Efficient methods for optimal control problems subject to partial differential equations with uncertain coefficients

Guth, Philipp Arthur

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URN: urn:nbn:de:bsz:180-madoc-636897
Document Type: Doctoral dissertation
Year of publication: 2022
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Schillings, Claudia
Date of oral examination: 21 December 2022
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematische Optimierung (Schillings 2017-2022)
Subject: 510 Mathematics
Subject headings (SWD): Optimierung , Stochastische Optimierung , Parametrische Optimierung , Numerische Integration , Partielle Differentialgleichung
Keywords (English): optimization under uncertainty , optimal control with partial differential equations under uncertainty , Quasi-Monte Carlo integration , multilevel optimization , high-dimensional integration
Abstract: In this thesis, we develop and analyze methods to efficiently solve optimization problems under uncertainty, constrained by partial differential equations (PDEs). The uncertainties may arise due to noisy measurements, unknown or unobservable parameters, model ambiguity, or intrinsic randomness of systems. The goal is to find a control which is robust with respect to variations in the uncertain parameters. We prove error bounds and convergence rates for the developed methods, confirm the theoretically derived results through numerical experiments, and examine the developed concepts with regard to their efficiency. The focus of this work is the application and analysis of quasi-Monte Carlo methods, as well as the use of surrogate models for computationally intensive systems in conjunction with a penalty strategy. We first analyze a general formulation of the optimal control problem for the existence and uniqueness of solutions, and then focus on three example problems of optimal control under uncertainty. The regularity of the problems with respect to the uncertain parameters plays a crucial role in the development and the error analysis of the methods. The numerical treatment of the considered problems requires different approximation methods. The total approximation error is decomposed into its components and each error contribution is then studied separately in a chapter. The error estimates and convergence results developed in these chapters are not limited to problems of optimal control subject to PDE constraints with uncertain coefficients. In addition, further strategies to increase the efficiency of the methods are investigated, such as multilevel strategies and the simultaneous solving of the optimal control problem and learning of surrogate models for computationally intensive models.

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