Toward Exascale Computation of Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo

Prof. Dr. Oleg Iliev

Feb. 23, 2018, 11 a.m. S2 416-2

Uncertainty quantification (UQ) is a critical issue in quantiative study of
many processes, in particular, of porous media
ows. An obstacle to advance
the knowledge in the area of stochastic PDEs, such as the considered here
one, is the extreme computational effort needed for solving realistic problems,
due to the high dimensionality of the problem.
In the frame of the DFG funded project EXA-DUNE, we shortly introduce
how the C++ based toolbox: Distributed and Unified Numerics Environment
DUNE can enable the handling of these computational challenges. We had
extended it by multiscale finite element methods (MsFEM) and by a parallel
framework for the multilevel Monte Carlo approach (MLMC). MLMC is a
general concept for computing expected values of simulation results depend-
ing on random fields, in our case these are the permeability of porous media.
MLMC belongs to the class of variance reduction methods and overcomes
the slow convergence of classical Monte Carlo, by combining cheap (and less
accurate) and expensive (and more accurate) solutions in an optimal ratio.
Selection of the levels in MLMC is an open question and it is a subject of
intensive research. Here we will present approach based on coarse/fine grids,
combined with Circulant Embedding algorithm for generating permeability
alongside heuristic algorithm for renormalization. For each realization of
permeability deterministic PDEs is solved using Finite Volume method or
MsFEM method. Results demonstrating the efficiency of MLMC will be
presented.
Each component of the algorithm: permeability generation, solving the
PDE and the variance reduction is computationally expensive and efficient
parallelization is an essential part, of the problem. We will present our
scaling experiments conducted on ITWM Beehive Cluster at Kaiserslautern,
Germany.