Numerical simulation of large-scale dynamical systems plays a crucial role and may be the only possibility in studying a great variety of complex physical phenomena with applications ranging from heat transfer to fluid dynamics, to signal propagation and interference in electronic circuits, and many more. However, these large-scale dynamical systems present significant computational difficulties when used in numerical simulation. Model reduction aims to reduce this computational burden by constructing simpler (reduced order) models, which are much easier and faster to simulate yet accurately represent the original system. These simpler reduced order models can then serve as efficient surrogates for the original, replacing them as components in larger systems; facilitating rapid development of controllers for real time applications; enabling optimal system design and uncertainty analysis.

In this talk, we will first review model reduction of dynamical systems by using rational interpolation as the underlying framework. The concept of transfer function will prove fundamental in this setting. We will discuss how interpolation, consequently model reduction, can be performed optimally in an appropriate system norm. We will illustrate the theoretical results on an example arising in flow control.  We will then discuss extending the interpolation framework to parametric dynamical systems and illustrate an example arising in nonlinear inversion, namely diffuse optical tomography.