The term ``Data Assimilation'' is used in the geosciences and refers to estimating the current state of a dynamical system using past and current observations in conjuction with a model of the dynamics which might be deterministic or stochastic. The observations may or may not be corrupted with noise. There is strong overlap with concepts such as optimal filtering, synchronisation, and observer design from control theory.

 

This contribution will consist of two parts. In the first part we will consider data assimilation in (infinite dimensional) dissipative PDE's, e.g. reaction diffusion, 2-dimensional Navier Stokes, but with finite rank observations. Relatively simple schemes employing linear error feedback will be discussed, and the long time behaviour of the error dynamics will be analysed, both in terms of individual realisations of the noise as well as on average. The relation to stochastic pullback attractors for the error dynamics will be discussed as well.

 

The second part will consider asymptotic properties of the optimal filtering process, which is the full conditional distribution of the underlying state given the observations. Results regarding the stability of the filtering process with regards to misspecifying the initial distribution will be presented, mostly in the context of deterministic (hyperbolic) dynamical systems, along with the relevance in applications.

 

This is joint work with Lea Oljaca, University of Exeter, and Tobias Kuna, University of L'Aquila, Italy.