Adaptive Finite Element Methods for Control Constrained Distributed and Boundary Optimal Control Problems PDF Download

Author: Michael Hintermüller
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Dieter Sirch
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832525572
Category : Mathematics
Languages : en
Pages : 166

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Book Description
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: Ronald H. W. Hoppe
Publisher:
ISBN:
Category :
Languages : en
Pages :

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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: Kristina Kohls
Publisher:
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Category :
Languages : en
Pages :

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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: W. Fitzgibbon
Publisher: Springer Science & Business Media
ISBN: 9048132398
Category : Science
Languages : en
Pages : 252

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Standing at the intersection of mathematics and scientific computing, this collection of state-of-the-art papers in nonlinear PDEs examines their applications to subjects as diverse as dynamical systems, computational mechanics, and the mathematics of finance.

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: Andrea Bonito
Publisher: Elsevier
ISBN: 0444643060
Category : Mathematics
Languages : en
Pages : 572

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Book Description
Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering. - About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization - Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading - The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs

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).