  Many studies in uncertainty quantification have been carried out under the
assumption of an input random field in which a countable number of independent
random variables are each uniformly distributed on an interval, with these
random variables entering linearly in the input random field (the so-called
affine model). In this paper we consider an alternative model of the random
field, in which the random variables have the same uniform distribution on an
interval, but the random variables enter the input field as periodic functions.
The field is constructed in such a way as to have the same mean and covariance
function as the affine random field. Higher moments differ from the affine
case, but in general the periodic model seems no less desirable. The new model
of the random field is used to compute expected values of a quantity of
interest arising from an elliptic PDE with random coefficients. The periodicity
is shown to yield a higher order cubature convergence rate of
$\mathcal{O}(n^{-1/p})$ independently of the dimension when used in conjunction
with rank-1 lattice cubature rules constructed using suitably chosen
smoothness-driven product and order dependent weights, where $n$ is the number
of lattice points and $p$ is the summability exponent of the fluctuations in
the series expansion of the random coefficient. We present numerical examples
that assess the performance of our method.
