Optimal Nudging in Particle Filters

We present an efficient particle filtering algorithm for multi-scale systems, that is adapted for dynamical systems which are inher- ently chaotic. We discuss the recent homogenization method developed by the authors that provides a Stochastic Partial Differential Equation (SPDE) for the evolution of the 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 drives the particles to locations most representative of the observations. The control is chosen as the one which mini- mizes certain cost functional that penalizes the particles that are far away from the observations. The measure change, needed to compensate for the addition of control in the “prognostic” equations, corresponds to that involved in optimal importance sampling.