Structure exploiting Galerkin schemes for optimization problems with pde constraints

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This project is concerned with the development of tailored discrete concepts and numerical algorithms for pde constrained optimization problems including control and state constraints. The mathematical analysis and numerical treatment of optimization problems with pde constraints necessitates the improvement of existing and the development of new mathematical concepts in algorithms, analysis and discretization. The major goal in pde constraint optimization consists in developing discrete concepts and algorithms which obey the relationeffort of optimization > constanteffort of simulationwith a constant of moderate size. In order to achieve this goal in this project we(a) propose a tailored discrete concept for optimization problems with nonlinear pdes including control constraints, and(b) develop a new discrete concept in pde constrained optimization with state constraints. For both cases we provide numerical analysis, including convergence proofs and adapted numerical algorithms.The key idea consists in conserving as much as possible 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. In a second application period we would be in position to combine the developed discretization strategies with hierarchical solution concepts for pde constrained optimization problems, such as multigrid methods, and to incorporate them into adaptive refinement strategies for pde constrained optimization strategies.