An $\mathcal H_2$-Type Error Bound for Time-Limited Balanced Truncation

TL;DR

This paper establishes the first $ ext{H}_2$-type error bound for time-limited balanced truncation, improving the theoretical understanding and practical accuracy of reduced models over finite time intervals.

Contribution

It provides the first $ ext{H}_2$ error bound for time-limited balanced truncation, enhancing the theoretical foundation of this model reduction technique.

Findings

The error bound is validated through numerical experiments.

The bound applies to two different representations of TLBT.

Results show improved accuracy of TLBT over traditional methods.

Abstract

When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an $\mathcal H_2$ error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments.