Shape optimization under uncertainty from a stochastic programming point of view

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Optimization problems whose constraints involve partial differential equations (PDEs) are relevant in many areas of technical, industrial, and economic app- cations. At the same time, they pose challenging mathematical research problems in numerical analysis and optimization. The present text is among the ? rst in the research literature addressing stochastic uncertainty in the context of PDE constrained optimization. The focus is on shape optimization for elastic bodies under stochastic loading. Analogies to ? nite dim- sional two-stage stochastic programming drive the treatment, with shapes taking the role of nonanticipative decisions. The main results concern level set-based s- chastic shape optimization with gradient methods involving shape and topological derivatives. The special structure of the elasticity PDE enables the numerical - lution of stochastic shape optimization problems with an arbitrary number of s- narios without increasing the computational effort signi? cantly. Both risk neutral and risk averse models are investigated. This monograph is based on a doctoral dissertation prepared during 2004-2008 at the Chair of Discrete Mathematics and Optimization in the Department of Ma- ematics of the University of Duisburg-Essen. The work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program “Optimi- tion with Partial Differential Equations”. Rüdiger Schultz Acknowledgments I owe a great deal to my supervisors, colleagues, and friends who have always supported, encouraged, andenlightenedmethroughtheirownresearch, comments, and questions.