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

• Published in: BIT Vol. 65; no. 3; p. 36
• Main Authors: Iserles, Arieh, Maierhofer, Georg
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.01.2025; Springer Nature B.V
• Subjects: Chebyshev approximation; Computational Mathematics and Numerical Analysis; Differential equations; Fast Fourier transformations; Fourier transforms; Integrals; Mathematics; Mathematics and Statistics; Numeric Computing; Operators (mathematics); Polynomials; Quadratures; Research Paper; 65D30; Numerical integration; Levin method; Chebyshev polynomials; Highly oscillatory quadrature
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-025-01079-4

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 the same cost as a Fast Fourier Transform in the 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.