Optimization with PDE Constraints

Abstract

Solving optimization problems subject to constraints given in terms of partial dif- 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 simu- lations to model-based design and optimal control is crucial. For the treatment of such optimization problems the interaction of optimization techniques and numer- ical simulation plays a central role. After proper discretization, the number of op- timization variables varies between 103 and 1010. 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 fur- ther explore the specific 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 amodern introduction to the rapidly developing math- ematical field of optimization with PDE constraints. The first chapter introduces to the analytical background and optimality theory for optimization problems with PDEs. Optimization problems with PDE-constraints are posed in infinite dimen- 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. These results form the foundation of efficient optimizationmethods in function space, their adequate numerical realization, mesh independence results and error estimators. The chapter starts with an introduction to the necessary background in functional analy- sis, Sobolev spaces and the theory ofweak solutions for elliptic and parabolic PDEs. These ingredients are then applied to study PDE-constrained optimization problems.