Data assimilation is the process by which observations are incorporated into a computer model
of a real system. Applications of data assimilation arise in many elds of geosciences, perhaps most
importantly in weather forecasting. In a joint work with M. Jolly and E. S. Titi, we present a
new continuous data assimilation algorithm for the two-dimensional Benard problem based on an
idea from control theory. Rather than inserting the observational measurements directly into the
equations, a feedback control term is introduced that forces the model towards the reference solution.
We show that the approximate solutions constructed using only observations in the velocity eld
and without any measurements on the temperature converge in time to the reference solution of the
two-dimensional Benard problem.
In a more recent joint work with E. Lunasin and E. S. Titi, we introduce an abridged continuous
data assimilation algorithm for the 2D Navier-Stokes, 2D Benard problem and 3D subgrid scale -
models of turbulence. The novelty of this improved algorithm is on the reduction on the components
of the observational data that needs to be measured and inserted into the model equation, in the
form of a feedback control term, to recover the unknown reference solution. We show that for the
2D Navier-Stokes equations the approximate solutions constructed using observations in only one
component of the velocity eld converge in time to the reference solution. In the case of the 3D
Leray- model, we show that the approximate solutions constructed using only observations any
two components, without any measurements on the third component, of the velocity eld converge
in time to the reference solution.
Back to Workshop I: Mathematical Analysis of Turbulence