Abstract of Ivan Graham

Quasi-Monte Carlo methods for computing flow in random porous media
In this talk we formulate and implement quasi-Monte Carlo (QMC) methods for computing the expectations of functionals of solutions of elliptic PDEs, with coefficients defined as Gaussian random fields. As we see, these methods outperform conventional Monte Carlo methods for such problems. Our main target application is the computation of several quantities of physical interest arising in the modeling of fluid flow in random porous media, such as the effective permeability or the exit time of a plume of pollutants. Such quantities are of great interest in uncertainty quantification in areas such as underground waste disposal, and here QMC is combined with a mixed finite element discretization in space. Our particular emphasis is on relatively high variance and low correlation length, leading to high stochastic dimension, where Karhunen-Loeve expansions converge slowly. In this case Monte Carlo is currently the method of choice but, as we demonstrate, QMC methods are more effective and efficient for a range of parameters and quantities of interest. The talk will discuss both theoretical and computational aspects of this problem and include some applications involving up to 106 stochastic dimensions.


Lars Grasedyck (MPI Leipzig, Germany)
Wolfgang Hackbusch (MPI Leipzig, Germany)
Boris Khoromskij (MPI Leipzig, Germany)