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Optimization problems are relevant in many areas of technical, industrial, and economic applications. At the same time, they pose challenging mathematical research problems in numerical analysis optimization. Harald Held considers an elastic body subjected to uncertain internal and external forces. Since simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings, he uses techniques from level set-based shape optimization and two-stage stochastic programming. Taking advantage of the PDE's linearity, he is able to compute solutions for an arbitrary number of scenarios without significantly increasing the computational effort. The author applies a gradient method using the shape derivative and the topological gradient to minimize, e.g, the compliance. The stochastic programming perspective also allows incorporating risk measures into the model which might be more appropriate objective in many practical applications.
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Über die Autorin bzw. den Autor
Dr. Harald Held completed his doctoral thesis at the Department of Mathematics at the University of Duisburg-Essen. He is now a Research Scientist at Siemens AG, Corporate Technology.
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Optimization problems are relevant in many areas of technical, industrial, and economic applications. At the same time, they pose challenging mathematical research problems in numerical analysis and optimization.
Harald Held considers an elastic body subjected to uncertain internal and external forces. Since simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings, he uses techniques from level set based shape optimization and two-stage stochastic programming. Taking advantage of the PDE s linearity, he is able to compute solutions for an arbitrary number of scenarios without significantly increasing the computational effort. The author applies a gradient method using the shape derivative and the topological gradient to minimize, e.g., the compliance . and shows that the obtained solutions strongly depend on the initial guess, in particular its topology. The stochastic programming perspective also allows incorporating risk measures into the model which might be a more appropriate objective in many practical applications.
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Shape Optimization Under Uncertainty From A Stochastic Programming Point Of View
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Shape Optimization under Uncertainty from a Stochastic Programming Point of View
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Shape Optimization under Uncertainty from a Stochastic Programming Point of View
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Vieweg+Teubner Verlag, Vieweg+Teubner Verlag Jul 2009, 2009
ISBN 10: 3834809098
ISBN 13: 9783834809094
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Taschenbuch. Zustand: Neu. Neuware -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.Springer Vieweg in Springer Science + Business Media, Abraham-Lincoln-Straße 46, 65189 Wiesbaden 148 pp. Englisch. Artikel-Nr. 9783834809094
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Shape Optimization under Uncertainty from a Stochastic Programming Point of View
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Taschenbuch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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. Artikel-Nr. 9783834809094
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Shape Optimization under Uncertainty from a Stochastic Programming Point of View
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