The variational Kalman filter and an efficient implementation using limited memory BFGS

In the field of state space estimation and data assimilation, the Kalman filter (KF) and the extended Kalman filter (EKF) are among the most reliable methods used. However, KF and EKF require the storage of, and operations with, matrices of size n×n, where n is the size of the state space. Furthermore, both methods include inversion operations for m×m matrices, where m is the size of the observation space. Thus, KF methods become impractical as the dimension of the system increases. In this paper, we introduce a variational Kalman filter (VKF) method to provide a low storage, and computationally efficient, approximation of the KF and EKF methods. Furthermore, we introduce a variational Kalman smoother (VKS) method to approximate the fixed-lag Kalman smoother (FLKS) method. Instead of using the KF formulae, we solve the underlying maximum a posteriori optimization problem using the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method. Moreover, the LBFGS optimization method is used to obtain a low storage approximation of state estimate covariances and prediction error covariances. A detailed description of the VKF and VKS methods with LBFGS is given. The methodology is tested on linear and nonlinear test examples. The simulated results of the VKF method are presented and compared with KF and EKF, respectively. The convergence of BFGS/LBFGS methods is tested and demonstrated numerically.