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 $\hat{I}“$, the Dirichlet-to-Neumann operator (DtN-map) $\hat{I}{›}_g:C^{\infty }(\hat{I}“)\to C^{\infty }(\hat{I}“)$ is defined by $\hat{I}{›}_{g}f={\frac{\partial u}{\partial \hat{I}½}|}_{\hat{I}}“$, where $\hat{I}½$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $\hat{I}{”}_{g}u=0,u{|}_{\hat{I}}“=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $\hat{I}{›}_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $(D,e)$, where $D=\{(x,y)\mid x^2+y^2\le 1\}$ is the unit disk and $e=d{x}^2+d{y}^2$ 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.

16 figures, 1 link to Google Drive