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Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element(FE) discretizations of elliptic partial differential equations (PDEs) with arandom coefficient, to estimate expected values of linear functionals of thesolution. The expected value is considered as an infinite-dimensional integralin the parameter space corresponding to the randomness induced by the randomcoefficient. We use a multi-level algorithm, with the number of QMC pointsdepending on the discretization level, and with a level-dependent dimensiontruncation strategy. In some scenarios, we show that the overall error is$\mathcal{O}(h^2)$, where $h$ is the finest FE mesh width, or$\mathcal{O}(N^{-1+\delta})$ for arbitrary $\delta>0$, where $N$ is the maximalnumber of QMC sampling points. For these scenarios, the total work isessentially of the order of one single PDE solve at the finest FEdiscretization level. The analysis exploits regularity of the parametricsolution with respect to both the physical variables (the variables in thephysical domain) and the parametric variables (the parameters corresponding torandomness). Families of QMC rules with "POD weights" ("product and orderdependent weights") which quantify the relative importance of subsets of thevariables are found to be natural for proving convergence rates of QMC errorsthat are independent of the number of parametric variables.

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