Conformal mapping for cavity inverse problem: an explicit reconstruction formula

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.
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Applicable Analysis, Taylor & Francis, 2016, 〈10.1080/00036811.2016.1208816〉
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Alexandre Munnier, Karim Ramdani. Conformal mapping for cavity inverse problem: an explicit reconstruction formula. Applicable Analysis, Taylor & Francis, 2016, 〈10.1080/00036811.2016.1208816〉. 〈hal-01196111v4〉

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