论文标题
在合奏Kalman滤波器中的最优性与稳定性权衡
Optimality vs Stability Trade-off in Ensemble Kalman Filters
论文作者
论文摘要
本文涉及集合卡尔曼过滤器（ENKF）算法的最佳和稳定性分析。 ENKF通常用作高维问题的卡尔曼过滤器的替代方案，在该问题上存储协方差矩阵在计算上很昂贵。该算法由反馈控制定律驱动的相互作用粒子组成。控制定律的设计使得，在无限多个粒子的线性高斯设置和渐近极限中，颗粒的平均值和协方差遵循卡尔曼滤波器的确切平均值和协方差。找到确切的控制法的问题不是独特的解决方案，让人想起在两个分布之间找到传输图的问题。可以通过引入控制成本功能来确定独特的控制法，这些功能是由最佳运输问题或Schrödinger桥问题引起的。本文的目的是研究精确控制定律家族的最优性与长期稳定性之间的关系。值得注意的是，在最佳运输意义上是最佳的控制定律会导致不稳定的ENKF算法。
This paper is concerned with optimality and stability analysis of a family of ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an alternative to the Kalman filter for high-dimensional problems, where storing the covariance matrix is computationally expensive. The algorithm consists of an ensemble of interacting particles driven by a feedback control law. The control law is designed such that, in the linear Gaussian setting and asymptotic limit of infinitely many particles, the mean and covariance of the particles follow the exact mean and covariance of the Kalman filter. The problem of finding a control law that is exact does not have a unique solution, reminiscent of the problem of finding a transport map between two distributions. A unique control law can be identified by introducing control cost functions, that are motivated by the optimal transportation problem or Schrödinger bridge problem. The objective of this paper is to study the relationship between optimality and long-term stability of a family of exact control laws. Remarkably, the control law that is optimal in the optimal transportation sense leads to an EnKF algorithm that is not stable.
