Inverse problems and conformal mapping

Abstract

In this exposition we give a unified presentation of the conformal mapping technique that was developed over the last decade by Akduman et al. [I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Probl. 18 (2002), pp. 1659–1672; R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simul. 66 (2004), pp. 255–265; H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Probl. 21 (2005), pp. 935–953; H. Haddar and R. Kress, Conformal mapping and an inverse impedance boundary value problem, J. Inverse Ill-Posed Probl. 14 (2006), pp. 785–804; H. Haddar and R. Kress, Conformal mapping and impedance tomography, Inverse Probl. 26 (2010), p. 074002] for the inverse problem to recover three different types of inclusions in a homogeneous conducting background medium from Cauchy data on the accessible exterior boundary. The main ingredient of this method is a nonlinear and nonlocal ordinary differential equation for boundary values of a holomorphic function in an annulus bounded by two concentric circles that maps this annulus conformally onto the unknown domain. Furthermore, in a concluding section we illustrate how this differential equation also can be applied to numerically construct conformal mappings for doubly connected domains including numerical examples.