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Title:Dimensional reduction in nonlinear estimation of multiscale systems
Author(s):Yeong, Hoong Chieh
Director of Research:Namachchivaya, Navaratnam S.
Doctoral Committee Chair(s):Namachchivaya, Navaratnam S.
Doctoral Committee Member(s):Chen, Yuguo; Chew, Huck B.; Perkowski, Nicolas; Rapti, Zoi; Voulgaris, Petros
Department / Program:Aerospace Engineering
Discipline:Aerospace Engineering
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Nonlinear filtering
Stochastic partial differential equation
Particle filter
Maximum likelihood estimation
Mutual information
Abstract:State or signal estimation of stochastic systems based on measurement data is an important problem in many areas of science and engineering. The true signal is usually hidden, evolving according to its own dynamics, and observations are usually corrupted and possibly incomplete. The goal is to obtain optimal estimates of the signal based on noisy observations. When the dynamical model of the signal is completely known, the theory of filtering provides a recursive algorithm for estimating the conditional density (the filter) of the signal. Particle filters have been well established for the implementation of nonlinear filtering in applications. However, computational issues arise in high dimensions due to large number of particles being required to represent the signal density. The work done in this research attempts to address this issue by combining stochastic averaging with filtering techniques to develop a reduced-dimension particle filtering method for partially observed multiscale diffusion processes. When the dynamical model contains unknown parameters, the parameters need to be estimated along with the hidden states. The parameter estimation problem overlaps with the filtering problem for state estimation. In this research, the theory of maximum likelihood estimation is used to study dimensional reduction in the parameter estimation problem. The main contribution of this work are 1) a theoretical basis for a reduced-dimension filter, 2) a proposed numerical scheme for the reduced-dimension filter, 3) a theoretical basis for reduced-dimension parameter estimation in a special multiscale setting, and 4) a time-varying characterization of the information shared between signal and observations in the reduced-dimension filter. The results of this research are in the context of slow-fast stochastic systems driven by Brownian motion, in which the timescales of the rates of change of different state/signal components differ by orders of magnitude. The multiscale filtering problem is studied via the Zakai equation that describes the time evolution of the nonlinear filter. We construct a lower dimensional Zakai equation for estimation of the slow signal component and show that the solution of the lower dimensional equation converges to that of the original Zakai equation in the wide timescales separation limit. The convergence is shown to be at a rate proportional to the square root of the timescales separation factor (ratio of characteristic timescale of the fast component to that of the slow). A numerical scheme to approximate the reduced-dimension filter (the solution to the lower dimensional Zakai equation) is also constructed. This scheme combines a particle filtering algorithm with an existing multiscale numerical integration scheme. The reduced filter dimension can restore the feasibility of particle filters in certain high dimensional problems and lowers computational costs by appropriately averaging out fast scale components. The particle filtering scheme is adapted to discrete-, sparse-time observations by constructing an optimal importance sampling (proposal) density. In between observation assimilation times, particles are gradually driven towards locations most representative of the next observation by solving a stochastic optimal control problem. This scheme is found to be beneficial especially when the signal dynamics is chaotic, and small errors in estimation can grow at exponentially rates in between observation assimilation times. The second aspect of nonlinear estimation in this work is in the setting in which stationary, deterministic model parameters are unknown. The theory of maximum likelihood estimation is combined with the reduced-dimension filtering results for the study of parameter estimation in the slow-fast dynamical system setting. Using the nonlinear filters convergence result, a lower-dimensional filtered likelihood function is constructed and shown to converge to the original filtered likelihood function in the wide timescales separation limit. For a special setting in which the slow diffusion is independent of the fast component, the maximum likelihood estimate using the reduced dimension filtered likelihood function is shown to be consistent, i.e. it converges to the true model parameter in the limit of sufficiently large observation set. The third aspect of this work concerns quantifying the uncertainty in the lower-dimensional state space of the reduced-dimension filter, given observations on the state space of the original multiscale signal. Well-known concepts of entropy and mutual information from information theory are utilized. Specifically, the time rate of change of uncertainty of the lower-dimensional state given observations is determined. The time rate of change of mutual information between the two then follows. From these, the effects of deterministic signal dynamics, diffusion effects, and information derived from observations on change in uncertainty and/or information over time can be identified and quantified. Uncertainty is found to grow according to the deterministic “volumetric growth” rate and the square of signal noise amplitude, while decreased by the square of the average information derived from observations.
Issue Date:2017-11-09
Rights Information:Copyright 2017 Hoong Chieh Yeong
Date Available in IDEALS:2018-03-13
Date Deposited:2017-12

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