# Model Order Reduction by Proper Orthogonal Decomposition

## TL;DR

This paper introduces Proper Orthogonal Decomposition (POD) for model order reduction, focusing on nonlinear parametric and time-dependent PDEs, with applications in PDE-constrained optimization and adaptive strategies.

## Contribution

It provides a comprehensive overview of POD-MOR, including theoretical foundations, error estimation, adaptivity, and basis update strategies, with practical numerical demonstrations.

## Key findings

- Effective reduction of nonlinear PDEs demonstrated
- Error estimates enable reliable surrogate models
- Adaptive basis updates improve control applications

## Abstract

We provide an introduction to POD-MOR with focus on (nonlinear) parametric PDEs and (nonlinear) time-dependent PDEs, and PDE constrained optimization with POD surrogate models as application. We cover the relation of POD and SVD, 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.