EECS Department, University of Michigan, Ann Arbor, Michigan, United States of America.
Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America.
PLoS One. 2021 Mar 18;16(3):e0248046. doi: 10.1371/journal.pone.0248046. eCollection 2021.
The ensemble Kalman filter (EnKF) is a data assimilation technique that uses an ensemble of models, updated with data, to track the time evolution of a usually non-linear system. It does so by using an empirical approximation to the well-known Kalman filter. However, its performance can suffer when the ensemble size is smaller than the state space, as is often necessary for computationally burdensome models. This scenario means that the empirical estimate of the state covariance is not full rank and possibly quite noisy. To solve this problem in this high dimensional regime, we propose a computationally fast and easy to implement algorithm called the penalized ensemble Kalman filter (PEnKF). Under certain conditions, it can be theoretically proven that the PEnKF will be accurate (the estimation error will converge to zero) despite having fewer ensemble members than state dimensions. Further, as contrasted to localization methods, the proposed approach learns the covariance structure associated with the dynamical system. These theoretical results are supported with simulations of several non-linear and high dimensional systems.
集合卡尔曼滤波器（EnKF）是一种数据同化技术，它使用一组模型（通过数据进行更新）来跟踪通常是非线性系统的时间演变。它通过使用著名的卡尔曼滤波器的经验近似来实现这一点。然而，当集合大小小于状态空间时，其性能可能会受到影响，因为对于计算负担大的模型来说，这通常是必要的。这种情况意味着状态协方差的经验估计不是满秩的，并且可能非常嘈杂。为了解决这个在高维情况下的问题，我们提出了一种计算快速且易于实现的算法，称为惩罚集合卡尔曼滤波器（PEnKF）。在某些条件下，可以从理论上证明，尽管 PEnKF 的集合成员数少于状态维度数，但它将是准确的（估计误差将收敛到零）。此外，与局域化方法不同，所提出的方法学习与动力系统相关的协方差结构。这些理论结果得到了几个非线性和高维系统的模拟的支持。
