Ensemble Filtering Methods for Nonlinear Dynamics

Abstract

The standard ensemble filtering schemes such as Ensemble Kalman Filter (EnKF) and Sequential Monte Carlo (SMC) do not properly represent states of low priori probability when the number of samples is too small and the dynamical system is high dimensional system with highly non-Gaussian statistics. For example, when the standard ensemble methods are applied to two well-known simple, but highly nonlinear systems such as a one-dimensional stochastic diffusion process in a double-well potential and the well-known three-dimensional chaotic dynamical system of Lorenz, they produce erroneous results to track transitions of the systems from one state to the other.In this dissertation, a set of new parametric resampling methods are introduced to overcome this problem. The new filtering methods are motivated by a general H-theorem for the relative entropy of Markov stochastic processes. The entropy-based filters first approximate a prior distribution of a given system by a mixture of Gaussians and the Gaussian components represent different regions of the system. Then the parameters in each Gaussian, i.e., weight, mean and covariance are determined sequentially as new measurements are available. These alternative filters yield a natural generalization of the EnKF method to systems with highly non-Gaussian statistics when the mixture model consists of one single Gaussian and measurements are taken on full states.In addition, the new filtering methods give the quantities of the relative entropy and log-likelihood as by-products with no extra cost. We examine the potential usage and qualitative behaviors of the relative entropy and log-likelihood for the new filters. Those results of EnKF and SMC are also included. We present results of the new methods on the applications to the above two ordinary differential equations and one partial differential equation with comparisons to the standard filters, EnKF and SMC. These results show that the entropy-based filters correctly track the transitions between likely states in both highly nonlinear systems even with small sample size N=100.