State estimation of random dynamical systems with noisy observations has been an important problem in many areas of science and engineering. Efficient new algorithms to estimate the present and future state of a dynamic signal based upon corrupted, distorted, and possibly partial observations of the signal are required. Since the true state is usually hidden and evolves according to its own dynamics, the objective of this work is to get an optimal estimation of the true state via noisy observations. The theory of filtering provides a recursive procedure for estimating an evolving signal or state from a noisy observation process. We consider a particle filter approach for nonlinear filtering in multiscale dynamical systems. Particle filters represent the posterior conditional distribution of the state variables by a system of particles, which evolves and adapts recursively as new information becomes available. Particle filters suffer from computational inefficiency when applied to high dimensional problems. In practice, large numbers of particles may be required to provide adequate approximations in higher dimensional poblems. In several high dimensional applications, after a sequence of updates, the particle system will often collapse to a single point. With the help of rigorous dimensional reduction methods, particle filters could regain their versatility. Based on our theoretical developments (Park, J. H., Sri Namachchivaya, N., and Sowers, R. B., 2008, “A Problem in Stochastic Averaging of Nonlinear Filters,” Stochastics Dyn., 8(3), pp. 543–560; Park, J. H., Sowers, R. B., and Sri Namachchivaya, N., 2010, “Dimensional Reduction in Nonlinear Filtering,” Nonlinearity, 23(2), pp. 305–324), we devise an efficient particle filter algorithm, which is applicable to high dimensional multiscale nonlinear filtering problems. In this paper, we present the homogenized hybrid particle filtering method that combines homogenization of random dynamical systems, reduced order nonlinear filtering, and particle methods.
Blended particle filters for large-dimensional chaotic dynamical systems