A Reduced Basis Ensemble Kalman Filter for State/parameter Identification in Large-scale Nonlinear Dynamical Systems

The ensemble Kalman filter is nowadays widely employed to solve state and/or parameter identification problems recast in the framework of Bayesian inversion. Unfortunately its cost becomes prohibitive when dealing with systems described by parametrized partial differential equations, because of the cost entailed by each PDE query. This is even worse for nonlinear time-dependent PDEs. In this paper we propose a reduced basis ensemble Kalman filter technique to speed up the numerical solution of Bayesian inverse problems arising from the discretization of nonlinear time dependent PDEs. The reduction stage yields intrinsic approximation errors, whose propagation through the filtering process might affect the accuracy of the identified state/parameters. Since their evaluation is computationally heavy, we equip our reduced basis ensemble Kalman filter with a reduction error model based on ordinary kriging for functional-valued data, to gauge the effect of state reduction on the whole filtering process. The accuracy and efficiency of our method is then verified on two numerical test cases, dealing with the identification of uncertain parameters or fields for a FitzHugh-Nagumo model and a Fisher-Kolmogorov model.