Structure exploiting Galerkin schemes for optimization problems with pde constraints

This project is concerned with the development and the analysis of discrete concepts and algorithms for pde constrained optimization problems including control and state constraints. We propose a tailored discrete concept for optimization problems with nonlinear pdes including control constraints and develop a new discrete concept in pde constrained optimization involving state constraints. The key idea consists in conserving as much as possible the structure of the infinite-dimensional KKT (Karush-Kuhn-Tucker) system on the discrete level, and to appropriately mimic the functional analytic relations of the KKT system through suitably chosen Ansätze for the variables involved.

For both cases we provide numerical analysis, including convergence proofs and adapted numerical algorithms. As a class of model problems we consider optimization with (nonlinear) elliptic and parabolic pdes. This allows to validate and compare the new concepts to be developed in this project against existing approaches for the class of elliptic control problems.