# A new type of singular perturbation approximation for stochastic bilinear systems

## TL;DR

This paper introduces a novel singular perturbation approximation method for stochastic bilinear systems, providing the first proven $L^2$-error bound for this class, enhancing model order reduction accuracy.

## Contribution

It extends existing MOR techniques to stochastic bilinear systems and establishes a new $L^2$-error bound for SPA, which was previously unproven even for deterministic cases.

## Key findings

- Proposed a modified reduced order model with a different reachability Gramian.
- Proved an $L^2$-error bound for SPA in stochastic bilinear systems.
- The error bound is novel even for deterministic bilinear systems.

## Abstract

Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized (stochastic) partial differential equations and hence reduce computational complexity. A particular class of MOR techniques is balancing related methods which rely on simultaneously diagonalizing the system Gramians. This has been extensively studied for deterministic linear systems. The balancing procedure has already been extended to bilinear equations [1], an important subclass of nonlinear systems. The choice of Gramians in [1] is referred to be the standard approach. In [18], a balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on the standard choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend the setting used in [18] by considering a stochastic system with bilinear drift and linear diffusion term. Moreover, we propose a modified reduced order model and choose a different reachability Gramian. Based on this new approach, an $L^2$-error bound is proved for SPA which is the main result of this paper. This bound is new even for deterministic bilinear systems.