SpecialityUniversity of Reading, UK

A dynamical systems' perspective on Data Assimilation: accuracy and asymptotic stability in finite and infinite dimensional problemsThe 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. We will study the error of data assimilation schemes in the presence of unbounded (e.g. Gaussian) noise on a wide class of finite and infinite dimensional dissipative dynamical systems (including the Lorenz models and the 2D incompressible Navier-Stokes equations), but with finite rank observations. Relatively simple schemes employing linear error feedback will be discussed, and it is proved that (depending on the rate of dissipation, the amount of error feedback and number of observed modes) the error exhibits a form of stationary behaviour, and in particular an accumulation of error does not occur. We derive bounds on the long-term error for individual realisations of the noise in time and demonstrate that the error is proportional to the standard deviation of the noise. This improves on previous results in which either the noise was bounded or the error was considered in expectation only. Time permitting, the relation to stochastic pullback attractors for the error dynamics will be discussed as well. This is joint work with former PhD student Lea Oljaca and Tobias Kuna, University of L'Aquila, Italy.