We present recent results on the existence of a continuous time limit for Ensemble Kalman Filter algorithms. In the setting of continuous signal and observation processes, we apply the original Ensemble Kalman Filter algorithm proposed by [1] as well as a recent variant [2] to the respective discretizations and show that in the limit of decreasing stepsize the filter equations converge to an ensemble of interacting (stochastic) differential equations in the ensemble-mean-square sense. Our analysis also allows for the derivation of convergence rates with respect to the stepsize. An application of our analysis is the rigorous derivation of continuous ensemble filtering algorithms consistent with discrete approximation schemes. Conversely, the continuous time limit allows for a better qualitative and quantitative analysis of the time-discrete counterparts using the rich theory of dynamical systems in continuous time. [1] Burgers, G., van Leeuwen, P. J., Evensen, G. (1998). Analysis scheme in the ensemble Kalman filter. Monthly weather review, 126(6), 1719-1724. [2] de Wiljes, J., Reich, S., Stannat, W. (2018). Long-Time Stability and Accuracy of the Ensemble Kalman-Bucy Filter for Fully Observed Processes and Small Measurement Noise. SIAM Journal on Applied Dynamical Systems, 17(2), 1152-1181.

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关于集合卡尔曼滤波器的连续时限

我们展示了关于集成卡尔曼滤波器算法存在连续时间限制的最新结果。在连续信号和观测过程的设置中，我们将 [1] 提出的原始 Ensemble Kalman Filter 算法以及最近的变体 [2] 应用于各自的离散化，并表明在减小步长的限制下，滤波器方程收敛到集合均方意义上的相互作用（随机）微分方程的集合。我们的分析还允许推导与步长有关的收敛率。我们的分析的一个应用是严格推导与离散近似方案一致的连续集成滤波算法。反过来，连续时间限制允许使用连续时间动态系统的丰富理论对时间离散对应物进行更好的定性和定量分析。[1] Burgers, G., van Leeuwen, PJ, Evensen, G. (1998)。集成卡尔曼滤波器中的分析方案。每月天气回顾，126(6)，1719-1724。[2] de Wiljes, J., Reich, S., Stannat, W. (2018)。用于完全观察过程和小测量噪声的集成卡尔曼布西滤波器的长期稳定性和准确性。SIAM 应用动力系统杂志，17(2)，1152-1181。用于完全观察过程和小测量噪声的集成卡尔曼布西滤波器的长期稳定性和准确性。SIAM 应用动力系统杂志，17(2)，1152-1181。用于完全观察过程和小测量噪声的集成卡尔曼布西滤波器的长期稳定性和准确性。SIAM 应用动力系统杂志，17(2)，1152-1181。