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International Journal for Uncertainty Quantification

Impact factor: 0.967

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2012003889
pages 357-370


Johnathan M. Bardsley
Department of Mathematical Sciences, The University of Montana, Missoula, Montana 59812-0864, USA
Antti Solonen
Lappeenranta University of Technology, Laboratory of Applied Mathematics
Albert Parker
Center for Biofilm Engineering, Montana State University, Bozeman, Montana, 59717, USA
Heikki Haario
Department of Mathematics and Physics, Lappeenranta University of Technology; Finnish Meteorological Institute, Helsinki, Finland
Marylesa Howard
Department of Mathematical Sciences, University of Montana, Missoula, Montana, 59812


The ensemble Kalman filter (EnKF) is a technique for dynamic state estimation. EnKF approximates the standard extended Kalman filter (EKF) by creating an ensemble of model states whose mean and empirical covariance are then used within the EKF formulas. The technique has a number of advantages for large-scale, nonlinear problems. First, large-scale covariance matrices required within EKF are replaced by low-rank and low-storage approximations, making implementation of EnKF more efficient. Moreover, for a nonlinear state space model, implementation of EKF requires the associated tangent linear and adjoint codes, while implementation of EnKF does not. However, for EnKF to be effective, the choice of the ensemble members is extremely important. In this paper, we show how to use the conjugate gradient (CG) method, and the recently introduced CG sampler, to create the ensemble members at each filtering step. This requires the use of a variational formulation of EKF. The effectiveness of the method is demonstrated on both a large-scale linear, and a small-scale, nonlinear, chaotic problem. In our examples, the CG-EnKF performs better than the standard EnKF, especially when the ensemble size is small.