INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2008; 15:1 Prepared using fldauth.cls [Version: 2002/09/18 v1.01] The Variational Kalman filter and an efficient implementation using limited memory BFGS H. Auvinen 1, * , J. M. Bardsley 2 , H. Haario 1 and T. Kauranne 1 1 Department of Mathematics and Physics Lappeenranta University of Technology Lappeenranta, Finland 2 Department of Mathematical Sciences University of Montana Missoula, Montana 59812, USA SUMMARY 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 Kalman filter 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 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 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 to KF and EKF, respectively. The convergence of BFGS/LBFGS methods is tested and demonstrated numerically. key words: Kalman filter, Bayesian inversion, large-scale optimization 1. Introduction Several variants of the Kalman filter (KF) and the extended Kalman filter (EKF) have been proposed to reduce their computational complexity for large-dimensional problems. The Reduced Rank Kalman Filter or Reduced Order extended Kalman filter (see, e.g., [23], [5], [30], [8], [11], [24], [28]) project the dynamical state vector of the model onto a low dimensional subspace. The success of the approach depends on a judicious choice of the reduction operator. Moreover, since the reduction operator * Correspondence to: H. Auvinen, Department of Mathematics and Physics, Lappeenranta University of Technology, email: harri.auvinen@lut.fi