 
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. noniterative) 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 MUSICtype 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 illposed. Re
cently new solution methods such as linear sampling methods and factor
