Numerical solution of the two-dimensional Calder\'{o}n problem based on the Hilbert transform of a planar domain

Abstract

Let $(M,g)$ be a $C^{\infty}$ smooth compact connected Riemannian manifold with boundary $\partial M$. Consider the Dirichlet problem: $\Delta_gu=0,\ u|_{\partial M}=f$ for $f\in C^{\infty}(\partial M)$. The Dirichlet-to-Neumann (DN) operator $\Lambda_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $\Lambda_gf=\left.\frac{\partial u}{\partial\nu}\right|_{\partial M}$, where $\nu$ is the unit outer normal to the boundary and $u$ is the unique solution to the Dirichlet problem. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calder\'{o}n problem is as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,\Lambda_g)$? In the two-dimensional case the surface $(M,g)$ is determined by the DN data uniquely up to conformal equivalence. Knowledge of the DN data is equivalent to knowledge of the Hilbert transform ${\mathcal H}_\Omega:C^\infty(\Gamma)\to C^\infty(\Gamma)$ on the boundary curve $\Gamma=\partial\Omega$ of a planar domain $\Omega$. We study properties of the Hilbert transform. In particular, we obtain an integral formula for ${\mathcal H}_\Omega$ for a simply connected $\Omega$ which generalizes the classical integral formula for the Hilbert transform on the unit circle. This formula is the base of our algorithm for reconstructing a simply connected planar domain from the DN data. Several numerical reconstructions are presented.