Model Order Reduction for PDE Constrained Optimization

Abstract

The optimization and control of systems governed by partial differential equations (PDEs) usually requires numerous evaluations of the forward problem or the optimality system. Despite the fact that many recent efforts, many of which are reported in this book, have been made to limit or reduce the number of evaluations to 5–10, this cannot be achieved in all situations and even if this is possible, these evaluations may still require a formidable computational effort. For situations where this effort is not acceptable, model order reduction can be a means to significantly reduce the required computational resources. Here, we will survey some of the most popular approaches that can be used for this purpose. In particular, we address the issues arising in the strategies discretize-then-optimize, in which the optimality system of the reduced-order model has to be solved, and optimize-then-discretize, where a reduced-order model of the optimality system has to be found. The methods discussed include versions of proper orthogonal decomposition (POD) adapted to PDE constrained optimization as well as system-theoretic methods.