Mathematical aspects of CFD-based optimization

There exist several computational strategies of different efficiency for the solution of model-based optimization problems - particularly, in the case of models based on challenging CFD problems. Applied mathematics provides means for their analysis and for advice on their proper usage. In this chapter, methods are mainly analyzed based on the explicit treatment of the underlying CFD-problem as a constraint of a nonlinear optimization problem, thus providing the potential for high computational efficiency. Methods of this form are termed optimization boundary value problem methods, simultaneous optimization methods or one-shot optimization methods. The necessary conditions of optimality play a key structural role in devising those strategies. Special attention is given to the following issues: modular sequential quadratic programming with approximate linear solvers, preconditioning of the Karush-Kuhn-Tucker (KKT) system and multigrid optimization in the case of stationary problems. In the case of unsteady problems, we will concentrate on time-domain decomposition such as by multiple shooting, and on algorithmic developments for real-time optimization. The aim of the presentation is to give a survey on advanced and fast methods for optimization within a CFD framework. For details, the reader is referred to the relevant literature.