Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Abstract

Abstract. Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H 2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H 2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.Key words. least squares, frequency response, model order reduction, vector fitting, transfer function AMS subject classifications. 34C20, 41A05, 49K15, 49M05, 93A15, 93C05, 93C151. Introduction. In many engineering applications, the dynamics that govern phenomenae of interest may be inaccessible to direct modeling, yet there may be an abundance of accurate frequency response measurements available. In such cases, one may build up an empirical dynamical system model that fits the measured frequency response data. This empirical system may then be used as a surrogate to predict behavior or derive control strategies.In other settings, one may have complete access to the underlying dynamical system of interest at least in principle (e.g., it may be an analytically derived computational model), however the full system may be a complex aggregate of many large subsystems, each perhaps representing diverse physics, and so it may be of such complexity that direct manipulation of the dynamical system is infeasible; potentially only simulation results would be available. Here, one may wish to capture the dominant dynamic features of the full aggregate system and realize them with a derived dynamical system (presumably of lower order) that can replicate the response characteristics of the full aggregate system. As before, this derived dynamical system may then be used as an efficient surrogate for the full system in contexts where performance is sensitive to model order.A natural formulation of this task leads one to a data fitting problem using rational functions and this ultimately is our principal focus. For convenience, we assume that the system of interest is a single-input/single-output...

The RKFIT Algorithm for Nonlinear Rational Approximation

“…Numerical issues arising in VFIT have been recently discussed and mitigated in [15,16]. Our approach avoids these problems altogether.…”

“…If they can be chosen freely, one can choose them as nodes of certain quadrature rules tailored to the application in the hope of improving both the numerical stability as well as the approximation quality. This idea is suggested in [15,16] for the discretized H 2 approximation of transfer function measurements, and it carries over straightforwardly to RKFIT.…”

Sampling-free parametric model reduction of systems with structured parameter variation

“…There are various ways to fit this frequency domain data. One can enforce H(s) to interpolate the data at every sampling point using the Loewner framework [5,46], or construct H(s) to fit the data in a least-squares (LS) sense [20,25,39], or force H(s) to interpolate some of the data and while minimizing the LS fit in the rest [50]. In this paper, we will fit data solely in a LS sense.…”

Tangential interpolation-based eigensystem realization algorithm for MIMO systems

“…In the frequency domain, this corresponds to sampling the transfer function and its derivates around infinity. For the cases where one has the exibility in choosing the frequency samples, a variety of techniques become available such as the Loewner framework [28], vector fitting [29,30], realization-independent IRKA (transfer function-IRKA [TF-IRKA]) [31] and various rational least-squares fitting methodologies [30,[32][33][34]. However, as stated earlier, our focus here is ERA and to make it computationally more efficient for MIMO systems with large input and output dimensions.…”