Optimization with PDE Constraints

1. Front Matter

   Pages i-xi

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2. Analytical Background and Optimality Theory

   • Stefan Ulbrich

   Pages 1-95

3. Optimization Methods in Banach Spaces

   • Michael Ulbrich

   Pages 97-156

4. Discrete Concepts in PDE Constrained Optimization

   • Michael Hinze

   Pages 157-232

5. Applications

   • René Pinnau

   Pages 233-263

6. Back Matter

   Pages 265-270

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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.

• Applications
• Banach spaces
• Optimal control
• Optimality theory
• PDE constrained optimization
• Tailored discrete concepts
• numerical analysis

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)

• Dept. Mathematik, Universität Hamburg, Hamburg, Germany

  Michael Hinze

• FB Mathematik, TU Kaiserslautern, Kaiserslautern Gebäude 48, Germany

  Rene Pinnau

• Lehrstuhl für Mathematische Optimierung, Fakultät für Mathematik, Technische Universität München, Garching, Germany

  Michael Ulbrich

• Fachbereich Mathematik, AG10, Technische Universität Darmstadt, Darmstadt, Germany

  Stefan Ulbrich

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