A time domain a posteriori error bound for balancing-related model order reduction

TL;DR

This paper introduces a rigorous, time-domain a posteriori error bound for balancing-related model order reduction methods, improving error estimation accuracy for linear systems over finite intervals.

Contribution

It develops a new a posteriori error bound for balanced truncation and singular perturbation, enhancing existing a priori bounds with rigorous, refined estimates applicable to systems with initial conditions.

Findings

The error bound is sharp and reliable in numerical tests.

The method accounts for nonzero initial conditions.

The approach improves error estimation accuracy in model reduction.

Abstract

The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model. While high fidelity can be achieved by a variety of methods, only a few of them allow for rigorous error control. In this paper, we propose a rigorous error bound for the reduction of linear systems with balancing-related reduction methods. More specifically, we consider the simulation over a finite time interval and provide an a posteriori adaption of the standard a priori bound for Balanced Truncation and Balanced Singular Perturbation Approximation in that setting, which improves the error estimation while still yielding a rigorous bound. Our result is based on an error splitting induced by a Fourier series approximation of the input and a subsequent refined error analysis. We make use of system-theoretic concepts, such as the notion of signal generator driven systems, steady-states and observability. Our bound is also applicable in the presence of nonzero initial conditions. Numerical evidence for the sharpness of the bound is given.