  In this work we consider the problem of approximating the statistics of a
given Quantity of Interest (QoI) that depends on the solution of a linear
elliptic PDE defined over a random domain parameterized by $N$ random
variables. The random domain is split into large and small variations
contributions. The large variations are approximated by applying a sparse grid
stochastic collocation method. The small variations are approximated with a
stochastic collocation-perturbation method. Convergence rates for the variance
of the QoI are derived and compared to those obtained in numerical experiments.
Our approach significantly reduces the dimensionality of the stochastic
problem. The computational cost of this method increases at most quadratically
with respect to the number of dimensions of the small variations. Moreover, for
the case that the small and large variations are independent the cost increases
linearly.
