A Priori Error Estimates for a State-constrained Elliptic Optimal Control Problem PDF Download
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Author: Arnd Rösch
Publisher:
ISBN:
Category :
Languages : en
Pages : 16
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Author: Arnd Rösch
Publisher:
ISBN:
Category :
Languages : en
Pages : 16
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Author: Klaus Krumbiegel
Publisher:
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Category :
Languages : en
Pages : 25
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Author:
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Languages : en
Pages :
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This thesis is concerned with the development, analysis, and implementation of an adaptive finite element method for distributed elliptic optimal control problems with pointwise unilateral constraints on the state. In particular, two residual-type a posteriori error estimators will be derived. The first one takes advantage of the modified adjoint state, which is defined as some kind of regularization of the adjoint state. Furthermore, this error estimator will, after minor modification, be transfered to the Lavrentiev regularization of the pure state constrained case. Up to a consistency error and data oscillation, reliability and efficiency results concerning the approximation of the state, the control, and the modified adjoint state can be provided for these error estimators. With two numerical examples, the performance of the adaptive algorithm will be investigated. A benefit compared to an uniform refinement strategy will be noticeable. The second developed a posteriori error estimator results from a measure extension of the discrete measure appearing in the right-hand side of the adjoint state equation to an element in the space of square integrable functions. This error estimator provides, again up to a consistency error and data oscillation, reliability and efficiency for the approximation error in the control, in the state, and in a semi-continuous auxiliary adjoint state. Another numerical example will show that this error estimator might be advantageous.
Author: Michael Hinze
Publisher: Springer Science & Business Media
ISBN: 1402088396
Category : Mathematics
Languages : en
Pages : 279
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Solving optimization problems subject to constraints given in terms of partial d- ferential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of industrial, medical and economical applications, where the transition from model-based numerical si- lations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and num- ical simulation plays a central role. After proper discretization, the number of op- 3 10 timization variables varies between 10 and 10 . It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilize and f- ther explore the speci?c mathematical structure of optimization problems with PDE constraints, and to develop new mathematical approaches concerning mathematical analysis, structure exploiting algorithms, and discretization, with a special focus on prototype applications. The present book provides a modern introduction to the rapidly developing ma- ematical ?eld of optimization with PDE constraints. The ?rst chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in in?nite dim- sional spaces. Therefore, functional analytic techniques, function space theory, as well as existence- and uniqueness results for the underlying PDE are essential to study the existence of optimal solutions and to derive optimality conditions.
Author: Günter Leugering
Publisher: Springer Science & Business Media
ISBN: 3034801335
Category : Mathematics
Languages : en
Pages : 622
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This special volume focuses on optimization and control of processes governed by partial differential equations. The contributors are mostly participants of the DFG-priority program 1253: Optimization with PDE-constraints which is active since 2006. The book is organized in sections which cover almost the entire spectrum of modern research in this emerging field. Indeed, even though the field of optimal control and optimization for PDE-constrained problems has undergone a dramatic increase of interest during the last four decades, a full theory for nonlinear problems is still lacking. The contributions of this volume, some of which have the character of survey articles, therefore, aim at creating and developing further new ideas for optimization, control and corresponding numerical simulations of systems of possibly coupled nonlinear partial differential equations. The research conducted within this unique network of groups in more than fifteen German universities focuses on novel methods of optimization, control and identification for problems in infinite-dimensional spaces, shape and topology problems, model reduction and adaptivity, discretization concepts and important applications. Besides the theoretical interest, the most prominent question is about the effectiveness of model-based numerical optimization methods for PDEs versus a black-box approach that uses existing codes, often heuristic-based, for optimization.
Author: Dieter Sirch
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832525572
Category : Mathematics
Languages : en
Pages : 166
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Subject of this work is the analysis of numerical methods for the solution of optimal control problems governed by elliptic partial differential equations. Such problems arise, if one does not only want to simulate technical or physical processes but also wants to optimize them with the help of one or more influence variables. In many practical applications these influence variables, so called controls, cannot be chosen arbitrarily, but have to fulfill certain inequality constraints. The numerical treatment of such control constrained optimal control problems requires a discretization of the underlying infinite dimensional function spaces. To guarantee the quality of the numerical solution one has to estimate and to quantify the resulting approximation errors. In this thesis a priori error estimates for finite element discretizations are proved in case of corners or edges in the underlying domain and nonsmooth coefficients in the partial differential equation. These facts influence the regularity properties of the solution and require adapted meshes to get optimal convergence rates. Isotropic and anisotropic refinement strategies are given and error estimates in polygonal and prismatic domains are proved. The theoretical results are confirmed by numerical tests.
Author: Michael Hinze
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Günter Leugering
Publisher: Springer
ISBN: 3319050834
Category : Mathematics
Languages : en
Pages : 539
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Optimization problems subject to constraints governed by partial differential equations (PDEs) are among the most challenging problems in the context of industrial, economical and medical applications. Almost the entire range of problems in this field of research was studied and further explored as part of the Deutsche Forschungsgemeinschaft (DFG) priority program 1253 on “Optimization with Partial Differential Equations” from 2006 to 2013. The investigations were motivated by the fascinating potential applications and challenging mathematical problems that arise in the field of PDE constrained optimization. New analytic and algorithmic paradigms have been developed, implemented and validated in the context of real-world applications. In this special volume, contributions from more than fifteen German universities combine the results of this interdisciplinary program with a focus on applied mathematics. The book is divided into five sections on “Constrained Optimization, Identification and Control”, “Shape and Topology Optimization”, “Adaptivity and Model Reduction”, “Discretization: Concepts and Analysis” and “Applications”. Peer-reviewed research articles present the most recent results in the field of PDE constrained optimization and control problems. Informative survey articles give an overview of topics that set sustainable trends for future research. This makes this special volume interesting not only for mathematicians, but also for engineers and for natural and medical scientists working on processes that can be modeled by PDEs.
Author: Kristian Bredies
Publisher: Springer Science & Business Media
ISBN: 3034806310
Category : Mathematics
Languages : en
Pages : 221
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Book Description
Many mathematical models of physical, biological and social systems involve partial differential equations (PDEs). The desire to understand and influence these systems naturally leads to considering problems of control and optimization. This book presents important topics in the areas of control of PDEs and of PDE-constrained optimization, covering the full spectrum from analysis to numerical realization and applications. Leading scientists address current topics such as non-smooth optimization, Hamilton–Jacobi–Bellmann equations, issues in optimization and control of stochastic partial differential equations, reduced-order models and domain decomposition, discretization error estimates for optimal control problems, and control of quantum-dynamical systems. These contributions originate from the “International Workshop on Control and Optimization of PDEs” in Mariatrost in October 2011. This book is an excellent resource for students and researchers in control or optimization of differential equations. Readers interested in theory or in numerical algorithms will find this book equally useful.
Author: Michael Kieweg
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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