  Essential features of the Multigrid Ensemble Kalman Filter (G. Moldovan, G.
Lehnasch, L. Cordier, M. Meldi, A multigrid/ensemble Kalman filter strategy for
assimilation of unsteady flows, Journal of Computational Physics 443-110481)
recently proposed for Data Assimilation of fluid flows are investigated and
assessed in this article. The analysis is focused on the improvement in
performance due to the inner loop. In this step, data from solutions calculated
on the higher resolution levels of the multigrid approach are used as surrogate
observations to improve the model prediction on the coarsest levels of the
grid. The latter represents the level of resolution used to run the ensemble
members for global Data Assimilation. The method is tested over two classical
one-dimensional problems, namely the linear advection problem and the Burgers'
equation. The analyses encompass a number of different aspects, such as
different grid resolutions. The results indicate that the contribution of the
inner loop is essential in obtaining accurate flow reconstruction and global
parametric optimization. These findings open exciting perspectives of
application to grid-dependent reduced-order models extensively used in fluid
mechanics applications for complex flows, such as Large Eddy Simulation (LES).
