SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 946-973, April 2024.
Abstract. We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler–Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler–Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles, and noisier systems.

中文翻译：

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一类随机微分方程的序贯离散化方案及其在贝叶斯滤波中的应用

SIAM 数值分析杂志，第 62 卷，第 2 期，第 946-973 页，2024 年 4 月
。摘要。我们引入了一种用于一类随机微分方程数值积分的预测校正离散化方案，并证明它以弱阶 1.0 收敛。新方案的关键特征是它在解的状态空间维度上顺序（递归地）构建，因此适合高维状态空间模型的近似。我们使用随机 Lorenz 96 系统作为测试模型表明，所提出的方法可以使用比标准 Euler-Maruyama 方案更大的时间步长进行操作，因此可以以较小的计算成本生成有效的近似值。我们还介绍了当新的预测校正器方案用作连续时间系统的离散时间贝叶斯滤波器的构建块时所产生的误差的理论分析。最后，我们评估了几个集成卡尔曼滤波器的性能，这些滤波器结合了所提出的顺序预测校正欧拉方案和标准欧拉-丸山方法。数值实验表明，采用新的顺序方案的滤波器可以在更大的时间步长、更小的蒙特卡罗系综和噪声更大的系统下运行。