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An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals

Arieh Iserles, Georg Maierhofer

TL;DR

This work addresses the efficient evaluation of highly oscillatory integrals by developing an accelerated Levin method based on Clenshaw–Curtis collocation and Chebyshev polynomials. By revealing a banded representation of the Levin operator on the Chebyshev basis, the authors reduce the dominant cost from dense solves to essentially $O(\nu \log \nu)$ operations, while accommodating vector-valued weights and endpoint derivatives. The approach extends naturally from scalar to vector-weighted cases and demonstrates robust performance for exponential and Bessel-type oscillators, with numerical evidence showing spectral convergence in the node count $\nu$ and substantial speed-ups over traditional methods. These findings offer a scalable and practical tool for high-frequency integral evaluation in physics, engineering, and applied analysis. The method broadens the applicability of Levin-type quadrature and paves the way for higher-dimensional extensions and alternative polynomial bases.

Abstract

The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just $\mathcal{O}(ν\logν)$ operations, where $ν$ is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples.

An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals

TL;DR

This work addresses the efficient evaluation of highly oscillatory integrals by developing an accelerated Levin method based on Clenshaw–Curtis collocation and Chebyshev polynomials. By revealing a banded representation of the Levin operator on the Chebyshev basis, the authors reduce the dominant cost from dense solves to essentially operations, while accommodating vector-valued weights and endpoint derivatives. The approach extends naturally from scalar to vector-weighted cases and demonstrates robust performance for exponential and Bessel-type oscillators, with numerical evidence showing spectral convergence in the node count and substantial speed-ups over traditional methods. These findings offer a scalable and practical tool for high-frequency integral evaluation in physics, engineering, and applied analysis. The method broadens the applicability of Levin-type quadrature and paves the way for higher-dimensional extensions and alternative polynomial bases.

Abstract

The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just operations, where is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples.
Paper Structure (14 sections, 8 theorems, 93 equations, 5 figures)

This paper contains 14 sections, 8 theorems, 93 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. The types of integrals under consideration
  4. The Levin method
  5. The accelerated Levin--Clenshaw--Curtis method
  6. The accelerated algorithm in the simple scalar case, $\mathbf{M=1,s=0}$
  7. The inclusion of endpoint derivative values in the scalar case, $\mathbf{M=1,s\geq 1}$
  8. Vectorial oscillatory weight functions, $\mathbf{M\geq2,s=0}$
  9. Vectorial oscillatory weight functions with endpoint derivative values, $\mathbf{M\geq 2,s\geq 1}$
  10. Numerical Experiments
  11. Example 1: Finite-length Fourier transform
  12. Example 2: Hankel transform
  13. Concluding remarks
  14. Conflict of interest

Key Result

Theorem 1

Suppose we are in the case of integrals described in Example ex:exponential_oscillator, i.e. with $g'(x)\neq 0,\, \forall x\in[-1,1]$ and $g^{(j)}(\pm1)\neq0,\,j=2,\dots,s$. Then the Levin method based on eqn:general_levin_collocation_equations has an asymptotic error of the form

Figures (5)

  • Figure 1: The absolute error, $|\mathcal{Q}_\omega^{{L},[\nu]}[f]-I^{(1)}_\omega[f]|$, in the Levin--Clenshaw--Curtis method for $I_\omega^{(1)}[f]$ as a function of $\omega$ for fixed $\nu=4,64,128$.
  • Figure 2: The error and timing of the Levin--Clenshaw--Curtis method as a function of the number of collocation points $\nu$ for fixed frequency $\omega=100$.
  • Figure 3: The condition number of the full collocation system compared against the systems appearing in our present methodology.
  • Figure 4: The absolute error, $|\mathcal{Q}_\omega^{{L},[\nu]}[f]-I^{(2)}_\omega[f]|$, in the Levin--Clenshaw--Curtis method for $I_\omega^{(2)}[f]$ as a function of $\omega$ for fixed $\nu=4,64,128$.
  • Figure 5: The error and timing of the Levin--Clenshaw--Curtis method for $I^{(2)}_\omega[f]$ as a function of the number of collocation points $\nu$ for fixed frequency $\omega=100$.

Theorems & Definitions (17)

  • Example 1: Exponential oscillator
  • Example 2: Bessel weight function, see Example 1 in levin1996fast
  • Theorem 1: Thm 4.1 in olver2006, see also Thm. 3.5 in deano2017computing
  • Theorem 2: Thm. 4.1 in Olver2007
  • Lemma 1: Eqs. 22.7.4 & 22.8.3 in abramowitz1965handbook
  • Proposition 1
  • proof
  • Claim 1
  • proof : Proof of Claim \ref{['claim6:special_properties_of_DCTI_in_this_space']}
  • Proposition 2
  • ...and 7 more