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

Zhizhang Wu, Ivan G. Graham, Dingjiong Ma, Zhiwen Zhang

TL;DR

The paper introduces the Filon-Clenshaw-Curtis-Smolyak (FCCS) rule, a sparse-grid, high-frequency quadrature for multi-dimensional oscillatory integrals with linear phase, by combining the 1D Filon-Clenshaw-Curtis rule with Smolyak interpolation. It provides explicit error bounds that couple a Smolyak-type estimate with a frequency-dependent decay factor, and shows that the normalized error can decay rapidly with frequency when regularity conditions hold. The authors apply FCCS to a 1D Helmholtz forward UQ problem with affine random refractive index, using a hybrid numerical-asymptotic solution to reduce spatial oscillations and then compute the random integrals via FCCS, including a dimension-adaptive sparse-grid variant. Numerical experiments demonstrate improved accuracy with increasing wavenumber and adaptive level, and illustrate efficiency gains as dimension grows and when dimension importance decays. The work offers a robust, scalable approach for high-dimensional, high-frequency UQ in wave problems and suggests several future extensions to nonlinear phases and higher spatial 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.

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

TL;DR

The paper introduces the Filon-Clenshaw-Curtis-Smolyak (FCCS) rule, a sparse-grid, high-frequency quadrature for multi-dimensional oscillatory integrals with linear phase, by combining the 1D Filon-Clenshaw-Curtis rule with Smolyak interpolation. It provides explicit error bounds that couple a Smolyak-type estimate with a frequency-dependent decay factor, and shows that the normalized error can decay rapidly with frequency when regularity conditions hold. The authors apply FCCS to a 1D Helmholtz forward UQ problem with affine random refractive index, using a hybrid numerical-asymptotic solution to reduce spatial oscillations and then compute the random integrals via FCCS, including a dimension-adaptive sparse-grid variant. Numerical experiments demonstrate improved accuracy with increasing wavenumber and adaptive level, and illustrate efficiency gains as dimension grows and when dimension importance decays. The work offers a robust, scalable approach for high-dimensional, high-frequency UQ in wave problems and suggests several future extensions to nonlinear phases and higher spatial 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 dimensional cube . 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 , where is the wavenumber and is the number of oscillatory dimensions. If all dimensions are oscillatory, a higher negative power of 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 and a random heterogeneous refractive index, depending in an affine way on 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 , 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 and the order of the rule increase. We also give results for dimension-adaptive sparse grid FCCS quadrature showing its efficiency as dimension increases.
Paper Structure (28 sections, 9 theorems, 138 equations, 15 tables, 1 algorithm)

This paper contains 28 sections, 9 theorems, 138 equations, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of Algorithm
  4. Main results of the paper
  5. The FCC quadrature rule for 1D problems
  6. An FCCS rule for multi-dimensional integrals
  7. Error analysis of the FCCS rule
  8. Application to a UQ problem for the Helmholtz equation
  9. The deterministic problem and its hybrid numerical-asymptotic solution
  10. Computation of $\widetilde{u}^1$
  11. Partition of $[0, 1]$
  12. Computation of $\mu_0$ and $\nu_0$
  13. Computation of $\mu_1$ and $\nu_1$
  14. Computation of $\mu_2$ and $\nu_2$
  15. The random problem
  16. ...and 13 more sections

Key Result

Theorem 1.2

For each $p\geq 1$ and $d \geq 1$ there is a constant $C_{d,p}>0$ such that, for all $\mathbf{a} \in \mathbb{R}^d$, $k > 0$, and $r$ sufficiently large, we have where $N(r,d)$ is the number of point evaluations of $f$ used in the quadrature rule FCCSmolyak_intro.

Theorems & Definitions (26)

  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5: Dependence on $p$
  • Remark 2.6: Relation to results from complexity theory
  • Example 3.1
  • ...and 16 more