Functional Approach to a Posteriori Error Estimation for Elliptic Optimal Control Problems with Distributed Control PDF Download
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Author: Alexandra Gaevskaya
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Alexandra Gaevskaya
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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: Ronald Hoppe
Publisher: Springer
ISBN: 3319080253
Category : Computers
Languages : en
Pages : 422
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Book Description
This book on PDE Constrained Optimization contains contributions on the mathematical analysis and numerical solution of constrained optimal control and optimization problems where a partial differential equation (PDE) or a system of PDEs appears as an essential part of the constraints. The appropriate treatment of such problems requires a fundamental understanding of the subtle interplay between optimization in function spaces and numerical discretization techniques and relies on advanced methodologies from the theory of PDEs and numerical analysis as well as scientific computing. The contributions reflect the work of the European Science Foundation Networking Programme ’Optimization with PDEs’ (OPTPDE).
Author: Michael Kieweg
Publisher:
ISBN:
Category :
Languages : en
Pages : 128
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Book Description
Author: Johannes Kraus
Publisher: Walter de Gruyter
ISBN: 3110927098
Category : Mathematics
Languages : en
Pages : 241
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Book Description
This book contains four survey papers related to different topics in computational mechanics, in particular (1) novel discretization and solver techniques in mechanics and (2) inverse, control, and optimization problems in mechanics. These topics were considered in lectures, seminars, tutorials, and workshops at the Special Semester on Computational Mechanics held at the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, in December 2005.
Author: Michael Hinze
Publisher: Springer Science & Business Media
ISBN: 1402088396
Category : Mathematics
Languages : en
Pages : 279
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Book Description
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: Olli Mali
Publisher: Springer Science & Business Media
ISBN: 9400775814
Category : Computers
Languages : en
Pages : 366
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Book Description
The importance of accuracy verification methods was understood at the very beginning of the development of numerical analysis. Recent decades have seen a rapid growth of results related to adaptive numerical methods and a posteriori estimates. However, in this important area there often exists a noticeable gap between mathematicians creating the theory and researchers developing applied algorithms that could be used in engineering and scientific computations for guaranteed and efficient error control. The goals of the book are to (1) give a transparent explanation of the underlying mathematical theory in a style accessible not only to advanced numerical analysts but also to engineers and students; (2) present detailed step-by-step algorithms that follow from a theory; (3) discuss their advantages and drawbacks, areas of applicability, give recommendations and examples.
Author: Mark Ainsworth
Publisher: John Wiley & Sons
ISBN: 1118031075
Category : Mathematics
Languages : en
Pages : 266
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Book Description
An up-to-date, one-stop reference-complete with applications This volume presents the most up-to-date information available on aposteriori error estimation for finite element approximation inmechanics and mathematics. It emphasizes methods for ellipticboundary value problems and includes applications to incompressibleflow and nonlinear problems. Recent years have seen an explosion in the study of a posteriorierror estimators due to their remarkable influence on improvingboth accuracy and reliability in scientific computing. In an effortto provide an accessible source, the authors have sought to presentkey ideas and common principles on a sound mathematicalfooting. Topics covered in this timely reference include: * Implicit and explicit a posteriori error estimators * Recovery-based error estimators * Estimators, indicators, and hierarchic bases * The equilibrated residual method * Methodology for the comparison of estimators * Estimation of errors in quantities of interest A Posteriori Error Estimation in Finite Element Analysis is a lucidand convenient resource for researchers in almost any field offinite element methods, and for applied mathematicians andengineers who have an interest in error estimation and/or finiteelements.
Author: Hans Georg Bock
Publisher: Springer Science & Business Media
ISBN: 3642303676
Category : Mathematics
Languages : en
Pages : 342
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Book Description
This judicious selection of articles combines mathematical and numerical methods to apply parameter estimation and optimum experimental design in a range of contexts. These include fields as diverse as biology, medicine, chemistry, environmental physics, image processing and computer vision. The material chosen was presented at a multidisciplinary workshop on parameter estimation held in 2009 in Heidelberg. The contributions show how indispensable efficient methods of applied mathematics and computer-based modeling can be to enhancing the quality of interdisciplinary research. The use of scientific computing to model, simulate, and optimize complex processes has become a standard methodology in many scientific fields, as well as in industry. Demonstrating that the use of state-of-the-art optimization techniques in a number of research areas has much potential for improvement, this book provides advanced numerical methods and the very latest results for the applications under consideration.
Author: Sergey I. Repin
Publisher: Walter de Gruyter
ISBN: 3110203049
Category : Mathematics
Languages : en
Pages : 329
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Book Description
This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis. After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.
