论文标题
用于多维振荡积分的FILON-CLENSHAW-CURTIS-SMOLYAK规则，并应用于Helmholtz方程的UQ问题
A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals with application to a UQ problem for the Helmholtz equation
论文作者
论文摘要
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 $[-1,1]^d$.通过将FCC规则的稳定性和收敛性估计与Smolyak插值操作员的错误估计结合在一起，我们获得了FCCS规则的错误估算，由Smolyak-type错误估计的乘积组成，乘以$ \ \ Mathcal {O}（k^o}（k^waverum waven）和waver iS $ waven和waven} = per， $ \ tilde {d} $是振荡尺寸的数量。如果所有维度都是振荡的，则估计中会出现较高的$ K $负功率。作为一个应用程序，我们考虑了与WaveNumber $ K $和随机的异质折射率的一维赫尔姆霍尔兹问题的不确定性量化（UQ）的远期问题，这取决于$ d $ i.i.d.d.均匀的随机变量。在应用经典的杂化数值 - 反应近似之后，可以将该问题解决方案的功能的期望作为超过$ [-1,1]^d $的振荡积分总和，我们使用FCCS规则来计算。我们给出了FCC规则的数值结果，UQ算法表明，当$ K $和规则的顺序增加时，准确性会提高。我们还为维度自适应稀疏网格FCC的尺寸较大的结果提供了结果，显示其效率随着尺寸的增加而效率。
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.
