Optimization with PDE constraints
 Responsibility
 M. Hinze ... [et al.].
 Language
 English.
 Imprint
 [New York?] : Springer, c2009.
 Physical description
 xi, 270 p. : ill. ; 24 cm.
 Series
 Mathematical modellingtheory and applications ; v. 23.
Access
Available online
 dx.doi.org SpringerLink
 dx.doi.org SpringerLink
Math & Statistics Library

Stacks

Unknown
QA402.5 .O677 2009

Unknown
QA402.5 .O677 2009
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Creators/Contributors
 Contributor
 Hinze, Michael.
Contents/Summary
 Bibliography
 Includes bibliographical references (p. 265270).
 Contents

 1 Analytical Background and Optimality Theory ... 7 1.1 Introduction and examples ... 7 1.1.1 Introduction ... 7 1.1.2 Examples for optimization problems with PDEs ... 10 1.1.3 Optimization of a stationary heating process ... 10 1.1.4 Optimization of an unsteady heating processes ... 13 1.1.5 Optimal design ... 13 1.2 Linear functional analysis and Sobolev spaces ... 14 1.2.1 Banach and Hilbert spaces ... 15 1.2.2 Sobolev spaces ... 18 1.2.3 Weak convergence ... 27 1.3 Weak solutions of elliptic and parabolic PDEs ... 29 1.3.1 Weak solutions of elliptic PDEs ... 29 1.3.2 Weak solutions of parabolic PDEs ... 37 1.4 Gateaux and Fr'echet Differentiability ... 48 1.4.1 Basic definitions ... 48 1.4.2 Implicit function theorem ... 50 1.5 Existence of optimalcontrols ... 50 1.5.1 Existence result for a general linearquadratic problem ... 50 1.5.2 Existence results for nonlinear problems ... 52 1.5.3 Applications ... 53 1.6 Reduced problem, sensitivities and adjoints ... 55 1.6.1 Sensitivity approach ... 55 1.6.2 Adjoint approach ... 56 1.6.3 Application to a linearquadratic optimal control problem ... 57 1.6.4 A Lagrangianbased view of the adjoint approach ... 59 3 4 Contents 1.6.5 Second derivatives ... 60 1.7 Optimality conditions ... 61 1.7.1 Optimality conditions for simply constrained problems ... 61 1.7.2 Optimality conditions for controlconstrained problems ... 66 1.7.3 Optimality conditions for problems with general constraints ... 74 1.8 Optimal control of instationary incompressible NavierStokes flow ... 80 1.8.1 Functional analytic setting ... 81 1.8.2 Analysis of the flow control problem ... 83 1.8.3 Reduced Optimal Control Problem ... 85 2 Optimization.
 (source: Nielsen Book Data)
 Publisher's Summary
 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 modelbased numerical si lations to modelbased 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 PDEconstraints 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2009
 Series
 Mathematical modelling: theory and applications ; v. 23
 Note
 Based on the lecture notes of the autumn school Modellierung und Optimierung mit Partiellen Differentialgleichungen, which was held in Sept. 2005 at the Universität Hamburg.
 ISBN
 1402088388
 9781402088384