Numerical methods for the shape reconstruction of electrical anomalies using single or double boundary measurements

A solution of the conductivity problem with anomalies of piecewise constant conductivities in a homogeneous medium can be represented as a single layer potential. We propose a simple disk reconstruction method based on the Laurent expansion of the single layer potential to estimate anomalies that can be used as an initial guess for an iterative searching algorithm. Using a simple linear relationship between the normal domain perturbation distance and the boundary perturbation, we develop an iterative algorithm to find anomalies within the domain. The performance of the algorithm is illustrated via numerical examples. The iterative searching algorithm works well in many cases using a single boundary measurement. However, some features of the anomalies require that the algorithm utilize multiple measurements. We discuss the limitations of the inverse conductivity problem using a single Cauchy data set and present a modified version of the algorithm for use with double boundary measurements. The improved performance of this modified numerical scheme is also illustrated with various examples.