Towards Time-Limited $\mathcal H_2$-Optimal Model Order Reduction

TL;DR

This paper develops a new model order reduction method tailored for finite time intervals, improving upon traditional $ ext{H}_2$-optimal techniques that focus on infinite horizons, and demonstrates its effectiveness through numerical tests.

Contribution

It introduces a time-limited $ ext{H}_2$-norm, derives optimality conditions, and proposes an iterative scheme to produce better reduced models within finite time intervals.

Findings

The new method outperforms traditional IRKA in finite time intervals.

The proposed iterative scheme effectively satisfies the time-limited optimality conditions.

Numerical examples confirm improved accuracy of the reduced models.

Abstract

In order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are $\mathcal H_2$-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited $\mathcal H_2$-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited $\mathcal H_2$-norm. Based on these optimality conditions, we propose an iterative scheme, which, upon convergence, aims at satisfying these conditions approximately. Then, we analyze how far away the obtained ROM due to the proposed algorithm is from satisfying the optimality conditions. We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to the unrestricted IRKA in the finite time interval of interest.