Subject:  Efficient Solvers for Optimal Design Problems with PDE Constraints 
Author:  Dipl.Ing. Wolfram Mühlhuber 
Supervisor:  Prof. Dr. Ulrich Langer 
External Referee:  Prof. Dr. Andreas Griewank 
In optimal design we try to improve an object by modifying its shape. These optimization problems are located on the interface to partial differential equations, numerical analysis and scientific computing. This makes the solution of optimal design problems very challenging. During recent years, the importance of optimal design has been growing, especially in the commercial market. But still nowadays, changes in the design are most often based on long lasting experience, rather than optimization methods. The main spectiality of optimal design problems is that they are optimization problems governed by differential equations where we consider only the case of partial differential equations. We present strategies for the numerical solution of these optimization problems, and discuss the arising problems. For optimal design problems with only a few design parameters we consider an approach reducing the number of parameters by eliminating the state parameters. For this reduced problem we use standard optimization methods based on sequential quadratic programming. Here, the Hessian is usually approximated by quasiNewton formulas, like the wellknown BFGSformula. That is why, we need only function and gradient evaluations of the objective and the constraints. In most cases it is very difficult to implement gradient evaluation routines for real life optimal design problems as these involve the solution of the state problem. As an alternative to handcoded gradient routines we consider various blackbox approaches, like finite differences or automatic differentiation, and analyze their pros and cons. Combining the strengths of different approaches we realize a flexible, but also efficient method by combining automatic differentiation with handcoded gradient routines. We demonstrate the good performance using an optimal sizing problem coming from industry. For optimal design problems with many design parameters approaches eliminating the state equation are not suitable. We introduce an allatonce approach considering the optimal design problem in the product space of state and design parameters. This approach treats the state equation as an equality constraint during the optimization. Besides a method based on sequential quadratic programming we introduce methods based on sequential quadratic programming and iterative regularization as we use an illposed problem as model problem. We analyze the wellposedness of the ocurring quadratic programming subproblems in a continuous and discrete setting. For the considered model problem, the numerical approximation of the KarushKuhnTucker systems of the quadratic subproblems leads to equation systems with large, sparse, symmetric but indefinite matrices. We consider the numerical solution of these problems using Uzawatype methods, reduced SQP methods or simultaneous methods. A nested iteration approach additionally accelerates the proposed method. The considered examples show the good numerical performance of the proposed method. 
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