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

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 H2 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 H2 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.