  In this paper, we combine the Smolyak technique for multi-dimensional
interpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensional
oscillatory integration, to obtain a new Filon-Clenshaw-Curtis-Smolyak (FCCS)
rule for oscillatory integrals with linear phase over the $d-$dimensional cube
$[-1,1]^d$. By combining stability and convergence estimates for the FCC rule
with error estimates for the Smolyak interpolation operator, we obtain an error
estimate for the FCCS rule, consisting of the product of a Smolyak-type error
estimate multiplied by a term that decreases with
$\mathcal{O}(k^{-\tilde{d}})$, where $k$ is the wavenumber and $\tilde{d}$ is
the number of oscillatory dimensions. If all dimensions are oscillatory, a
higher negative power of $k$ appears in the estimate. As an application, we
consider the forward problem of uncertainty quantification (UQ) for a
one-space-dimensional Helmholtz problem with wavenumber $k$ and a random
heterogeneous refractive index, depending in an affine way on $d$ i.i.d.
uniform random variables. After applying a classical hybrid
numerical-asymptotic approximation, expectations of functionals of the solution
of this problem can be formulated as a sum of oscillatory integrals over
$[-1,1]^d$, which we compute using the FCCS rule. We give numerical results for
the FCCS rule and the UQ algorithm showing that accuracy improves when both $k$
and the order of the rule increase. We also give results for dimension-adaptive
sparse grid FCCS quadrature showing its efficiency as dimension increases.
