Numerical solution of the two-dimensional Calderón 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: $\hat{I}{”}_{g}u=0,u{|}_{\partial M}=f$ for $f\in C^{\infty }(\partial M)$. The Dirichlet-to-Neumann (DN) operator $\hat{I}{›}_g:C^{\infty }(\partial M)⟶C^{\infty }(\partial M)$ is defined by $\hat{I}{›}_{g}f={\frac{\partial u}{\partial \hat{I}½}|}_{\partial M}$, where $\hat{I}½$ 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 Calderón problem is as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_{\partial },\hat{I}{›}_g)$? 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 ${\mathcal{H}}_{\hat{I}}\text{c◯}:C^{\infty }(\hat{I}“)\to C^{\infty }(\hat{I}“)$ on the boundary curve $\hat{I}“=\partial \hat{I}\text{c◯}$ of a planar domain $\hat{I}\text{c◯}$. We study properties of the Hilbert transform. In particular, we obtain an integral formula for ${\mathcal{H}}_{\hat{I}}\text{c◯}$ for a simply connected $\hat{I}\text{c◯}$ 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.