Optimization with Pde Constraints

ISBN-13: 9781402088384 / Angielski / Twarda / 2008 / 270 str.

Michael Hinze; Rene Pinnau; Michael Ulbrich

Opis

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.

Recenzje

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)

Spis treści

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 Gˆateaux- 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

Blurb

This book presents a modern introduction of pde constrained optimization. It provides a precise functional analytic treatment via optimality conditions and a state-of-the-art, non-smooth algorithmical framework. Furthermore, new structure-exploiting discrete concepts and large scale, practically relevant applications are presented. The main focus is on the algorithmical and numerical treatment of pde constrained optimization problems on the infinite dimensional level. A particular emphasis is on simple constraints, such as pointwise bounds on controls and states. For these practically important situations, tailored Newton- and SQP-type solution algorithms are proposed and a general convergence framework is developed. This is complemented with the numerical analysis of structure-preserving Galerkin schemes for optimization problems with elliptic and parabolic equations. Finally, along with the optimization of semiconductor devices and the optimization of glass cooling processes, two challenging applications of pde constrained optimization are presented. They demonstrate the scope of this emerging research field for future engineering applications.