Principal aim of this project is to develop efficient numerical techniques for the computation of the Pareto set of PDE-constrained multiobjective optimal control problems. This set encompasses the optimal compromises between several, typically conflicting objectives. The solution of the partial differential equations contained in the problem formulation requires a considerable numerical effort. Therefore, already the solution of a single-objective optimization problem often becomes a challenge, and the solution of a multiobjective optimization problem with PDE constraints quickly becomes practically infeasible. To solve this problem, in this project we will combine model reduction techniques for partial differential equations with algorithms for the set oriented solution of multiobjective optimization problems. To develop effizient numerical methods, we will first analyse existing methods for the solution of multiobjective optimization problems with respect to their behavior in the presence of inexact function and derivative evaluations. Secondly, we will extend these algorithms to enable them to deal with reduced models. Usage of such models, which are derived on the basis of the well-known proper orthogonal decomposition (POD), introduces an approximation error that propagates into errors in further quantities that are essential for the application of the above-mentioned methods. For this reason we will derive analytical bounds for the magnitude of these errors which allow us to understand the dependence of the errors on parameters used in the model reduction procedure. Subsequently, this makes it possible to develop algorithms adapted to the considered problem class that utilize automatically chosen model reduction strategies. We will first realize the above-sketched research program for a specific problem class. In the later stages of the project, more complex problem formulations will be considered that include e.g. state constraints or non-smooth objective functions.
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