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Title: Asymptotic analysis and computations of probability measures
Author: Lu, Yulong
ISNI:       0000 0004 6495 9755
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2017
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This thesis is devoted to asymptotic analysis and computations of probability measures. We are concerned with the probability measures arising from two classes of problems: Bayesian inverse problems and rare events in molecular dynamics. In the former we are interested in the concentration phenomenon of the posterior measures such as posterior consistency, and the computational methods for sampling the posterior, such as the Markov Chain Monte Carlo (MCMC). In the latter we want to describe the most probable transition paths on molecular energy landscapes in the small temperature regime. First, we examine the asymptotic normality of a general family of finite dimensional probability measures indexed by a small parameter. We begin this by studying the best Gaussian approximation to the target measure with respect to the Kullback-Leibler divergence, and then analyse the asymptotic behaviour of such approximation via Γ-convergence. This abstract theory is employed to study the posterior consistency of a finite dimensional Bayesian inverse problem. Next, we are concerned with a Bayesian inverse problem arising from barcode denoising, namely reconstructing a binary signal from finite many noisy pointwise evaluations. By choosing the prior appropriately, we show that in the small noise limit the resulting posterior concentrates on a manifold which consists of a family of parametrized binary profiles. Furthermore, we extend the use of Gaussian approximation in the context of the (infinite dimensional) transition path problem. In particular, we characterize the most probable paths as an ensemble of paths which fluctuates within an optimal Gaussian tube. The low temperature limit of these optimal paths is also identified via the Γ-convergence of some relevant variational problem. Finally, we introduce, analyze and implement a novel Bayesian level set method for solving geometric inverse problems. This Bayesian approach not only removes some draw- backs of classical level set methods but also enables quantifying geometric uncertainties induced by noisy measurements.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council ; University of Warwick
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics