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）规则，提出了一种新的Filon-Clenshaw-Curtis-Smolyak（FCCS）规则，用于计算定义在$d$维立方体$[-1,1]^d$上、具有线性相位的振荡积分。通过将FCC规则的稳定性和收敛性估计与Smolyak插值算子的误差估计相结合，我们得到了FCCS规则的一个误差估计，其形式为Smolyak型误差估计乘以一个随波数$k$和振荡维度数$\tilde{d}$以$\mathcal{O}(k^{-\tilde{d}})$衰减的项。若所有维度均为振荡性，则估计式中出现$k$的更高次负幂。作为应用，我们考虑一维亥姆霍兹问题的不确定量化（UQ）正向问题，该问题具有波数$k$和依赖于$d$个独立同分布均匀随机变量仿射形式的随机非均匀折射率。在应用经典混合数值-渐近近似后，该问题解的泛函期望可表示为$[-1,1]^d$上振荡积分之和，我们使用FCCS规则进行计算。数值结果表明，FCCS规则及UQ算法的精度随$k$和规则阶数的增加而提高。此外，维度自适应稀疏网格FCCS求积的结果展示了其在高维情形下的效率。