A conformal mapping method in inverse obstacle scattering

Abstract

Akduman, Haddar and Kress [Akduman I, Kress R. Electrostatic imaging via conformal mapping. Inverse Prob. 2002;18:1659–1672; Haddar H, Kress R. Conformal mappings and inverse boundary value problems. Inverse Prob. 2005;21:935–953; Kress R. Inverse Dirichlet problem and conformal mapping. Math.Comput. Simul. 2004;66:255–265] have employed a conformal mapping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for electrostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to obtain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.