Sequential Discretization Schemes for a Class of Stochastic Differential Equations and their Application to Bayesian Filtering

IF 2.8 2区 数学 Q1 MATHEMATICS, APPLIED SIAM Journal on Numerical Analysis Pub Date : 2024-04-08 DOI:10.1137/23m1560124
Ö. Deniz Akyildiz, Dan Crisan, Joaquin Miguez
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Abstract

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.
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一类随机微分方程的序列离散化方案及其在贝叶斯过滤中的应用
SIAM 数值分析期刊》第 62 卷第 2 期第 946-973 页,2024 年 4 月。 摘要。我们为一类随机微分方程的数值积分引入了一种预测器-校正器离散化方案,并证明它以弱阶 1.0 收敛。新方案的主要特点是在解的状态空间维度上依次建立(和递归),因此适用于高维状态空间模型的逼近。我们以随机洛伦兹 96 系统为测试模型,证明了与标准欧拉-马鲁山方案相比,所提出的方法能以更大的时间步长运行,因此能以更小的计算成本生成有效的近似值。我们还介绍了新预测器-校正器方案作为连续时间系统离散时间贝叶斯滤波器构建模块时产生误差的理论分析。最后,我们评估了几种集合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器-校正器欧拉方案和标准欧拉-Maruyama 方法。数值实验表明,采用新序列方案的滤波器可以在更大的时间步长、更小的蒙特卡罗集合和更嘈杂的系统中运行。
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来源期刊
CiteScore
4.80
自引率
6.90%
发文量
110
审稿时长
4-8 weeks
期刊介绍: SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.
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