Seminar in Numerical Analysis: Rüdiger Schultz (Universität Duisburg-Essen)

Stochastic Shape Optimization by Analogues from Operations Research

This talk aims at demonstrating how concepts and techniques which are well-established in operations research may serve as blueprints for approaching shape optimization with linearized  elasticity and stochastic loading. Stochastic shape optimization problems are considered from a two-stage viewpoint: In a first stage, without anticipation of the random loading, the shape has to be fixed. After realization  of the load, the displacement obtained from solving the elasticity boundary value problem then may be seen as a second-stage (or recourse) action, and the variational problem of the weak formulation as a second-stage optimization problem.

At this point, there is a perfect match with two-stage stochastic programming: after having taken a non-anticipative decision in  the first stage, and having observed the random data, a well-defined second-stage problem remains and is solved to optimality. Suitable objective functions complete the formal descriptions of the models, for instance, costs in the stochastic-programming setting and compliance or tracking functionals in shape optimization.

Stochastic programming now offers a wide collection of models to address shape optimization under uncertainty. This starts with risk neutral models, is continued by mean-risk  optimization involving different risk measures, and will finally lead to analogues in shape optimization of decision problems with stochastic-order (or dominance) constraints.

In the talk we will present these models, discuss solution methods, and report some computational tests.

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