Conformal mapping for cavity inverse problem: an explicit reconstruction formula

TL;DR

This paper presents an explicit formula for reconstructing the shape and location of a cavity in a domain using conformal mapping techniques, based on the Dirichlet-to-Neumann map and generalized Pólia-Szeg"o tensors.

Contribution

It introduces a new explicit reconstruction formula for cavities in inverse problems using conformal mapping and a novel factorization of the DtN map.

Findings

Explicit formula for the Riemann map coefficients derived

The method demonstrates efficiency and simplicity in numerical tests

Analytic dependence of coefficients on the DtN map established

Abstract

In this paper, we address a classical case of the Calder\'on (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity $\omega$ (with boundary $\gamma$) contained in a domain $\Omega$ (with boundary $\Gamma$) from the knowledge of the Dirichlet-to-Neumann (DtN) map $\Lambda_\gamma: f \longmapsto \partial_n u^f|_{\Gamma}$, where $u^f$ is harmonic in $\Omega\setminus\overline{\omega}$, $u^f|_{\Gamma}=f$ and $u^f|_{\gamma}=c^f$, $c^f$ being the constant such that $\int_{\gamma}\partial_n u^f\,{\rm d}s=0$. We obtain an explicit formula for the complex coefficients $a_m$ arising in the expression of the Riemann map $z\longmapsto a_1 z + a_0 + \sum_{m\leqslant -1} a_m z^{m}$ that conformally maps the exterior of the unit disk onto the exterior of $\omega$. This formula is derived by using two ingredients: a new factorization result of the DtN map and the so-called generalized P\'olia-Szeg\"o tensors (GPST) of the cavity. As a byproduct of our analysis, we also prove the analytic dependence of the coefficients $a_m$ with respect to the DtN. Numerical results are provided to illustrate the efficiency and simplicity of the method.