Transfer function interpolation remainder formula of rational Krylov subspace methods

TL;DR

This paper derives an explicit error formula for rational Krylov subspace methods in model order reduction, revealing new insights into the interpolation remainder and proposing an algorithm for $H_{}$ norm MOR.

Contribution

It introduces a novel explicit MOR error formula involving shifts and Ritz values, enhancing understanding of transfer function approximation in Krylov methods.

Findings

Derived an explicit MOR error expression involving shifts and Ritz values

Provided two interpretations of the error as an interpolation and quadrature remainder

Suggested a greedy algorithm for $H_{}$ norm MOR based on the error formula

Abstract

Rational Krylov subspace projection methods have proven to be a highly successful approach in the field of model order reduction (MOR), primarily due to the fact that some derivatives of the approximate and original transfer functions are identical.This is the well-known theory of moments matching. Nevertheless, the properties of points situated at considerable distances from the interplation nodes remain underexplored. In this paper, we present the explicit expression of the MOR error, which involves both the shifts and the Ritz values.The superiority of our discoveries over the known moments matching theory can be likened to the disparity between the Lagrange and Peano type remainder formulas in Taylor's theorem. Furthermore, two explanations are provided for the error formula with respect to the two parameters in the resolvent function. One explanation reveals that the MOR error is an interplation remainder, while the other explanation implies that the error is also a Gauss-Christoffel quadrature remainder.By applying the error formula, we suggest a greedy algorithm for the interpolatory $H_{\infty}$ norm MOR.