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.

Van Mieses stresses for 1078 subdomains

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 quasi-Newton formulas, like the well-known BFGS-formula. 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 hand-coded gradient routines we consider various black-box 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 hand-coded gradient routines. We demonstrate the good performance using an optimal sizing problem coming from industry.

parameter distribution for exact data at level 4, qmin = 0.59, qmax = 1.4

For optimal design problems with many design parameters approaches eliminating the state equation are not suitable. We introduce an all-at-once 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 ill-posed problem as model problem. We analyze the well-posedness of the ocurring quadratic programming subproblems in a continuous and discrete setting. For the considered model problem, the numerical approximation of the Karush-Kuhn-Tucker 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 Uzawa-type 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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