Tan, Xiaosi (2015-11). Multilevel Uncertainty Quantification Techniques Using Multiscale Methods. Doctoral Dissertation. Thesis uri icon

abstract

  • In this dissertation, we focus on the uncertainty quantification problems in sub-surface flow models which can be computationally demanding because of the large number of unknowns in forward simulations. First, we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the Generalized Multiscale Finite Element Method (GMsFEM) and Multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations at different resolutions, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. By suitably choosing the number of samples at different levels, one can use less of expensive forward simulations on the fine grid, while more of inexpensive forward simulations on the coarse grid in Monte Carlo simulations. Further, we describe a Multilevel Markov Chain Monte Carlo (MLMCMC) method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on the fine grid, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale feature of the GMsFEM with the multilevel feature of the MLMC methods, and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. We also propose a multiscale space-parameter separation model reduction method for handling uncertainties in forward problems. The method is based on the idea of separation of variables. This involves seeking the solution in terms of an expansion, where each term is a separable function of space and parameter variables. To find each term in the expansion, we solve a minimization problem associated with the forward problem. The minimization is performed successively for each term consisting of a separable function. In this proposed approach, we need to solve the PDE repeatedly, where we use GMsFEM to speed up the computation. We discuss how the GMsFEM can be used in this context and how the computational gain can be achieved. We present numerical results, which illustrate the efficiency and accuracy of our method. We also discuss efficient sampling techniques for uncertainty quantification in inverse problems. In particular, we consider Approximate Bayesian computation (ABC) and develop a Multilevel Approximate Bayesian computation (MLABC) by using a hierarchy of forward simulation models within the MLMC framework. This approach improves the MLMCMC approach. In this part of the dissertation, we develop a mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving parameter-dependent two-phase flow problems with transport model. A hierarchy of approximations at different resolutions can be provided by this mixed GMsFEM. ABC can be incorporated in different levels to reduce the computational cost, and to produce an approximate solution by ensembling at different levels.

publication date

  • November 2015