STABILIZED BFGS APPROXIMATE KALMAN FILTER

被引:2
作者
Bibov, Alexander [1 ]
Haario, Heikki [2 ,3 ]
Solonen, Antti [2 ,3 ]
机构
[1] Lappeenranta Univ Technol, LUT Mafy Dept Math & Phys, FI-53851 Lappeenranta, Finland
[2] Lappeenranta Univ Technol, Dept Math & Phys, FI-53851 Lappeenranta, Finland
[3] MIT, Dept Aeronaut & Astronaut, Cambridge, MA 02139 USA
关键词
Extended Kalman filter; approximate Kalman filter; low-memory storage; BFGS update; observation-deficient inversion; chaotic dynamics; QUASI-GEOSTROPHIC MODEL; DATA ASSIMILATION;
D O I
10.3934/ipi.2015.9.1003
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
The Kalman filter (KF) and Extended Kalman filter (EKF) are well-known tools for assimilating data and model predictions. The filters require storage and multiplication of n x n and n x m matrices and inversion of m x m matrices, where n is the dimension of the state space and m is dimension of the observation space. Therefore, implementation of KF or EKF becomes impractical when dimensions increase. The earlier works provide optimization-based approximative low-memory approaches that enable filtering in high dimensions. However, these versions ignore numerical issues that deteriorate performance of the approximations: accumulating errors may cause the covariance approximations to lose non-negative definiteness, and approximative inversion of large close-to-singular covariances gets tedious. Here we introduce a formulation that avoids these problems. We employ L-BFGS formula to get low-memory representations of the large matrices that appear in EKF, but inject a stabilizing correction to ensure that the resulting approximative representations remain non-negative definite. The correction applies to any symmetric covariance approximation, and can be seen as a generalization of the Joseph covariance update. We prove that the stabilizing correction enhances convergence rate of the covariance approximations. Moreover, we generalize the idea by the means of Newton-Schultz matrix inversion formulae, which allows to employ them and their generalizations as stabilizing corrections.
引用
收藏
页码:1003 / 1024
页数:22
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