An output error bound for time-limited balanced truncation

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 spatiallydiscretized 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 H 2 error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments.

Time‐limited pseudo‐optimal ‐model order reduction

“…The computation of ‐norm can become a computationally demanding task in a large‐scale setting as it requires the computation of or . In this scenario, or may be replaced with its low‐rank approximation as suggested in [43, 45].…”

“…We solved the large‐scale Lyapunov equations (5)–(10) exactly for the sake of accuracy and the fairness of the comparison. Practically, however, these can be replaced with their low‐rank approximations, as suggested in [45]. All the experiments are conducted on a computer with Intel(R) Core(TM) i7‐8550U processors and 16 GB memory using MATLAB 2016.…”

“…However, Gugercin and Antoulas's TLBT [44] is computationally more expensive than the original TLBT, and the accuracy is also poor. An ‐norm error bound for TLBT is also proposed in [45], which is based on the error bound in [46]. TLBT is extended to more general classes of linear systems in [47–50] and bilinear systems in [51].…”

A new model order reduction method for the design of compensator by using moment matching algorithm

“…The time and frequency responses of any dynamic systems depend upon the number, position and density (numbers of poles lie within a specific region) of poles. The poles of higher order system which are far away from the imaginary axis are directly neglected in most of the MOR techniques (Prajapati and Prasad, 2018e; Redmann and Kürschner, 2018; Sabir and Ibrir, 2018; Yang and Jiang, 2018). In the proposed approach, the poles that are far away from the origin of s-plane are cumulated in the dominant poles for the determination of the lower order system.…”

Cross-Gramian-based dominant subspaces

“…In this section an error indicator for the cross-Gramian-based dominant subspace method is developed. Previous works, such as [39,46,35,44], already introduced error bounds for the Hardy H 2 -norm. Here, an H 2 -error indicator of simple structure using time-domain quantities is proposed, which is loosely related to the simplified balanced gains approach from [11].…”