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The discretization of optimal control problems governed by partial differential equations typically leads to large-scale optimization problems. We consider flow control involving the time-dependent Navier-Stokes equations as state equation which is stamped by exactly this property. In order to avoid the difficulties of dealing with large-scale (discretized) state equations during the optimization process, a reduction of the number of state variables can be achieved by employing a reduced order modelling technique. Using the snapshot proper orthogonal decomposition method, one obtains a low-dimensional model for the computation of an approximate solution to the state equation. In fact, often a small number of POD basis functions suffices to obtain a satisfactory level of accuracy in the reduced order solution. However, the small number of degrees of freedom in a POD based reduced order model also constitutes its main weakness for optimal control purposes. Since a single reduced order model is based on the solution of the Navier-Stokes equations for a specified control, it might be an inadequate model when the control (and consequently also the actual corresponding flow behaviour) is altered, implying that the range of validity of a reduced order model, in general, is limited. Thus, it is likely to meet unreliable reduced order solutions during a control problem solution based on one single reduced order model. In order to get out of this dilemma, we propose to use a trust-region proper orthogonal decomposition (TRPOD) approach. By embedding the POD based reduced order modelling technique into a trust-region framework with general model functions, we obtain a mechanism for updating the reduced order models during the optimization process, enabling the reduced order models to represent the flow dynamics as altered by the control. In fact, a rigorous convergence theory for the TRPOD method is obtained which justifies this procedure also from a theoretical point of view. Benefiting from the trust-region philosophy, the TRPOD method guarantees to save a lot of computational work during the control problem solution, since the original state equation only has to be solved if we intend to update our model function in the trust-region framework. The optimization process itself is completely based on reduced order information only.

This work is concerned with the numerical solution of optimization problems that arise in the context of ground water modeling. Both ground water hydraulic and quality management problems are considered. The considered problems are discretized problems of optimal control that are governed by discretized partial differential equations. Aspects of special interest in this work are inaccurate function evaluations and the ensuing numerical treatment within an optimization algorithm. Methods for noisy functions are appropriate for the considered practical application. Also, block preconditioners are constructed and analyzed that exploit the structure of the underlying linear system. Specifically, KKT systems are considered, and the preconditioners are tested for use within Krylov subspace methods. The project was financed by the foundation Stiftung Rheinland-Pfalz für Innovation and carried out in joint work with TGU GmbH, a company of consulting engineers for ground water and water resources.

The goal of this thesis is to transfer the logarithmic barrier approach, which led to very efficient interior-point methods for convex optimization problems in recent years, to convex semi-infinite programming problems. Based on a reformulation of the constraints into a nondifferentiable form this can be directly done for convex semi- infinite programming problems with nonempty compact sets of optimal solutions. But, by means of an involved max-term this reformulation leads to nondifferentiable barrier problems which can be solved with an extension of a bundle method of Kiwiel. This extension allows to deal with inexact objective values and subgradient information which occur due to the inexact evaluation of the maxima. Nevertheless we are able to prove similar convergence results as for the logarithmic barrier approach in the finite optimization. In the further course of the thesis the logarithmic barrier approach is coupled with the proximal point regularization technique in order to solve ill-posed convex semi-infinite programming problems too. Moreover this coupled algorithm generates sequences converging to an optimal solution of the given semi-infinite problem whereas the pure logarithmic barrier only produces sequences whose accumulation points are such optimal solutions. If there are certain additional conditions fulfilled we are further able to prove convergence rate results up to linear convergence of the iterates. Finally, besides hints for the implementation of the methods we present numerous numerical results for model examples as well as applications in finance and digital filter design.