Quality of Approximate Balanced Truncation

TL;DR

The paper addresses reliable model reduction for very large-scale LTI systems by focusing on approximate balanced truncation using low-rank Gramian factors. It notes that classical global error bounds derived under exact Gramians do not apply to reduced models built from approximate Gramians, and it establishes global error bounds for reduced models obtained via approximate balanced truncation. This work bridges theory and practice, providing a framework to quantify the accuracy of reduced models when only low-rank Gramian information is available. The results enable safer deployment of BT-based reductions in large-scale applications where exact Gramian computation is infeasible.

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.