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.

翻译：囊括 Kalman 过滤器( EnKF) 是一种数据同化技术,它使用各种模型,并用数据加以更新,来跟踪通常非线性系统的时间演变。它使用著名的 Kalman 过滤器的经验近似值来跟踪通常非线性系统的时间演变。然而,当共括体大小小于国家空间时,其性能就会受到影响,而计算繁琐模型往往需要这样。这个假设意味着,对州差值的经验性估计并不是完全的,而且可能相当吵闹。为了解决这一高维系统的问题,我们建议快速和容易地使用称为受罚的共括体 Kalman 过滤器的算法。 在某些条件下,理论上可以证明,尽管共括体成员少于国家维度,但PEnKF 的准确性(估计误差会达到零) 。 此外,与本地化方法不同, 拟议的方法学习了与动态系统相关的共变体结构。这些理论结果得到若干非线性和高维系统模拟的支持。