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Identification of small inhomogeneities: Asymptotic factorization

Authors: Habib Ammari, Roland Griesmaier and Martin Hanke
Journal: Math. Comp. 76 (2007), 1425-1448
MSC (2000): Primary 35R30, 35C20
Published electronically: February 19, 2007
MathSciNet review: 2299781
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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

Roland Griesmaier
Affiliation: Institut für Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Germany

Martin Hanke
Affiliation: Institut für Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Germany

Keywords: Electrical impedance tomography, small conductivity inhomogeneities, asymptotic expansions
Received by editor(s): January 7, 2006
Received by editor(s) in revised form: May 15, 2006
Published electronically: February 19, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.