On the numerical solution of a shape optimization problem for the heat equation

The present paper is concerned with the numerical solution of a shape identification problem for the heat equation. The goal is to determine of the shape of a void or inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional, (b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nyström discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations.