The University of Montana

Department of Mathematical Sciences

Technical report #33/2008

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

**H. Auvinen, H. Haario, and T. Kauranne**

Department of Mathematics and Physics

Lappeenranta University of Technology

Lappeenranta, Finland

J. M. Bardsley

Department of Mathematical Sciences

University of Montana

Missoula, Montana 59812, USA

**Abstract**

The standard formulations of the Kalman filter (KF) and extended
Kalman filter (EKF) require storing 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. 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.

**Keywords:**Kalman filter, Bayesian inversion, large-scale optimization

**AMS Subject Classification:**

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