contrast) of such inclusion. To access to the period of themicrostructure, higher-order homogenization is finally addressed, with emphasis on the need for suitable boundary conditions.
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Submitted on : Thursday, August 10, 2017 - 4:25:06 PM
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Rémi Cornaggia. On the development and use of higher-order asymptotics for solving inverse scattering problems.. Numerical Analysis [math.NA]. Université Paris Saclay (COmUE); University of Minnesota, 2016. English. ⟨NNT : 2016SACLY012⟩. ⟨tel-01573831⟩



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