On the numerical solution of an inverse boundary value problem for the heat equation

We consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of our method.