An Optimal Transport Formulation of the Ensemble Kalman Filter
An Optimal Transport Formulation of the Ensemble Kalman Filter
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DOI：
10.1109/tac.2020.3015410
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发表时间：
2019-10
影响因子：
6.8
通讯作者：
A. Taghvaei;P. Mehta
中科院分区：
文献类型：
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作者：
A. Taghvaei;P. Mehta
受控相互作用粒子系统，如系综卡尔曼滤波(EnKF)和反馈粒子滤波(FPF)，是逼近连续时间非线性滤波问题解的数值算法。这些算法的特点是贝叶斯更新步骤是使用反馈控制律来实现的。文献中已经注意到，控制律并不是唯一的。这是本文讨论的主要问题。为了获得唯一的控制律，这里将滤波问题表示为最优运输问题。在线性高斯条件下，给出了(平均场型)最优控制律的显式表达式。与文献中描述的不同类型的EnKF算法的控制律进行了比较。通过对平均场控制律的经验逼近，得到了有限元N元控制的相互作用粒子算法。对于该算法，推导了经验均值和协方差方程，并证明了该方程与卡尔曼滤波是一致的。这使得基于卡尔曼滤波的经典滤波稳定性理论的收敛和误差性质得到了强有力的结论。结果表明，在一定的工艺条件下，即使在粒子数有限的情况下，均方误差也收敛到零。对有限元N元算法进行了详细的混沌传播分析。利用这一分析证明了经验分布为$N\right tarrow\inty$的弱收敛。针对一类简化的滤波问题，对均方误差算法和基于重要性抽样的算法进行了分析比较。这一分析有助于解释最近文献中几个数值研究中报告的基于控制的算法的良好伸缩特性。
Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this article. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (mean-field type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the mean-field control law, a finite-$N$ controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finite-$N$ algorithm. The analysis is used to prove weak convergence of the empirical distribution as $N\rightarrow \infty$. For a certain simplified filtering problem, analytical comparison of the mse with the importance sampling-based algorithms is described. The analysis helps explain the favorable scaling properties of the control-based algorithms reported in several numerical studies in recent literature.