A0277
Title: Filtering dynamical systems using observations of statistics
Authors: Eviatar Bach - University of Reading (United Kingdom) [presenting]
Tim Colonius - California Institute of Technology (United Kingdom)
Isabel Scherl - California Institute of Technology (United States)
Andrew Stuart - California Institute of Technology (United States)
Abstract: The problem of filtering dynamical systems is considered, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density $\rho(v, t)$ given noisy observations of the true density $\rho^\dagger$; this contrasts with the standard filtering problem based on observations of the state $v$. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities $\rho$. However, for the purposes of tractability, algorithms are sought in state space; specifically, a mean-field state-space model is introduced, and using interacting particle system approximations to this model, an ensemble method is proposed. The resulting methodology is referred to as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, it is shown that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, the numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. Possible applications of the EnFPF are discussed in climate ensembles and in turbulence modeling.
