Optimization with PDE constraints.

*(English)*Zbl 1167.49001
Mathematical Modelling: Theory and Applications 23. Dordrecht: Springer (ISBN 978-1-4020-8838-4/hbk; 978-1-4020-8839-1/ebook). xi, 270 p. (2009).

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.

Chapter 1 provides an introduction to analytical background and optimality theory for optimization problems with PDEs. First, necessary background in functional analysis, Sobolev spaces, the theory of week solutions for elliptic and parabolic equations and Gáteaux and Fréchet differentiability are presented. Next, existence of optimal controls both for linear quadratic and nonlinear abstract problems in Hilbert spaces are considered. Further, first order optimality conditions for problems described by PDEs with control and state constraints are derived. Elliptic, parabolic and Navier-Stokes optimal control problems are used as illustrative examples.

The second chapter presents a selection of important algorithms for optimization problems with PDEs. Several variants of generalized Newton methods in Banach spaces are derived and analysed. Problems of convergence of the considered algorithms are discussed in details. Elliptic and Navier-Stokes optimal control problems are used as illustrative examples.

The third chapter gives an introduction to discrete concepts for optimization problems with PDEs constraints. Two approaches: “First discretize, then optimize” and “First optimize, then discretize” are compared and discussed. Several numerical examples together with error analysis are presented.

The final fourth chapter is devoted to the study of two industrial applications in which optimization with PDEs plays a crucial role. Two problems from modern semiconductor design and glass industry are described in details together with their numerical studies of related optimal control problems.

Every chapter of the book was written by another author. In spite of this, four chapters of the book are logically and smoothly connected. Notations and terminology used throughout the book are consequent. 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.

Chapter 1 provides an introduction to analytical background and optimality theory for optimization problems with PDEs. First, necessary background in functional analysis, Sobolev spaces, the theory of week solutions for elliptic and parabolic equations and Gáteaux and Fréchet differentiability are presented. Next, existence of optimal controls both for linear quadratic and nonlinear abstract problems in Hilbert spaces are considered. Further, first order optimality conditions for problems described by PDEs with control and state constraints are derived. Elliptic, parabolic and Navier-Stokes optimal control problems are used as illustrative examples.

The second chapter presents a selection of important algorithms for optimization problems with PDEs. Several variants of generalized Newton methods in Banach spaces are derived and analysed. Problems of convergence of the considered algorithms are discussed in details. Elliptic and Navier-Stokes optimal control problems are used as illustrative examples.

The third chapter gives an introduction to discrete concepts for optimization problems with PDEs constraints. Two approaches: “First discretize, then optimize” and “First optimize, then discretize” are compared and discussed. Several numerical examples together with error analysis are presented.

The final fourth chapter is devoted to the study of two industrial applications in which optimization with PDEs plays a crucial role. Two problems from modern semiconductor design and glass industry are described in details together with their numerical studies of related optimal control problems.

Every chapter of the book was written by another author. In spite of this, four chapters of the book are logically and smoothly connected. Notations and terminology used throughout the book are consequent. 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.

Reviewer: Wiesław Kotarski (Sosnowiec)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |