Optimization with PDE Constraints by Michael HinzeOptimization with PDE Constraints by Michael Hinze

Optimization with PDE Constraints

byMichael Hinze

Hardcover | November 14, 2008

Pricing and Purchase Info

$146.30 online 
$151.95 list price
Earn 732 plum® points

Prices and offers may vary in store

Quantity:

In stock online

Ships free on orders over $25

Not available in stores

about

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.
Title:Optimization with PDE ConstraintsFormat:HardcoverDimensions:270 pagesPublished:November 14, 2008Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:1402088388

ISBN - 13:9781402088384

Reviews

Table of 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 GËateaux- 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 linear-quadratic 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 linear-quadratic optimal control problem . . . . . . . . . . 57 1.6.4 A Lagrangian-based 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 control-constrained problems . . . . . . . . . . . . 66 1.7.3 Optimality conditions for problems with general constraints . . . . . . . . 74 1.8 Optimal control of instationary incompressible Navier-Stokes 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

Editorial Reviews

From the reviews:"The book presents a state-of-the-art of optimization problems described by partial differential equations (PDEs) and algorithms for obtaining their solutions. Solving optimization problems with constraints given in terms of PDEs is one of the most challenging problems appearing, e.g., in industry, medical and economical applications. The book consists of four chapters. . This well-written book can be recommended to scientists and graduate students working in the fields of optimal control theory, optimization algorithms and numerical solving of optimization problems described by PDEs." (Wieslaw Kotarski, Zentralblatt MATH, Vol. 1167, 2009)