Toward Exascale Computation of Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo
Prof. Dr. Oleg IlievFeb. 23, 2018, 10 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 depending 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.