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IDENTIFICATION OF SMALL INHOMOGENEITIES: ASYMPTOTIC FACTORIZATION
 

Summary: IDENTIFICATION OF SMALL INHOMOGENEITIES:
ASYMPTOTIC FACTORIZATION
HABIB AMMARI, ROLAND GRIESMAIER, AND MARTIN HANKE
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 asymp-
totic 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. non-iterative) reconstruc-
tion algorithms for the determination of the location of the small inclu-
sions 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.
1. Introduction
Inverse boundary value problems for partial differential equations, in prin-
ciple, are difficult to solve since they are both nonlinear and ill-posed. Re-
cently new solution methods such as linear sampling methods and factor-

  

Source: Ammari, Habib - Centre de Mathématique Appliquées, École Polytechnique

 

Collections: Mathematics