Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients

Abstract

Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Ele- ment (FE) discretizations of elliptic partial differentia l equations (PDEs) with a ran- dom coefficient. The representation of the random coefficien t is assumed to require a countably infinite number of terms. The multi-level FE discretizations are combined with families of QMC meth- ods (specifically, randomly shifted lattice rules) to estim ate expected values of linear functionals of the solution, as in (17, 18, 23) in the single- level setting. Here, the ex- pected value is considered as an infinite-dimensional integ ral in the parameter space corresponding to the randomness induced by the random coeffi cient. In this paper we study the same model as in (23). The error analysis of (23) is generalized to a multi-level scheme, with the number of QMC points depending on the discretization level, and with a level-dependent dimension truncation strategy. In some scenarios, it is shown that the overall error of the expected value of the functionals of the solution (i.e., the root-mean-square error averaged over all shifts ) is of order O(h 2 ), where h is the finest FE mesh width, or O(N −1+δ ) for arbitrary δ > 0, where N denotes the maximal number of QMC sampling points in the parameter space. For these scenar- ios, the total work for all PDE solves in the multi-level QMC-FE method is shown to be essentially of the order of one single PDE solve at the finest FE discretization level, for spatial dimension d ≥ 2 with linear elements. The analysis exploits regularity of the parametric solutio n with respect to both the physical variables (the variables in the physical domai n) and the parametric vari-