Constrained Programming for Optimization Problems in PDE

Optimization problems in PDE models are often approached by considering the PDE model as a black-box input-output relation and thus solving an unconstrained optimization problem. In contrast to that, considering the PDE model as a side condition of a resulting constrained programming problem enables us to simultaneously solve the PDE model equations together with the optimization problem. This leads to high computational gains. In order to fully exploit the constrained programming structure, special algorithmic features have to be dovetailed, like partially reduced SQP methods or tailored multigrid methods for the solution of QP subproblems. Applications of these features are demonstrated in practical examples from shape optimization of turbine blades and inverse modeling in environmental engineering. 1 The problem class Practical optimization problems in partially diierential equations (PDE) typically evolve from a simulation task for a PDE model of a physical or technical process. This PDE model involves state variables y to be determined by a presumably uniquely solvable system of model equations c(y) = 0 (i.e. @c=@y is nonsingular). Additionally, there are innuence variables u in the model equations|which therefore depend on u as well|c(y; u) = 0, which are to be chosen in such a way that an objective functional f(y; u) 2 R has an optimal value. The states y can be considered the output of the PDE model corresponding to the input u via the model equations. This point of view provides a natural separation of variables into \free" control variables u and \depen-dent" state variables y. In most cases, the optimal solution has to satisfy additional restrictions, r(y; u) 0 or = 0. The standard numerical optimization approach is to conclude from the implicit function theorem the existence of a unique function y(u), which represents the input-output relationship mentioned above. Thus, the optimization problem is transformed into an unconstrained one (ignoring restrictions r 0; = 0 for a moment) for the objective functional ~ f(u) := f(y(u); u) which involves only the u-variable space. Since the space of the u-variables