Numerical PDE-Constrained Optimization

verfasst von: Juan Carlos De los Reyes

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Optimization

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book introduces, in an accessible way, the basic elements of Numerical PDE-Constrained Optimization, from the derivation of optimality conditions to the design of solution algorithms. Numerical optimization methods in function-spaces and their application to PDE-constrained problems are carefully presented. The developed results are illustrated with several examples, including linear and nonlinear ones. In addition, MATLAB codes, for representative problems, are included. Furthermore, recent results in the emerging field of nonsmooth numerical PDE constrained optimization are also covered. The book provides an overview on the derivation of optimality conditions and on some solution algorithms for problems involving bound constraints, state-constraints, sparse cost functionals and variational inequality constraints.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

Let Ω be a bounded three-dimensional domain with boundary Γ, which represents a body that has to be heated. We may act along the boundary by setting a temperature \(u=u(x)\) and, in that manner, change the temperature distribution inside the body. The goal of the problem consists in getting as close as possible to a given desired temperature distribution z _d(x) in Ω.

Juan Carlos De los Reyes

2. Basic Theory of Partial Differential Equations and Their Discretization

Abstract

In this chapter we present some basic elements of the analysis of partial differential equations, and of their numerical discretization by finite differences. Our aim is to introduce some notions that enable the reader to follow the material developed in the subsequent chapters. Both the analysis and the numerical solution of partial differential equations (PDEs) are research areas by themselves, with a large amount of related literature. We refer, for instance, to the books [9,19] for the analysis of PDEs and to, e.g., [23, 52] for their numerical approximation.

Juan Carlos De los Reyes

3. Theory of PDE-Constrained Optimization

Abstract

In this chapter we present optimality conditions of first and second order for optimization problems with PDE constraints. The theory is illustrated with several examples.

Juan Carlos De los Reyes

4. Numerical Optimization Methods

Abstract

In this chapter we present and analyze some infinite dimensional optimization methods for the solution of problem (.1). Once the infinite dimensional algorithm is posed, the discretization of the partial differential equations is carried out. Such an approach is known as optimize-then-discretize in contrast to the discretize-then—optimize one, where the equations and the cost functional are first discretized and the problem is then solved using large-scale optimization tools.

Juan Carlos De los Reyes

5. Box-Constrained Problems

Abstract

In this chapter we present optimality conditions and solution methods for PDE-constrained optimization problems with so-called box constraints.

Juan Carlos De los Reyes

6. Nonsmooth PDE-Constrained Optimization

Abstract

In the last chapter, some representative nonsmooth PDE-constrained optimization problems are addressed. Problems with cost functionals involving the L1-norm, with state constraints, or with variational inequality constraints are considered.

Juan Carlos De los Reyes

Backmatter

Titel: Numerical PDE-Constrained Optimization
verfasst von: Juan Carlos De los Reyes
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-13395-9
Print ISBN: 978-3-319-13394-2
DOI: https://doi.org/10.1007/978-3-319-13395-9