Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications

Abstract

We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen–Loève expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O ( N - 1 / 2 ) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 10 6 are reported.