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Vollmer, Sebastian
Languages: English
Types: Doctoral thesis
Subjects: QA

Classified by OpenAIRE into

arxiv: Statistics::Computation
Many mathematical models used in science and technology often contain parameters that are not known a priori. In order to match a model to a physical phenomenon, the parameters have to be adapted on the basis of the available data. One of the most important statistical concepts applied to inverse problems is the Bayesian approach which models the a priori and a posteriori uncertainty through probability distributions, called the prior and posterior, respectively. However, computational methods such as Markov Chain Monte Carlo (MCMC) have to be used because these probability measures are only given implicitly. This thesis deals with two major tasks in the area of Bayesian inverse problems: the improvement of the computational methods, in particular, different kinds of MCMC algorithms, and the properties of the Bayesian approach to inverse problems such as posterior consistency. In inverse problems, the unknown parameters are often functions and therefore elements of infinite dimensional spaces. For this reason, we have to discretise the underlying problem in order to apply MCMC methods to it. Finer discretisations lead to a higher\ud dimensional state space and usually to a slower convergence rate of the Markov chain. We study these convergence rates rigorously and show how they deteriorate for standard methods. Moreover, we prove that slightly modified methods exhibit a dimension independent performance constituting one of the first dimension independent convergence results for locally moving MCMC algorithms. The second part of the thesis concerns numerical and analytical investigations of the posterior based on artificially generated data corresponding to a true set of parameters.\ud In particular, we study the behaviour of the posterior as the amount of data increases or the noise in the data decreases. Posterior consistency describes the phenomenon that a sequence of posteriors concentrates around the truth. In this thesis, we present one of the first posterior consistency results for non-linear infinite dimensional inverse problems. We also study a multiscale elliptic inverse problem in detail. In particular, we show that it is not posterior consistent but the posterior concentrates around a manifold.

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