Quasi-Monte Carlo Finite Element Methods for a Class of Elliptic Partial Differential Equations with Random Coefficients

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen-Loève expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The analysis exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). As is common for the analysis of QMC methods, “weights”, describing the varying difficulty of different subsets of the variables, are needed in the analysis in order to make sure that the infinite dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a non-standard kind, being of neither product nor order-dependent form, but instead a hybrid of the two — we refer to these as “product and order-dependent weights”, or “POD weights” in short. Nevertheless these POD weights are of a simple enough form to permit a component-by-component construction of a randomly shifted lattice rule that has optimal convergence properties for the given weighted space setting. If the terms in the expansion for the random coefficient have an appropriate decay property, and if we choose (POD) weights that minimize a certain upper bound on the error, then the solution of the PDE belongs to the joint function space needed for the analysis, and the QMC error (in the sense of a root-mean-square error averaged over shifts) is of order O(N−1+δ) for arbitrary δ > 0, where N denotes the number of sampling points in the parameter space. Moreover, for convergence rates less than 1, the conditions under which various convergence rates are achieved are exactly those found in a recent study by Cohen, De Vore and Schwab of the same model by best N -term approximations. We analyze the impact of a finite element (FE) discretization on the overall efficiency of the scheme, in terms of accuracy versus overall cost, with results that are comparable to those of the best N -term approximation.