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.

翻译：本文结合多维插值的Smolyak技术与一维振荡积分的Filon-Clenshaw-Curtis（FCC）方法，提出一种新的Filon-Clenshaw-Curtis-Smolyak（FCCS）方法，用于求解$d$维立方体$[-1,1]^d$上具有线性相位的振荡积分。通过将FCC方法的稳定性与收敛性估计与Smolyak插值算子的误差估计相结合，我们得到了FCCS方法的误差估计，该估计由Smolyak型误差估计项与一个随$\mathcal{O}(k^{-\tilde{d}})$衰减的项的乘积构成，其中$k$为波数，$\tilde{d}$为振荡维数。若所有维度均为振荡维度，则估计式中会出现更高阶的$k$负幂次项。作为应用，我们考虑一个一维空间亥姆霍兹问题的前向不确定性量化（UQ）问题：波数为$k$，折射率为随机非均匀介质，且该折射率以仿射形式依赖于$d$个独立同分布均匀随机变量。通过应用经典的混合数值-渐近近似，该问题解泛函的期望可表示为$[-1,1]^d$上振荡积分之和，我们采用FCCS方法进行计算。针对FCCS方法与UQ算法的数值结果表明，随着$k$与规则阶数的增加，精度得到提升。此外，我们还给出了维度自适应稀疏网格FCCS求积的数值结果，展示了其在高维情况下的计算效率。