Numerical approximation of highly oscillatory integrals

TLDR: Olver et al. as discussed by the authors investigated efficient methods for numerical integration of highly oscillatory functions, over both univariate and multivariate domains, and demonstrated that high oscillation is in fact beneficial: the methods discussed in this paper improve with accuracy as the frequency of oscillation increases.

Abstract: The purpose of this essay is the investigation of efficient methods for the numerical integration of highly oscillatory functions, over both univariate and multivariate domains. Such integrals have an unwarranted reputation for being difficult to compute. We will demonstrate that high oscillation is in fact beneficial: the methods discussed in this paper improve with accuracy as the frequency of oscillation increases. The asymptotic expansion will provide a point of departure, allowing us to prove that other, convergent methods have the same asymptotic behaviour, up to arbitrarily high order. This includes Filon-type methods, which require moments and Levin-type methods, which do not require moments but are typically less accurate and are not available in certain situations. Though we focus on the exponential oscillator, we also demonstrate the effectiveness of these methods for other oscillators such as the Bessel and Airy functions. The methods are also applicable in certain cases where the integral is badly behaved; such as integrating over an infinite interval or when the integrand has an infinite number of oscillations. Extent of original research. Section 2 is a review section: only Corollary 2.2 and the example in Figure 1 are due to me. All of the research is my own in Section 3 through Section 8. In Section 9, the paragraphs on changing the interval of integration are my own research. This starts with the sentence that begins “At first sight, . . .” on the top of page 30, and ends on the middle of page 31 with the sentence “. . .Levin-type method, see Figure 19.”. The rest of Section 9 consists of quoted results. All of my research was done on my own, except for Theorem 7.1, which is based on conversations with David Levin for the asymptotic expansion of the integral of the Airy function. ∗ Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Rd, Cambridge CB3 0WA, UK, email: S.Olver@damtp.cam.ac.uk