Abstract:The talk is concerned with the solution of inverse shape optimization problems for the heat equation. The goal is to determine the shape of an inclusion or a heat source from measurements of temperatures and heat fluxes on the exterior boundary. This is done by minimizing a least-squares cost functional for the difference of computed and measured boundary data. The shape gradient of the cost functional is computed by means of the adjoint method. Thus a gradient based nonlinear Ritz-Galerkin scheme can be applied to discretize the shape optimization problem.
The state equation and its adjoint are boundary value problems of the heat equation, which are reformulated as parabolic boundary integral equations. The direct evaluation of discretized thermal potential operators has O(N^2 M^2) cost, where N is the number of temporal and M is the number of spatial nodes. The complexity can be reduced to nearly O(N M) operations by using a space-time fast multipole method for the rapid evaluation of thermal layer potentials.
The talk will conclude with numerical experiments that illustrate the efficiency and reconstruction quality of the approach.