Optimal control (motion planning) of the free interface in classical two-phase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The first-order optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function.Suitable discretization schemes for the forward and adjoint systems are described. Numerical examples verify the correctness and flexibility of the proposed scheme.

The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) Runge-Kutta scheme are employed. Polynomial spaces of quadratic order are used. A numerical example with a change of topology is provided, and the order of convergence is studied on the Frank sphere example.

Abstract. Unconstrained convex quadratic optimization problems subject to parameter perturbations are considered. A robustification approach is proposed and analyzed which reduces the sensitivity of the optimal function value with respect to the parameter. Since reducing the sensitivity and maintaining a small objective value are competing goals, strategies for balancing these two objectives are discussed. Numerical examples illustrate the approach.