Multilevel Monte Carlo Samplers in Bayesian Inverse problems

This thesis explores and implements a new multilevel MonteCarlo sampler in Bayesian inverse problems. It combines the multilevel MonteCarlo andthe optimization-based samplers, including Randomized-and-Then-Optimize (RTO)and Implicit Sampling, to increase the efficiency of the standard Monte Calo,and implements the samplers in computationally costly Bayesian inverse problemsincluding the ODE and PDE problems. The thesis also develops the complexitytheorem for multilevel self-normalizing estimators to adapt the multilevelmethod on the optimization-based samplers. This research study has contributed to both the faster solutionof Bayesian inverse problems and an extension of multilevel Monte Carlo.