  Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element
(FE) discretizations of elliptic partial differential equations (PDEs) with a
random coefficient, to estimate expected values of linear functionals of the
solution. The expected value is considered as an infinite-dimensional integral
in the parameter space corresponding to the randomness induced by the random
coefficient. We use a multi-level algorithm, with the number of QMC points
depending on the discretization level, and with a level-dependent dimension
truncation 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 maximal
number of QMC sampling points. For these scenarios, the total work is
essentially of the order of one single PDE solve at the finest FE
discretization level. The analysis exploits regularity of the parametric
solution with respect to both the physical variables (the variables in the
physical domain) and the parametric variables (the parameters corresponding to
randomness). Families of QMC rules with "POD weights" ("product and order
dependent weights") which quantify the relative importance of subsets of the
variables are found to be natural for proving convergence rates of QMC errors
that are independent of the number of parametric variables.
