Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers

Houssem Haddar 1 Thi-Phong Nguyen 1
1 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : This paper is dedicated to the design and analysis of sampling methods to reconstruct the shape of a local perturbation in a periodic layer from measurements of scattered waves at a fixed frequency. We first introduce the model problem that corresponds with the semi-discretized version of the continous model with respect to the Floquet-Bloch variable. We then present the inverse problem setting where (propagative and evanescent) plane waves are used to illuminate the structure and measurements of the scattered wave at a parallel plane to the periodicity directions are performed. We introduce the near field operator and analyze two possible factorizations of this operator. We then establish sampling methods to identify the defect and the periodic background geometry from this operator measurement. We also show how one can recover the geometry of the background independently from the defect. We then introduce and analyze the single Floquet-Bloch mode measurement operators and show how one can exploit them to built an indicator function of the defect independently from the background geometry. Numerical validating results are provided for simple and complex backgrounds.
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Submitted on : Thursday, November 23, 2017 - 10:31:17 AM
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Houssem Haddar, Thi-Phong Nguyen. Sampling methods for reconstructing the geometry of a local perturbation in unknown periodic layers. Computers and Mathematics with Applications, Elsevier, 2017, 74 (11), pp.2831-2855. ⟨10.1016/j.camwa.2017.07.015⟩. ⟨hal-01645674⟩

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