Shape optimization under uncertainty from a stochastic programming point of view

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Optimization problems involving partial differential equations (PDEs) are crucial in various technical, industrial, and economic fields, presenting significant challenges in numerical analysis and optimization. This text is among the first to explore stochastic uncertainty in PDE-constrained optimization, specifically focusing on shape optimization for elastic bodies under stochastic loading. The approach draws parallels to finite-dimensional two-stage stochastic programming, where shapes represent nonanticipative decisions. Key findings include level set-based stochastic shape optimization utilizing gradient methods that incorporate shape and topological derivatives. The unique structure of the elasticity PDE allows for the numerical resolution of stochastic shape optimization problems with numerous scenarios without significantly increasing computational demands. Both risk-neutral and risk-averse models are examined. This work is derived from a doctoral dissertation completed between 2004 and 2008 at the Chair of Discrete Mathematics and Optimization at the University of Duisburg-Essen, supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program “Optimization with Partial Differential Equations.” Acknowledgments express gratitude to supervisors, colleagues, and friends for their invaluable support and insights throughout the research process.