A Structure-preserving Model Reduction Algorithm for Dynamical Systems with Nonlinear Frequency Dependence

Very large-scale dynamical systems, even linear time-invariant systems, can present significant computational difficulties when used in numerical simulation. Model reduction is one response to this challenge but standard methods often are restricted to systems that are presented as standard first-order realizations; in the frequency domain such systems will be linear in the frequency parameter. We consider here dynamical systems with a nonlinear frequency dependence; systems for which either a standard first-order realization is unknown or inconvenient to obtain. Such systems may nonetheless have realizations that reflect important structural features of the model and we may wish to retain this structure in any reduced model used as a surrogate. In this work, we present a structure-preserving model reduction algorithm for systems having quite general nonlinear frequency dependence. We take advantage of recent algorithms that produce high quality rational interpolants to transfer functions that only require transfer function evaluation, thus allowing for nonstandard realizations that are nonlinear in the frequency parameter. However, our final reduced model will have a structure that reflects the structure of the original system, and indeed, may not have a rational transfer function. We illustrate our approach on a benchmark problem that offers a transcendental transfer function.