Bayesian inference and uncertainty propagation in dynamical systems
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Decision making is based on our models of what we expect to observe. Apart from the underlying models, the real-time observations or measurements are another important source for predictive inference. Typically, these models and observations are noisy, whose uncertainty can be characterized using probability distributions. The Bayesian framework provides a principled basis to combine such uncertain models and noisy measurements. The dissertation focuses mainly on data assimilation methods in puff-based Lagrangian dispersion models and uncertainty propagation in dynamic models. Real-time tracking and prediction of chemical and biological releases are important for fast response to chemical and biological accidents and attacks. In a chemical release incident, the important questions that arise in hazard prediction and assessment are: Where are the sources? Where are the toxic plumes going? The two sources for predictive inference with which to address these questions are the relevant dispersion models and concentration data. In this dissertation, data assimilation is studied in the context of puff-based Lagrangian chem-bio atmospheric dispersion models. For the estimation of plume evolution, an Extended Kalman filter and a particle filter are designed for a representative puff-based dispersion model and the results are discussed. The application of particle filters in a variable dimension state space model and its potential for various puff-based atmospheric dispersion modeling packages is demonstrated. A novel grid-based algorithm is presented for efficient source identification, where the number of sources, locations and strengths are unknown. The source identification problem is formulated as a convex optimization problem in the l 1 metric, which exploits the sparse nature of the solution to efficiently estimate the source characteristics, even when the number of sources is large. Dispersion is a complex nonlinear physical process with numerous uncertainties in model parameters, source parameters, and initial conditions. Accurate propagation of these uncertainties through the models is crucial for a reliable prediction of the probability distribution of the states and assessment of risk. The problem of uncertainty propagation and sensitivity analysis in nonlinear puff-based dispersion models is addressed using stochastic spectral methods based on polynomial chaos (PC) series expansions of random processes. A wide class of probability distributions can be represented using this approach. While the existing methods work very well and even provide exact description of the uncertainty propagation for linear dynamical systems subject to either initial condition and temporal stochastic disturbance modeled as white noise process or time-invariant parametric uncertainty, the main challenge lies in characterizing the uncertainty in the system states due to both parametric and temporal stochastic uncertainties simultaneously. A hybrid Bayesian-PC based approach is proposed to address this general problem of uncertainty propagation in linear dynamical systems with random inputs and uncertain model parameters. This approach is further extended to address the filtering problem, when there is parametric uncertainty.