On shape optimization with parabolic state equation

The present paper intends to summarize the main results of [17, 18] on the numerical solution of shape optimization problems for the heat equation. This is carried out by means of a specific problem, namely the reconstruction of a heat source which is located inside the computational domain under consideration from measurements of the heat flux through the boundary. We arrive at a shape optimization problem by tracking the mismatch of the heat flux at the boundary. For this shape functional, the Hadamard representation of the shape gradient is derived by use of the adjoint method. The state and its adjoint equation 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. To demonstrate the similarities to shape optimization problems for elliptic state equations, we consider also the related stationary shape optimization problem which involves the Poisson equation. Numerical results are given to illustrate the theoretical findings.