# EU Regional School - Benner Seminar

Location: AICES Seminar Room 115, 1st floor, Schinkelstr. 2, 52062 Aachen

## Prof. Dr. Benner - Model Reduction Using Rational Approximation Techniques

Computational Methods in Systems and Control Theory
Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

### Abstract

Model Order Reduction (MOR) is an ubiquitous tool in analysis, simulation and in particular design of dynamical systems, controllers and regulators, circuit simulation, structural dynamics, CFD, etc. In the past decades, different science and engineering communities have developed a number of methods for reducing the order of a given model. We will focus on classes of methods treating the model reduction problem as rational approximation. Linear systems can be represented by their transfer function in frequency domain. This is a scalar or matrix-valued function, the entries of which are rational functions of a complex variable. Approximating the transfer function of a linear system by an analogous (matrix-valued) rational function of lower degree can therefore be seen as a rational approximation problem. It is less obvious how the reduction of a nonlinear system can be understood as rational approximation. We will show how nonlinear model reduction methods can be developed using similar ideas as in the linear case by gradually increasing the degree of nonlinearity in the dynamical systems to be reduced.

In this lecture, we will discuss some of the most prominent methods used for linear systems: interpolatory methods which construct an approximate model by rational interpolation of the system's transfer function, and balanced truncation - a method based on a best approximation of a certain energy transfer operator related to the system. We will also compare the properties of these approaches and highlight similarities. We will demonstrate the effectiveness of the different approaches by showing numerical results obtained for real-world applications from various application areas.

Furthermore, we will discuss the extension of some of these approaches to nonlinear systems. We will also briefly discuss differences and similarities to other popular MOR methods for nonlinear systems like Proper Orthogonal Decomposition (POD) and the Reduced Basis Method. The applicability of all methods for nonlinear systems discussed depends much more on the application than for linear MOR methods. We conclude by presenting some open questions and challenges for future research on nonlinear MOR methods and their application.