Set-oriented multi objective optimal control of elliptic non-linear partial differential equations using POD objectives and gradient

Abstract

In this thesis we use the Reduced Basis (RB) method to sol- ve multiobjective optimization problems controlled by parameter dependent semilinear elliptic partial differential equations. First, we show the existence and uniqueness of solutions of the partial differential equations under con- sideration. Subsequently, we introduce the RB method to be able to signifi- cantly reduce the computational effort compared to standard approximation schemes like the Finite Element (FE) method. Here we work with the true error and an a-posteriori error estimator. In addition, we use the Discrete Empirical Interpolation method to solve the surrogate model independently of the dimension of the FE space so we can save even more computational effort. In the optimization step we use a set-oriented subdivision algorithm based on gradient calculations of the objective functions to solve the problem. To evaluate the objective functions and their gradients in the optimization step we use the surrogate model and through this we save a lot of compu- tational effort. In order to take inexactness of the gradients into account, e.g. caused by the surrogate model, we derive an additional condition for the descent direction. We prove the convergence with this new descent direction to a tight superset of the Pareto set in this thesis. Finally we examine our results using numerical examples.