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The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion


Authors: Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi
Journal: Math. Comp. 81 (2012), 367-386
MSC (2010): Primary 35R30, 49Q10, 49Q12
DOI: https://doi.org/10.1090/S0025-5718-2011-02533-0
Published electronically: August 15, 2011
MathSciNet review: 2833499
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Abstract: With each $ \mathcal{C}^2$-domain and material parameter, an infinite number of tensors, called the Generalized Polarization Tensors (GPTs), is associated. The GPTs contain significant information on the shape of the domain and its material parameter. They generalize the concept of Polarization Tensor (PT), which can be seen as the first-order GPT. It is known that given an arbitrary shape, one can find an equivalent ellipse or ellipsoid with the same PT. In this paper we consider the problem of recovering finer details of the shape of a given domain using higher-order polarization tensors. We design an optimization approach which solves the problem by minimizing a weighted discrepancy functional. In order to compute the shape derivative of this functional, we rigorously derive an asymptotic expansion of the perturbations of the GPTs that are due to a small deformation of the boundary of the domain. Our derivations are based on the theory of layer potentials. We perform some numerical experiments to demonstrate the validity and the limitations of the proposed method. The results clearly show that our approach is very promising in recovering fine shape details.


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Additional Information

Habib Ammari
Affiliation: Department of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France
Email: habib.ammari@ens.fr

Hyeonbae Kang
Affiliation: Department of Mathematics, Inha University, Incheon 402-751, Korea
Email: hbkang@inha.ac.kr

Mikyoung Lim
Affiliation: Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
Email: mklim@kaist.ac.kr

Habib Zribi
Affiliation: Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
Email: zribi@cmapx.polytechnique.fr

DOI: https://doi.org/10.1090/S0025-5718-2011-02533-0
Keywords: Generalized polarization tensor, asymptotic expansions, shape recovery
Received by editor(s): August 18, 2010
Received by editor(s) in revised form: December 2, 2010
Published electronically: August 15, 2011
Article copyright: © Copyright 2011 American Mathematical Society