Optimal nudging in particle filters

We present an efficient particle filtering algorithm for multi-scale systems, that is adapted for dynamical systems which are inherently chaotic. We discuss the recent homogenization method developed by the authors that provides a Stochastic Partial Differential Equation (SPDE) for the evolution of the conditional distribution of the coarse-grained variables given the observations. Particle methods are used for approximating the solution to the SPDE. Importance sampling and control methods are then used as a basic and flexible tool for the construction of the proposal density inherent in particle filtering. We superimpose a control on the particle dynamics which aims to drive the particles to locations most representative of the observations, while still trying to remain faithful to the original signal dynamics. This control is obtained by minimizing certain cost functional. The measure change, needed to compensate for the addition of control in the "prognostic" equations, corresponds to that involved in optimal importance sampling.