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In this paper, we combine the Smolyak technique for multi-dimensionalinterpolation with the Filon-Clenshaw-Curtis (FCC) rule for one-dimensionaloscillatory 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 rulewith error estimates for the Smolyak interpolation operator, we obtain an errorestimate for the FCCS rule, consisting of the product of a Smolyak-type errorestimate multiplied by a term that decreases with$\mathcal{O}(k^{-\tilde{d}})$, where $k$ is the wavenumber and $\tilde{d}$ isthe number of oscillatory dimensions. If all dimensions are oscillatory, ahigher negative power of $k$ appears in the estimate. As an application, weconsider the forward problem of uncertainty quantification (UQ) for aone-space-dimensional Helmholtz problem with wavenumber $k$ and a randomheterogeneous refractive index, depending in an affine way on $d$ i.i.d.uniform random variables. After applying a classical hybridnumerical-asymptotic approximation, expectations of functionals of the solutionof 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 forthe 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-adaptivesparse grid FCCS quadrature showing its efficiency as dimension increases.

A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation