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, an important subclass of nonlinear systems. The choice of Gramians in Al-Baiyat and Bettayeb (In: Proceedings of the 32nd IEEE conference on decision and control, 1993) is the most frequently used approach. A balancing related MOR scheme for bilinear systems called singular perturbation approximation (SPA) has been described that relies on this choice of Gramians. However, no error bound for this method could be proved. In this paper, we extend SPA to stochastic systems with bilinear drift and linear diffusion term. However, we propose a slightly modified reduced order model in comparison to previous work 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.

模型阶数减少（MOR）技术通常用于减少空间离散（随机）偏微分方程的阶次，从而降低计算复杂度。MOR的一类特殊技术是平衡相关方法，这些方法依赖于同时对角化系统Gramians。对于确定性线性系统已对此进行了广泛的研究。平衡过程已经扩展到双线性方程组，这是非线性系统的重要子类。在Al-Baiyat和Bettayeb中选择Gramians（见：第32届IEEE决策与控制会议记录，1993年）是最常用的方法。已经描述了依赖于Gramians的这种双线性系统的一种与平衡有关的MOR方案，称为奇异摄动近似（SPA）。然而，此方法没有错误界限。在本文中，我们将SPA扩展到具有双线性漂移和线性扩散项的随机系统。但是，与先前的工作相比，我们提出了稍微修改的降阶模型，并选择了不同的可达性Gramian。基于这种新方法，证明了对SPA的（（L ^ 2 \） - ）误差范围，这是本文的主要结果。即使对于确定性双线性系统，此界限也是新的。