| We consider evolution problems for which the initial conditions are not known precisely and some solution components are observed subject to measurement errors. The solution to this problem is provided by the celebrated Kalman filter in case the system is linear and all errors and the initial conditions are normally distributed. Various extensions to nonlinear differential equations have been proposed in the past. One of the currently most popular approaches is provided by the ensemble Kalman filter technique which is now widely being used, for example, in meteorology. However, a number of open questions need to be addressed. Most of these questions relate to small ensemble sizes and poor approximations to the ensemble covariance matrix. In this talk, I will describe regularization techniques, such a ensemble inflation and localization, to overcome artifacts of small ensemble sizes in the context a continuous update formulation of the ensemble deviation matrix. Finally, this will point us to connections between algorithms for computing Lyapunov exponents and ensemble propagation under an ensemble Kalman filter. |
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