  Quasi-Monte Carlo (QMC) integration of output functionals of solutions of the
diffusion problem with a log-normal random coefficient is considered. The
random coefficient is assumed to be given by an exponential of a Gaussian
random field that is represented by a series expansion of some system of
functions. Graham et al. [16] developed a lattice-based QMC theory for this
problem and established a quadrature error decay rate $\approx 1$ with respect
to the number of quadrature points. The key assumption there was a suitable
summability condition on the aforementioned system of functions. As a
consequence, product-order-dependent (POD) weights were used to construct the
lattice rule. In this paper, a different assumption on the system is
considered. This assumption, originally considered by Bachmayr et al. [3] to
utilise the locality of support of basis functions in the context of polynomial
approximations applied to the same type of the diffusion problem, is shown to
work well in the same lattice-based QMC method considered by Graham et al.: the
assumption leads us to product weights, which enables the construction of the
QMC method with a smaller computational cost than Graham et al. A quadrature
error decay rate $\approx 1$ is established, and the theory developed here is
applied to a wavelet stochastic model. By a characterisation of the Besov
smoothness, it is shown that a wide class of path smoothness can be treated
with this framework.
