Large-scale Kalman filtering using the limited memory BFGS method

The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require the storage and multiplication of matrices of size n X n, where n is the size of the state space, and the inversion of matrices of size m X m, where m is the size of the observation space. Thus when both m and n are large, implementation issues arise. In this paper, we advocate the use of the limited memory BFGS method (LBFGS) to address these issues. A detailed description of how to use LBFGS within both the KF and EKF methods is given. The methodology is then tested on two examples: the first is large-scale and linear, and the second is small scale and nonlinear. Our results indicate that the resulting methods, which we will denote LBFGS-KF and LBFGS-EKF, yield results that are comparable with those obtained using KF and EKF, respectively, and can be used on much larger scale problems. Key words. Kalman filter, Bayesian estimation, large-scale optimization AMS subject classifications. 65K10, 15A29