Applying the numerical method of steepest descent on multivariate oscillatory integrals in scattering theory

TL;DR

This paper shows that the straightforward steepest descent method fails for oscillatory integrals in scattering theory, but a polar change of variables enables efficient numerical solutions using combined techniques.

Contribution

The paper introduces a polar transformation approach that improves numerical integration of highly oscillatory integrals in scattering theory.

Findings

Straightforward steepest descent fails on these integrals.

Polar change of variables enables efficient computation.

Numerical demonstrations confirm the effectiveness of the method.

Abstract

In this paper we demonstrate that the numerical method of steepest descent fails when applied in a straight forward fashion to the most commonly occurring highly oscillatory integrals in scattering theory. Through a polar change of variables, however, the integral can be brought on a form that can be solved efficiently using a mix of oscillatory integration techniques and classical quadrature. The approach is described in detail and demonstrated numerically on integration problems taken from applications.