Metody pro zefektivnění vzorkování v bayesovských inverzních úlohách

Abstract

The Bayesian approach provides a powerful framework for formulating inverse problems with uncertainty, where solving such problems requires the use of efficient numerical methods in many practical applications. In this thesis, we focus on the approximation of the posterior probability distribution using Markov Chain Monte Carlo (MCMC) methods, mainly analyzing the sampling process and the quality of the resulting estimates. After briefly reviewing the fundamentals of probability theory and introducing a motivating Bayesian inverse problem, the main part of the work is devoted to MCMC methods. We first establish the theoretical foundations of these methods, followed by a detailed discussion of the Metropolis-Hastings (MH) algorithm and its advanced modifications: Adaptive Metropolis (AM), Delayed Rejection Adaptive Metropolis (DRAM), and DiffeRential Evolution Adaptive Metropolis (DREAM). For each algorithm, we present the theoretical background, the implementation, and extensive numerical experiments on illustrative problems of varying complexity. The results are analyzed using established metrics, allowing for a comparison of the efficiency and characteristic properties of the individual methods.