Robust optimization of PDE-constrained problems using second-order models and nonsmooth approaches

This thesis considers nonconvex PDE-constrained optimization problems that depend on uncertain data. Following the worst-case approach of the well known robust optimization paradigm, the uncertain optimization problem is transformed into a challenging deterministic bilevel problem. We examine an approach that approximates the effects of the uncertain parameters on the objective and constraint functions by Taylor expansions of second order. The approximation transforms the maximization programs on the lower level to trust-region subproblems, for which efficient solution methods exist despite their nonconvexity. We formulate the resulting approximated robust counterpart as a nonsmooth optimization problem, which naturally allows for nonunique lower-level solutions. We employ matrix-free methods to deal with larger numbers of uncertain parameters. We also study an inexact optimization framework that helps to improve the approximation quality by gradually shifting the Taylor expansion points towards local maxima. The methods developed here are applied to two shape optimization problems of load-carrying structures in mechanical engineering, both of which originate from the author’s work within the Collaborative Research Center 805 funded by the German Research Foundation (DFG). The first example is a two-dimensional truss structure under uncertain time-dependent load, which is optimized to reduce vibrations. The second example is a three-dimensional smart structure, optimized so as to exhibit desirable sensitivity properties despite inaccurate sensor placement.