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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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Identification of small inhomogeneities: Asymptotic factorization
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by Habib Ammari, Roland Griesmaier and Martin Hanke PDF
Math. Comp. 76 (2007), 1425-1448 Request permission

Abstract:

We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral equation methods. It demonstrates the strong relation between factorization methods and MUSIC-type methods for the solution of this inverse problem.
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Additional Information
  • Habib Ammari
  • Affiliation: Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France
  • MR Author ID: 353050
  • Email: ammari@cmapx.polytechnique.fr
  • Roland Griesmaier
  • Affiliation: Institut für Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Germany
  • Email: griesmaier@math.uni-mainz.de
  • Martin Hanke
  • Affiliation: Institut für Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Germany
  • Email: hanke@math.uni-mainz.de
  • Received by editor(s): January 7, 2006
  • Received by editor(s) in revised form: May 15, 2006
  • Published electronically: February 19, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 1425-1448
  • MSC (2000): Primary 35R30, 35C20
  • DOI: https://doi.org/10.1090/S0025-5718-07-01946-1
  • MathSciNet review: 2299781