Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo

Uncertainty quantification (UQ) for porous media flow is of great importance for many societal, environmental and industrial problems. An obstacle to the progress in solving such problems, as well as in solving other stochastic PDEs, SPDEs, is the extreme computational effort needed for solving realistic problems. It is expected that the computers will open the door for a significant progress in this area. We shortly introduce the Distributed and Unified Numerics Environment DUNE [www.dune-project.org], and demonstrate how new features, developed in the last few years, can enable the handling of these computational challenges. In the frame of the DFG funded project EXA-DUNE, the software has been extended by multiscale finite element methods (MsFEM) and by a parallel framework for the multilevel Monte Carlo approach (MLMC). This is a general concept for computing expected values of simulation results depending on random fields, e.g. the permeability of porous media. It belongs to the class of variance reduction methods and overcomes the slow convergence of classical Monte Carlo by combining cheap/inexact and expensive/accurate solutions in an optimal ratio.