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

TL;DR

This paper introduces a new numerical integration rule combining Smolyak and FCC methods for multi-dimensional oscillatory integrals, with applications to uncertainty quantification in Helmholtz problems, demonstrating improved accuracy and efficiency.

Contribution

The paper develops the FCCS rule that merges Smolyak interpolation with FCC for oscillatory integrals, providing error estimates and demonstrating its application to UQ for Helmholtz equations.

Findings

Error estimate shows decay with increasing wavenumber and dimensions.

Numerical results confirm improved accuracy with higher wavenumber and rule order.

Dimension-adaptive sparse grid FCCS quadrature is efficient for high dimensions.

Abstract

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.