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Description:
In this thesis we consider different multiobjective optimization problems constrained by elliptic PDEs. For such problems the computational effort can be challenging due to (1) the presence of many objectives and the uncountableness of the Pareto set and due to (2) the presence of PDE constraints, which make the objective function evaluation expensive. To overcome these two challenges, we use two reduction techniques, which are (i) Hierarchical Multiobjective Optimization, which aims at a efficient description of the Pareto set, and (ii) Reduced Order Modelling (ROM) techniques, to speed up the PDE solves. To be precise, we are using the Reduced Basis (RB) method as a tool for reduced order modelling in combination with hierarchical variants of Continuation methods (CM) and Weighted sum methods (WSM) for the multiobjective optimization. Those variants aim at computing the boundary of the Pareto (critical) set by considering subsets of the objective functions and are based on a theoretical description of the hierarchical structure of the Pareto (critical) set. This has the advantage that objective components can be neglected for the computation of certain Pareto critical points and that the Pareto (critical) set is described completely by a smaller amount of points needed. In the case of a strictly convex, quadratic, coercive objective we proof that the Pareto set is completely described as the convex hull of the minimizers of the components and we apply this fact to a special class of abstract multiobjective optimal control problems. Further, we consider how the inexactness due to the RB approximation in the objective translates into an error in the Pareto (critical) set. The hierarchical CM and the RB method are applied to a non-convex multiobjective parameter optimization problem and the RB method in combination with the WSM is applied to a non-smooth multiobjective parameter-optimization problem with $l^1$-regularization. Numerical tests confirm the benefit from the hierarchical optimization and the reduced ...