Applying the numerical method of steepest descent on multivariate oscillatory integrals in scattering theory
For researchers in scattering theory, this provides a practical fix for a known numerical bottleneck, though the approach is domain-specific.
The paper shows that the standard steepest descent method fails for common oscillatory integrals in scattering theory, but a polar coordinate transformation enables efficient solution using oscillatory integration and quadrature, with numerical demonstrations.
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.