Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation

In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.Comment: 26 page

Application of multilevel concepts for uncertainty quantification in reservoir simulation

Uncertainty quantification is an important task in reservoir simulation and is an active area of research. The main idea of uncertainty quantification is to compute the distribution of a quantity of interest, for example oil rate. That uncertainty, then feeds into the decision making process. A statistically valid way of quantifying the uncertainty is a Markov Chain Monte Carlo (MCMC) method, such as Random Walk Metropolis (RWM). MCMC is a robust technique for estimating the distribution of the quantity of interest. RWM is can be prohibitively expensive, due to the need to run a huge number of realizations, 45% - 70% of these may be rejected and, even for a simple reservoir model it may take 15 minutes for each realization. Hamiltonian Monte Carlo accelerates the convergence for RWM but may lead to a large increase computational cost because it requires the gradient. In this thesis, we present how to use the multilevel concept to accelerate convergence for RWM. The thesis discusses how to apply Multilevel Markov Chain Monte Carlo (MLMCMC) to uncertainty quantification. It proposes two new techniques, one for improving the proxy based on multilevel idea called Multilevel proxy (MLproxy) and the second one for accelerating the convergence of Hamiltonian Monte Carlo is called Multilevel Hamiltonian Monte Carlo (MLHMC). The idea behind the multilevel concept is a simple telescoping sum: which represents the expensive solution (e.g., estimating the distribution for oil rate on finest grid) in terms of a cheap solution (e.g., estimating the distribution for oil rate on coarse grid) and `correction terms', which are the difference between the high resolution solution and a low resolution solution. A small fraction of realizations is then run on the finer grids to compute correction terms. This reduces the computational cost and simulation errors significantly. MLMCMC is a combination between RWM and multilevel concept, it greatly reduces the computational cost compared to the RWM for uncertainty quantification. It makes Monte Carlo estimation a feasible technique for uncertainty quantification in reservoir simulation applications. In this thesis, MLMCMC has been implemented on two reservoir models based on real fields in the central Gulf of Mexico and in North Sea. MLproxy is another way for decreasing the computational cost based on constructing an emulator and then improving it by adding the correction term between the proxy and simulated results. MLHMC is a combination of Multilevel Monte Carlo method with a Hamiltonian Monte Carlo algorithm. It accelerates Hamiltonian Monte Carlo (HMC) and is faster than HMC. In the thesis, it has been implemented on a real field called Teal South to assess the uncertainty