Abstract. 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 an 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. 1. Introduction. Recently shape optimization for elliptic boundary value problems has become a well-established mathematical and computational tool; see, e.g., [3,15,19,26,30] and the references therein. In this context boundary element methods have proved themselves to be an extremely effective tool for the solution of the associated state equation; see, e.g., [8,13,23,25,27]. The main reason is that no domain triangulation has to be computed. Moreover, fast methods to handle the associated dense matrix problems have reached a mature state and can be readily applied within the iterative solution of the minimization problem.In contrast to elliptic shape optimization, the literature on parabolic shape optimization problems is rather limited. There are theoretical results (see, e.g., [5,16,17,29] and the references therein), but the development of efficient numerical methods for shape optimization problems with a parabolic state equation is still in its beginning stages, especially for three-dimensional geometries. With the goal to develop such efficient methods, we consider in the present paper a shape identification problem for the heat equation. Specifically, we detect an inclusion or void of zero temperature inside a solid or liquid body by measurements of the temperature and the transient heat flux at the accessible outer boundary.This problem is severely ill-posed and was discussed in a two-dimensional setting in [1,2], where the uniqueness of the inclusion and the differentiability with respect to its boundary was established. The approach in these papers is to use Newton's method to solve the nonlinear functional equation that maps the shape of the inclusion to the measured data. Thus each step of the iteration involves one forward problem
