[NA] Andrew Gibbs: Evaluating Oscillatory Integrals using Automatic Steepest Descent

16 June 2023 12:30 till 13:15 - Location: 01.150 Lipkenszaal | Add to my calendar

Highly oscillatory integrals arise across physical and engineering applications, particularly when modelling wave phenomena. When using standard numerical quadrature rules to evaluate highly oscillatory integrals, one requires a fixed number of points per wavelength to maintain accuracy across all frequencies of interest. Several oscillatory quadrature methods exist, but in contrast to standard quadrature rules (such as Gauss and Clenshaw-Curtis), effective use requires a priori analysis of the integral and, thus, a strong understanding of the method. This makes highly oscillatory quadrature rules inaccessible to non-experts.

A popular approach for evaluating highly oscillatory integrals is "Steepest Descent". The idea behind Steepest Descent methods is to deform the integration range onto complex contours where the integrand is non-oscillatory and exponentially decaying. By Cauchy's Theorem, the value of the integral is unchanged. Practically, this reformulation is advantageous, because exponential decay is far more amenable to asymptotic and numerical evaluation. As with other oscillatory quadrature rules, if naively applied, Steepest Descent methods can break down when the phase function contains coalescing stationary points.

In this talk, I will present a new algorithm based on Steepest Descent, which evaluates oscillatory integrals with a cost independent of frequency. The two main novelties are: (1) robustness - cost and accuracy are unaffected by coalescing stationary points, and (2) automation - no expertise or a priori analysis is required to use the algorithm.