Conformal mappings and inverse boundary value problems

Recently, Akduman and Kress (2002 Inverse Problems 18 1659–72) proposed a conformal mapping technique for reconstructing the interior boundary curve of a two-dimensional doubly connected bounded domain from the Cauchy data of a harmonic function on the exterior boundary curve. The approach is based on determining a holomorphic function that maps an annulus bounded by two concentric circles bijectively onto the unknown domain. First, the boundary values of this holomorphic function on the exterior circle are obtained from solving a nonlocal ordinary differential equation by successive approximations. Then the unknown boundary is found as the image of the interior circle by solving an ill-posed Cauchy problem for holomorphic functions via a regularized Laurent expansion. In the paper of Akduman and Kress only the case of a homogeneous Dirichlet condition on the interior boundary was considered. In addition, the Cauchy data were restricted to a nonvanishing total flux through the exterior boundary and to normal derivatives on the exterior boundary without zeros. In order to extend the applicability of the method and also to allow a homogeneous Neumann boundary condition on the interior boundary, in the present paper we propose modifications of the scheme that relax the above restrictions. Our analysis includes convergence results on one of our modified schemes and a series of numerical examples illustrating the performance of the method.