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

The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require storing and multiplication of matrices of size n×n, where n is the size of the state space, and the inversion of matrices of size m×m, where m is the size of the observation space. For large dimensions implementation issues arise. In this paper we introduce a Variational Kalman Filter (VKF) method to provide a low storage approximation of KF/EKF methods. In stead of using the KF formulae, we solve the underlying maximum a posteriori optimization problem using the limited memory BFGS (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 method with LBFGS is given. The methodology is tested on linear and nonlinear test examples. Our simulations indicate that the approach yields results that are comparable with those obtained using KF and EKF, respectively, and can be used on much larger scale problems.