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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The generalized polarization tensors for resolved imaging. Part I: Shape reconstruction of a conductivity inclusion
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by Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi PDF
Math. Comp. 81 (2012), 367-386 Request permission

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
  • MR Author ID: 353050
  • Email: habib.ammari@ens.fr
  • Hyeonbae Kang
  • Affiliation: Department of Mathematics, Inha University, Incheon 402-751, Korea
  • MR Author ID: 268781
  • Email: hbkang@inha.ac.kr
  • Mikyoung Lim
  • Affiliation: Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, Daejeon 305-701, Korea
  • MR Author ID: 689036
  • 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
  • Received by editor(s): August 18, 2010
  • Received by editor(s) in revised form: December 2, 2010
  • Published electronically: August 15, 2011
  • © Copyright 2011 American Mathematical Society
  • Journal: Math. Comp. 81 (2012), 367-386
  • MSC (2010): Primary 35R30, 49Q10, 49Q12
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02533-0
  • MathSciNet review: 2833499