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 Title
 NONGAUSSIAN STOCHASTIC DYNAMICS: MODELING, SIMULATION, QUANTIFICATION AND ASSIMILATION
 Creator
 Gao, Ting
 Date
 2015, 201505
 Description

Motivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are...
Show moreMotivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are considered for nonGaussian stochastic dynamics. More specifically, three problems are formulated to investigate nonGaussian dynamics: (i) exit problem and timedependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and timedependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non Gaussian stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on nonGaussian models is studied. For continuousdiscrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, timedependent transition probability between metastable states are investigated. xiMotivated by real world applications, three topics  deterministic quantities, uncertainty quantification and data assimilation, are considered for nonGaussian stochastic dynamics. More specifically, three problems are formulated to investigate nonGaussian dynamics: (i) exit problem and timedependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and timedependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non Gaussian stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on nonGaussian models is studied. For continuousdiscrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, timedependent transition probability between metastable states are investigated.
Ph.D. in Applied Mathematics, May 2015
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