2
HABIB AMMARI AND HYEONBAE KANG
severely ill-posed and nonlinear
[1].
These major and fundamental difficulties can
be understood by means of a mean value type theorem in elliptic partial differen-
tial equations. The value of the voltage potential at each point inside the region
can be expressed as a weighted average of its neighborhood potential where the
weight is determined by the conductivity distribution. In this weighted averaging
way, the conductivity distribution is conveyed to the boundary potential. There-
fore, the boundary data is entangled in the global structure of the conductivity
distribution in a highly nonlinear way. This is the main obstacle to finding non-
iterative reconstruction algorithms with limited data. If, however, in advance we
have additional structural information about the conductivity profile, then we may
be able to determine specific features about the conductivity distribution with a
satisfactory resolution. One such type of knowledge could be that the body consists
of a smooth background containing a number of unknown small inclusions with a
significantly different conductivity. The inclusions might in a medical application
represent potential tumors, in a material science application they might represent
impurities in the material, and finally in a war or post-war situation they could
represent anti-personnel mines.
Over the last 10 years or so a considerable amount of interesting work has been
dedicated to the imaging of such low volume fraction inclusions [71, 72, 73, 55,
52, 54, 45, 18]. In this article we shall not attempt to give an exhaustive survey
of all work of this nature, rather we shall focus attention on certain asymptotic
representation formulae and their implications and applications. The method of
asymptotic expansions of small volume inclusions provides a useful framework to
accurately and efficiently reconstruct the location and geometric features of the
inclusions in a stable way, even for moderately noisy data (18]. The higher-order
terms are essential when the background voltage has some critical points inside
the conductor (15]. The first-order perturbations due to the presence of the inclu-
sions are of dipole-type. The dipole-type expansion is only valid when the potential
within the inclusion is nearly constant. On decreasing the distance between the in-
clusion and the boundary of the background medium this assumption begins to fail
because higher-order multi-poles become significant due to the interaction between
the inclusion and the boundary of the background medium. A more complicated
asymptotic formula should be used instead of dipole-type expansion when the in-
clusion is close to the boundary of the background medium.
The new concepts of GPT's associated with a bounded Lipschitz domain and
an isotropic/or anisotropic conductivity are central in this asymptotic approach.
The GPT's are the basic building blocks for the full asymptotic expansions of
the boundary voltage perturbations due to the presence of a small conductivity
inclusion inside a conductor. It is then important from an imaging point of view
to precisely characterize these GPT's and derive some of their properties, such as
symmetry, positivity, and optimal bounds on their elements, for developing efficient
algorithms to reconstruct conductivity inclusions of small volume. The GPT's seem
to contain significant information on the domain and its conductivity which are yet
to be investigated. On the other hand, the use of these GPT's leads to stable and
accurate algorithms for the numerical computations of the steady-state voltage in
the presence of small conductivity inclusions. It is known that small size features
cause difficulties in the numerical solution of the conductivity problem by the finite
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