Numerical integration of functions of a rapidly rotating phase

TL;DR

This paper introduces a black-box numerical algorithm for efficiently evaluating integrals with rapidly oscillating phases, maintaining uniform performance for large frequencies and demonstrating exponential convergence for analytic functions.

Contribution

The paper presents a novel, general-purpose numerical method that avoids pre-computations and handles high-frequency oscillatory integrals with proven exponential convergence.

Findings

Method achieves uniform performance for large 

Converges exponentially for analytic functions

Demonstrated effective error control in numerical tests

Abstract

We present an algorithm for the efficient numerical evaluation of integrals of the form \[ I(\omega) = \int_0^1 F( x,\mathrm e^{\mathrm i \omega x}; \omega) \, \mathrm d x \] for sufficiently smooth but otherwise arbitrary $F$ and $\omega \gg 1$. The method is entirely "black-box", i.e., does not require the explicit computation of moment integrals or other pre-computations involving $F$. Its performance is uniform in the frequency $\omega$. We prove that the method converges exponentially with respect to its order when $F$ is analytic and give a numerical demonstration of its error characteristics.