Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo

Mohring Jan, Milk Rene, Ngo Adrian, Klein Ole, Iliev Oleg, Ohlberger Mario, Bastian Peter

Abstract

Uncertainty quanti cation (UQ) for porous media flow is of great importance for many societal, environmental and industrial prob- lems. An obstacle for progress in this area is the extreme computational effort needed for solving realistic problems. It is expected that exa-scale computers will open the door for a signi cant progress in this area. We demonstrate how new features of the Distributed and Uni ed Numerics Environment DUNE address these challenges. In the frame of the DFG funded project EXA-DUNE the software has been extended by multi- scale nite 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 elds, 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.