Oleg Iliev
Fraunhofer Institute for Industrial Mathematics, ITWM
Karlsruhe, Germany
In collaboration with Jan Mohring, Nikolay Shegunov, Fraunhofer ITWM, Rene Milk, Mario Ohlberger, University Muenster, Peter Bastian, University of Heidelberg
In this contribution we present advances concerning efficient parallel multiscale methods and uncertainty quantification that have been obtained in the frame of the DFG priority program 1648 Software for Exascale Computing (SPPEXA) within the funded project EXA-DUNE. This project aims at the development of flexible but nevertheless hardware-specific software components and scalable high-level algorithms for the solution of partial differential equations based on the DUNE platform. We focus here on the development of scalable multiscale methods in the context of uncertainty quantification. Such problems add additional layers of coarse grained parallelism, as the underlying problems require the solution of many local or global partial differential equations in parallel that are only weakly coupled.
Uncertainty quantification (UQ) is a critical issue in quantiative study of many processes, in particular, of porous media flows. 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. We had extended DUNE 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, as well as results from scaling experiments will be presented.