确定性均场集合卡尔曼滤波
Deterministic Mean-field Ensemble Kalman Filtering
AI总结：
本文提出了一种基于密度的确定性近似均场极限集合卡尔曼滤波（DMFEnKF），通过偏微分方程求解器和求积规则扩展了EnKF的收敛性证明，适用于非高斯状态空间模型，并在低维情况下优于标准EnKF。
摘要：
Legland等人（2011）提出的标准集合卡尔曼滤波（EnKF）收敛性证明被扩展到非高斯状态空间模型。提出了一种基于密度的确定性近似均场极限EnKF（DMFEnKF），由偏微分方程求解器和求积规则组成。在两个近似之间具有一定收敛阶$κ$的情况下，该方法扩展到确定性滤波近似，因此当维度$d<2κ$时，该方法在渐进行为上优于标准EnKF。通过扩展总变差度量到随机测度，建立了对真实分布近似的保真度。这种保真度受限于由模型非线性和非高斯性引起的高斯偏差项，该偏差项存在于DMFEnKF和标准EnKF中。数值结果支持并扩展了该理论。
英文摘要：
The proof of convergence of the standard ensemble Kalman filter (EnKF) from Legland etal. (2011) is extended to non-Gaussian state space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence $κ$ between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF when the dimension $d<2κ$. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which exists for both DMFEnKF and standard EnKF. Numerical results support and extend the theory.
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