# Ensemble Filtering Methods for Nonlinear Dynamics

Abstract (Summary)

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 doublewell
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
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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 = 102.
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Bibliographical Information:

Advisor:

School:The University of Arizona

School Location:USA - Arizona

Source Type:Master's Thesis

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