## Large-scale Kalman filtering using the limited memory BFGS method

H. Auvinen, J. M. Bardsley, H. Haario, and T. Kauranne

### Abstract

The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require the storage and multiplication of matrices of size $n\times n$, where $n$ is the size of the state space, and the inversion of matrices of size $m\times m$, 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.

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### Key words

Kalman filter, Bayesian estimation, large-scale optimization

65K10, 15A29

### ETNA articles which cite this article

 Vol. 39 (2012), pp. 271-285 Antti Solonen, Heikki Haario, Janne Hakkarainen, Harri Auvinen, Idrissa Amour and Tuomo Kauranne: Variational ensemble Kalman filtering using limited memory BFGS

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