## Matrix Equations and Model Reduction

by P. Benner

Model reduction is a ubiquitous tool in analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the order of a given model. Often these methods have been derived in parallel in different disciplines with particular applications in mind. In this course, we will derive some of the most prominent methods used for linear systems: modal truncation based on eigenvalue algorithms, 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. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these new techniques, the range of applicability of some of the methods has considerably widened. Particular emphasis will be given to the numerical solution of matrix equations which is the main computational bottleneck in methods based on balanced truncation. We will also present some ideas in the direction towards nonlinear model reduction at the end of the course. Numerical experiments to be performed in the exercise session will show the efficiency of several approaches when applied to real-world examples from several disciplines.

An outline is as follows:

- Basics in numerical linear algebra: SVD and unitary matrix factorizations, Krylov subspace methods
- Basics in systems theory: reachability and observability and their algebraic characterization, transfer functions, system properties (stability, passivity, etc.)
- Model reduction principles, goals, and requirements, essentials of the approximation theory for dynamical systems ((Petrov-)Galerkin projection, measuring the approximation error)
- Modal truncation
- Derivation of balanced truncation
- Numerical computation of reduced-order models for by balanced truncation
- Numerical solution of algebraic Lyapunov equations
- Variants of balanced truncation
- Numerical solution of algebraic Riccati equations.
- Model reduction based on Krylov subspaces: moment-matching, rational interpolation, and the Lanczos-Pad\'e connection
- (Sub-)optimal reduced-order models using rational interpolation (''H2-optimal model reduction'')
- Nonlinear model reduction approaches: proper orthogonal decomposition (POD), (quadratic)-bilinearization.

*Contact email address: summerschool AT fudan.edu.cn*