Quality of Approximate Balanced Truncation

TL;DR

This paper investigates the accuracy of approximate balanced truncation in model reduction, establishing global error bounds when using approximate Gramians, which is crucial for large-scale systems where exact Gramians are infeasible.

Contribution

It provides the first rigorous global error bounds for reduced systems obtained through approximate balanced truncation, addressing a key gap in large-scale system reduction.

Findings

Derived global error bounds for approximate balanced truncation

Validated bounds through numerical experiments

Enhanced understanding of approximation quality in large systems

Abstract

Model reduction is a powerful tool in dealing with numerical simulation of large scale dynamic systems for studying complex physical systems. Two major types of model reduction methods for linear time-invariant dynamic systems are Krylov subspace-based methods and balanced truncation-based methods. The methods of the second type are much more theoretically sound than the first type in that there is a fairly tight global error bound on the approximation error between the original system and the reduced one. It is noted that the error bound is established based upon the availability of the exact controllability and observability Gramians. However, numerically, the Gramians are not available and have to be numerically calculated, and for a large scale system, a viable option is to compute low-rank approximations of the Gramians from which an approximate balanced truncation is then performed. Hence, rigorously speaking, the existing global error bound is not applicable to any reduced system obtained via approximate Gramians. The goal of this paper is to address this issue by establishing global error bounds for reduced systems via approximate balanced truncation.