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