Conformal mapping for cavity inverse problem: an explicit reconstruction formula

Abstract

In this paper, we address a classical case of the Calderón (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity (with boundary ) contained in a domain (with boundary ) from the knowledge of the Dirichlet-to-Neumann (DtN) map , where is harmonic in , and , being the constant such that . We obtain an explicit formula for the complex coefficients arising in the expression of the Riemann map that conformally maps the exterior of the unit disk onto the exterior of . This formula is derived using two ingredients: a new factorization result of the DtN map and the so-called generalized Pólia–Szegö tensors of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method.

Keywords:

• Inverse conductivity problem
• Laplace equation
• conformal mapping
• integral equations

AMS Subject Classifications:

• 30C30
• 31A25
• 45Q05
• 65N21

Notes

No potential conflict of interest was reported by the authors.