Particle filtering in high-dimensional chaotic systems.

We present an efficient particle filtering algorithm for multiscale systems, which is adapted for simple atmospheric dynamics models that are inherently chaotic. Particle filters represent the posterior conditional distribution of the state variables by a collection of particles, which evolves and adapts recursively as new information becomes available. The difference between the estimated state and the true state of the system constitutes the error in specifying or forecasting the state, which is amplified in chaotic systems that have a number of positive Lyapunov exponents. In this paper, we propose a reduced-order particle filtering algorithm based on the homogenized multiscale filtering framework developed in Imkeller et al. "Dimensional reduction in nonlinear filtering: A homogenization approach," Ann. Appl. Probab. (to be published). In order to adapt the proposed algorithm to chaotic signals, importance sampling and control theoretic methods are employed for the construction of the proposal density for the particle filter. Finally, we apply the general homogenized particle filtering algorithm developed here to the Lorenz'96 [E. N. Lorenz, "Predictability: A problem partly solved," in Predictability of Weather and Climate, ECMWF, 2006 (ECMWF, 2006), pp. 40-58] atmospheric model that mimics mid-latitude atmospheric dynamics with microscopic convective processes.