Numerical solution of the two-dimensional Calderon problem for domains close to a disk

Abstract

For a compact Riemannian surface $(M,g)$ with non-empty boundary $\Gamma$, the Dirichlet-to-Neumann operator (DtN-map) $\Lambda_g:C^\infty(\Gamma)\to C^\infty(\Gamma)$ is defined by $\Lambda_gf=\left.\frac{\partial u}{\partial\nu}\right|_\Gamma$, where $\nu$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $\Delta_gu=0,\ u|_\Gamma=f$. The Calder\'{o}n problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $\Lambda_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calder\'{o}n problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}=\{(x,y)\mid x^2+y^2\le1\}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain.