Energy estimates and model order reduction for stochastic bilinear systems

TL;DR

This paper develops energy estimates and an $L^2$-error bound for balanced truncation in large-scale stochastic bilinear systems, enabling effective model order reduction with quantifiable accuracy.

Contribution

It introduces a novel $L^2$-error bound for balanced truncation applied to stochastic bilinear systems, including a new technique for deriving these bounds.

Findings

Established energy estimates for stochastic bilinear systems.

Proved an $L^2$-error bound for model reduction.

Demonstrated the effectiveness of the reduction technique.

Abstract

In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an $L^2$-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.