We present several new, efficient algorithms that extract low complexity models from frequency response measurements of large-scale dynamical systems.

Our work is motivated by the fact that, in many applications, analytical models of a dynamical system are seldom available. Instead, we may only have access to its frequency response measurements. For example, for a system with multiple inputs and outputs, we may only have access to data sets of S-parameters. In this setting, our new approach extracts models that interpolate the given measurements. The extracted models have low complexity (or reduced order) and, thus, lead to short simulation times and low data storage requirements.

The main tool used by our approach is Lagrange rational interpolation -- a generalization of the classic result of Lagrange polynomial interpolation. We present an in-depth look at Lagrange rational interpolation and provide several new insights and simplified proofs. This analysis leads to new algorithms that rely on the singular value decomposition (SVD) of the Loewner matrix pencil formed directly from the measurements.

We show several new results on rational interpolation for measurements of linear, bi-linear and quadratic-linear systems. Furthermore, we generalize these results to parametrized measurements, that is, we show how to interpolate frequency response measurements that depend on parameters.

We showcase this new approach through a series of relevant numerical examples such as n-port systems and parametrized partial differential equations.