| Hörsaal D2
Vortrag: POD Reduced-Order Methods in Optimal Control and Design
Optimal control problems for partial differential equations (PDEs) are often hard to tackle numerically because their discretization by, e.g., finite elements or finite differences leads to very large scale optimization problems. Therefore, different techniques of model reduction were developed to approximate these problems by smaller ones that are tractable with less effort. Currently, Proper orthogonal decomposition (POD) is probably the mostly used and most successful model reduction technique for nonlinear optimal control problems, where the basis functions contain information from the solutions of the dynamical system at pre-specified time-instances, so-called snapshots. Due to a possible linear dependence or almost linear dependence the snapshots themselves are not appropriate as a basis. Hence a singular value decomposition is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. Therefore, a useful application is the use in optimization problems, where a PDE solver is part of the function evaluation. Obviously, thinking of a gradient evaluation or even a step-size rule in the optimization algorithm, an expensive function evaluation leads to an enormous amount of computing time. Here, the reduced order model can replace the system given by a PDE in the objective function. It is quite common that a PDE can be replaced by a five- or ten-dimensional system of ordinary differential equations. This results computationally in a very fast method for optimization compared to the effort for the computation of a single solution of a PDE.
In the talk we consider different optimization problems for PDEs (involving multiobjective optimal control problems) and discuss a-priori as well as a-posteriori error analysis for the reduced-order approximations.
The presented results are joint work with Stefan Banholzer, Dennis Beermann, Michael Dellnitz, Luca Mechelli and Sebastian Peitz.