Browsing by Author "Solonen, Antti"

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An ensemble Kalman filter using the conjugate gradient sampler
(2013) Bardsley, Johnathan Matheas; Solonen, Antti; Parker, Albert E.; Haario, Heikki; Howard, Marylesa
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.
Krylov space approximate Kalman filtering
(2013-03) Bardsley, Johnathan M.; Parker, Albert E.; Solonen, Antti; Howard, Marylesa
The Kalman filter is a technique for estimating a time-varying state given a dynamical model for and indirect measurements of the state. It is used, for example, on the control problems associated with a variety of navigation systems. Even in the case of nonlinear state and/or measurement models, standard implementations require only linear algebra. However, for sufficiently large-scale problems, such as arise in weather forecasting and oceanography, the matrix inversion and storage requirements of the Kalman filter are prohibitive, and hence, approximations must be made. In this paper, we describe how the conjugate gradient iteration can be used within the Kalman filter for quadratic minimization, as well as for obtaining low-rank approximations of the covariance and inverse-covariance matrices required for its implementation. The approach requires that we exploit the connection between the conjugate gradient and Lanczos iterations.