  A sequential estimator based on the Ensemble Kalman Filter for Data
Assimilation of fluid flows is presented in this research work. The main
feature of this estimator is that the Kalman filter update, which relies on the
determination of the Kalman gain, is performed exploiting the algorithmic
features of the numerical solver employed as a model. More precisely, the
multilevel resolution associated with the multigrid iterative approach for time
advancement is used to generate several low-resolution numerical simulations.
These results are used as ensemble members to determine the correction via
Kalman filter, which is then projected on the high-resolution grid to correct a
single simulation which corresponds to the numerical model. The assessment of
the method is performed via the analysis of one-dimensional and two-dimensional
test cases, using different dynamic equations. The results show an efficient
trade-off in terms of accuracy and computational costs required. In addition, a
physical regularization of the flow, which is not granted by classical KF
approaches, is naturally obtained owing to the multigrid iterative
calculations. The algorithm is also well suited for the analysis of unsteady
phenomena and, in particular,for potential application to in-streaming Data
Assimilation techniques.
