Reconstruction of an elliptical inclusion in the inverse conductivity problem

This study reports on a numerical investigation into the open problem of the unique reconstruction of an elliptical inclusion in the potential field from a single set of nontrivial Cauchy data. The investigation is based on approximating the potential fields of a composite material as a linear combination of fundamental solutions for the Laplace equation with sources shifted outside the solution domain and its boundary. The coefficients of these finite linear combinations are unknown along with the centre, the lengths of the semi-axes and the orientation of the sought ellipse. These are determined by minimizing the least-squares objective functional describing the gap between the given and computed data. The extension of the proposed technique for the reconstruction of two ellipses is also considered.