The subsurface flow is usually subject to uncertain porous media structures. However, in most cases we only have partial knowledge about the porous media properties. A common approach is to model the uncertain parameters as random fields, then the expectation of Quantity of Interest(QoI) can be evaluated by the Monte Carlo method. In this study, we develop a full multigrid-multilevel Monte Carlo (FMG-MLMC) method to speed up the evaluation of random parameters effects on single-phase porous flows. In general, MLMC method applies a series of discretization with increasing resolution and computes the QoI on each of them, the success of which lies in the effective variance reduction. We exploit the similar hierarchies of MLMC and multigrid methods, and obtain the solution on coarse mesh Qcl as a byproduct of the multigrid solution on fine mesh Qfl on each level l. In the cases considered in this thesis, the computational saving is 20% theoretically. In addition, a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals a smaller estimator variance and faster convergence rate of the latter method in this study.

This thesis studies novel and efficient computational sampling methods for appli- cations in three types of stochastic inversion problems: seismic waveform inversion, filtering problems, and static parameter estimation. A primary goal of a large class of seismic inverse problems is to detect parameters that characterize an earthquake. We are interested to solve this task by analyzing the full displacement time series at a given set of seismographs, but approaching the full waveform inversion with the standard Monte Carlo (MC) method is prohibitively expensive. So we study tools that can make this computation feasible. As part of the inversion problem, we must evaluate the misfit between recorded and synthetic seismograms efficiently. We employ as misfit function the Wasserstein metric origi- nally suggested to measure the distance between probability distributions, which is becoming increasingly popular in seismic inversion. To compute the expected values of the misfits, we use a sampling algorithm called Multi-Level Monte Carlo (MLMC). MLMC performs most of the sampling at a coarse space-time resolution, with only a few corrections at finer scales, without compromising the overall accuracy. We further investigate the Wasserstein metric and MLMC method in the context of filtering problems for partially observed diffusions with observations at periodic time intervals. Particle filters can be enhanced by considering hierarchies of discretizations to reduce the computational effort to achieve a given tolerance. This methodology is called Multi-Level Particle Filter (MLPF). However, particle filters, and consequently MLPFs, suffer from particle ensemble collapse, which requires the implementation of a resampling step. We suggest for one-dimensional processes a resampling procedure based on optimal Wasserstein coupling. We show that it is beneficial in terms of computational costs compared to standard resampling procedures. Finally, we consider static parameter estimation for a class of continuous-time state-space models. Unbiasedness of the gradient of the log-likelihood is an important property for gradient ascent (descent) methods to ensure their convergence. We propose a novel unbiased estimator of the gradient of the log-likelihood based on a double-randomization scheme. We use this estimator in the stochastic gradient ascent method to recover unknown parameters of the dynamics.