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Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. (English) Zbl 0916.35132
The non-destructive inspection technique known as electrical impedance imaging has recently received considerable attention in the mathematical, as well as in the engineering literature. Using this technique one seeks to determine information about the internal conductivity (or impedance) profile of an object based on boundary information about the applied steady-state currents and corresponding voltage potentials. The goal could be to image an entirely unknown internal conductivity profile, but frequently it may be somewhat more limited in scope: a priori one has some knowledge of the overall form of the conductivity profile and one then seeks to determine very specific features. Examples of the latter type are found in connection with the identification of cracks and the identification of one or more inhomogeneities.
Following this line of investigation the goal of this paper is to design an efficient method to determine the location and size of diametrically small conductivity imperfections inside a conductor of known background conductivity. The imperfections (inhomogeneities) have constant conductivities. These conductivities may be known or unknown, depending on the application. Unlike the inhomogeneities treated in other previous works, the ones considered here are of small size, and this allows for the design of an effective identification procedure. The identification of small inhomogeneities has previously been analyzed, but whereas those inhomogeneities were either perfectly insulating (voids) or perfectly conducting the ones considered here are just required to have a (finite) conductivity different from the background conductivity. The fundamental step in the design of the proposed identification procedure is the derivation of an asymptotic formula for the steady-state voltage potential for a conductor with a finite number of well separated, small inhomogeneities. This formula is used by the authors to establish continuous dependence estimates and to design an effective computational identification procedure.

MSC:
35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
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