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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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The generalized polarization tensors for resolved imaging Part II: Shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements
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by Habib Ammari, Hyeonbae Kang, Eunjoo Kim and June-Yub Lee PDF
Math. Comp. 81 (2012), 839-860 Request permission

Abstract:

This paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi-static response matrix measurements at a single frequency. The target is of characteristic size less than the operating wavelength. Using small-volume asymptotic expansions of the measurements of high-order, we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed. The asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors. Moreover, we extend the optimization approach proposed in Part I to image geometric details of an electromagnetic target that are finer than the equivalent ellipse. The equivalent ellipse still provides a good initial guess for the optimization procedure. However, compared to the conductivity case in Part I, the cost functional measures the discrepancy between the computed and measured high-order frequency dependent polarization tensors rather than between the generalized polarization tensors. The main reason for such a modification of the cost functional is the fact that the (measured) frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment. The proposed methods are numerically implemented to demonstrate their validity and efficiency.
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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
  • Eunjoo Kim
  • Affiliation: Institute of Mathematical Sciences and Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea
  • Email: kej@ewha.ac.kr
  • June-Yub Lee
  • Affiliation: Institute of Mathematical Sciences and Department of Mathematics, Ewha Womans University, Seoul 120-750, Korea
  • Email: jyllee@ewha.ac.kr
  • Received by editor(s): August 18, 2010
  • Received by editor(s) in revised form: December 2, 2010
  • Published electronically: September 20, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 81 (2012), 839-860
  • MSC (2010): Primary 35R30, 35B30, 78M50
  • DOI: https://doi.org/10.1090/S0025-5718-2011-02534-2
  • MathSciNet review: 2869039