Abstract

Co-author: Daniel Sanz (University of Warwick)

Many physical systems can be successfully modelled by a deterministic dynamical system for which, however, the initial conditions may contain uncertainty. In the presence of chaos this can lead to undesirable growth of uncertainty over time. However, when noisy observations of the system are present these may be used to compensate for the uncertainty in the initial state. This scenario is naturally modelled by viewing the initial state as given by a probability distribution, and to then condition this probability distribution on the noisy observations, thereby reducing uncertainty. Filtering refers to the situation where the conditional distribution on the system state is updated sequentially, at the time of each observation. In this talk we investigate the asymptotic behaviour of this filtering distribution for large time. We focus on a class of dissipative systems that includes the Lorenz ‘63 and ‘96 models, and the Navier-Stokes equations on a 2D torus. We first st udy the behaviour of a variant on the 3DVAR filter, creating a unified analysis which subsumes the existing work in [1,2] which, itself, builds on [3]. The optimality property of the true filtering distribution is then used, when combined with this modified 3DVAR analysis, to provide general conditions on the observation of our wide class of chaotic dissipative systems which ensure that the filtering distributions concentrate around the true state of the underlying system in the long-time asymptotic regime.

[1] C.E.A. Brett, K.F. Lam, K.J.H. Law, D.S. McCormick, M.R. Scott and A.M. Stuart, ``Accuracy and stability of filters for dissipative PDEs.’’ Physica D 245 (2013).

[2] K.J.H. Law, A. Shukla and A.M. Stuart, ``Analysis of the 3DVAR Filter for the Partially Observed Lorenz ‘63 Model.’’ Discrete and Continuous Dynamical Systems A, 34(2014).

[3] K. Hayden, E. Olsen and E.S. Titi, ``Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations.’’ Physica D 240 (2011).