A-priori Error Bounds for Finite Element Approximation of Elliptic Optimal Control Problems with Gradient Constraints PDF Download
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Author: Klaus Deckelnick
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
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Author: Klaus Deckelnick
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
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Author: Günter Leugering
Publisher: Springer
ISBN: 3319050834
Category : Mathematics
Languages : en
Pages : 539
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Book Description
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: 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:
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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: Klaus Deckelnick
Publisher:
ISBN:
Category :
Languages : en
Pages : 12
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Author: Wenbin Liu
Publisher: Alpha Science International Limited
ISBN: 9781842657157
Category : Mathematics
Languages : en
Pages : 197
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Summary: "This book emphasizes the discussions of some unique issues from the adaptive finite element approximation of optimal control. The main idea used in the approximation error analysis (both a priori and a posteriori) is to first combine convex analysis and interpolation error estimations of suitable interpolators, which much depend on the structure of the control constraints, to derive the error estimates for the control via the variational inequalities in the optimality conditions, and then to apply the standard techniques to derive the error estimates for the state equations. The need, the framework and the techniques of using multi adaptive meshes in developing efficient numerical algorithms for optimal control have been emphasized throughout the book. The book starts from several typical examples of optimal control problems and then discusses existence and optimality conditions for some optimal control problems. It is believed that these discussions are especially useful for the researchers and students who first entered this area. Then the finite element approximation schemes for several typical optimal control problems are set up, their a priori and a posteriori error estimates are derived following the main idea mentioned, and their computational methods are studied."-- Publisher website, viewed 13th July, 2012.
Author: Michal Krizek
Publisher: Springer Science & Business Media
ISBN: 3642185606
Category : Science
Languages : en
Pages : 405
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Book Description
The position taken in this collection of pedagogically written essays is that conjugate gradient algorithms and finite element methods complement each other extremely well. Via their combinations practitioners have been able to solve complicated, direct and inverse, multidemensional problems modeled by ordinary or partial differential equations and inequalities, not necessarily linear, optimal control and optimal design being part of these problems. The aim of this book is to present both methods in the context of complicated problems modeled by linear and nonlinear partial differential equations, to provide an in-depth discussion on their implementation aspects. The authors show that conjugate gradient methods and finite element methods apply to the solution of real-life problems. They address graduate students as well as experts in scientific computing.
Author: Jean-Pierre Aubin
Publisher: Courier Corporation
ISBN: 0486457915
Category : Mathematics
Languages : en
Pages : 386
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Book Description
A marriage of the finite-differences method with variational methods for solving boundary-value problems, the finite-element method is superior in many ways to finite-differences alone. This self-contained text for advanced undergraduates and graduate students is intended to imbed this combination of methods into the framework of functional analysis and to explain its applications to approximation of nonhomogeneous boundary-value problems for elliptic operators. The treatment begins with a summary of the main results established in the book. Chapter 1 introduces the variational method and the finite-difference method in the simple case of second-order differential equations. Chapters 2 and 3 concern abstract approximations of Hilbert spaces and linear operators, and Chapters 4 and 5 study finite-element approximations of Sobolev spaces. The remaining four chapters consider several methods for approximating nonhomogeneous boundary-value problems for elliptic operators.
Author: Zi-Cai Li
Publisher: World Scientific
ISBN: 9789810202927
Category : Mathematics
Languages : en
Pages : 286
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Book Description
This book presents two kinds of numerical methods for solving elliptic boundary value problems with singularities. Part I gives the boundary methods which use analytic and singular expansions, and Part II the nonconforming methods combining finite element methods (FEM) (or finite difference methods (FDM)) and singular (or analytic) expansions. The advantage of these methods over the standard FEM and FDM is that they can cope with complicated geometrical boundaries and boundary conditions as well as singularity. Therefore, accurate numerical solutions near singularities can be obtained. The description of methods, error bounds, stability analysis and numerical experiments are provided for the typical problems with angular, interface and infinity singularities. However, the approximate techniques and coupling strategy given can be applied to solving other PDE and engineering problems with singularities as well. This book is derived from the author's Ph. D. thesis which won the 1987 best doctoral dissertation award given by the Canadian Applied Mathematics Society.
Author: Christian Meyer
Publisher:
ISBN:
Category :
Languages : en
Pages : 22
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