Large-time behaviour of Data Assimilation algorithms for deterministic systems

"Data assimilation" is a term used in the geosciences to refer to reconstructing the current state of a dynamical system from observations. The system itself as well as the observations might be perturbed by noise. In geophysical applications, the underlying dynamics are typically infinite dimensional, while observations are finite dimensional (albeit with very large dimensions).

Data assimilation involves a wide range of different mathematical concepts. In this presentation I will focus on the relation to nonlinear filtering and observer design. Then several results will be presented on the large--time behaviour of data assimilation algorithms in the context of deterministic dynamics but with noisy observations.

For certain finite--dimensional hyperbolic dynamics, the asymptotic behaviour of the full nonlinear filter can be characterised. For the 2-dimensional Navier--Stokes or transport--diffusion (and similar) equations, we analyse conceptually much simpler algorithms that use linear error feedback.

This presentation is based on joint work with Giulia Carigi, Tobias Kuna, and Vincent Martinez