An efficient numerical method for a shape-identification problem arising from the heat equation

This paper is dedicated to the determination of the shape of a compactly supported constant source in the heat equation from measurements of the heat flux through the boundary. This shape-identification problem is formulated as the minimization of a least-squares cost functional for the desired heat flux at the boundary. The shape gradient of the shape functional under consideration is computed by means of the adjoint method. A gradient-based nonlinear Ritz–Galerkin scheme is applied to discretize the shape optimization problem. The state equation and its adjoint are computed by a fast space-time multipole method for the heat equation. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach