Model Order Reduction by Proper Orthogonal Decomposition

We provide an introduction to proper orthogonal decomposition (POD) model order reduction with focus on (nonlinear) parametric partial differential equations (PDEs) and (nonlinear) time-dependent PDEs, and PDE-constrained optimization with POD surrogate models as application. We cover the relation of POD and singular value decomposition, POD from the infinite-dimensional perspective, reduction of nonlinearities, certification with a priori and a posteriori error estimates, spatial and temporal adaptivity, input dependency of the POD surrogate model, POD basis update strategies in optimal control with surrogate models, and sketch related algorithmic frameworks. The perspective of the method is demonstrated with several numerical examples.publishe