TU Delft Institute for Computational Science and Engineering (DCSE)

Computational Science and Engineering (CSE) is rapidly developing field that brings together applied mathematics, engineering and (social) science. DCSE is represented within all eight faculties of TU Delft. About forty research groups and more than three hundred faculty members are connected to, and actively involved in DCSE and its activities. Over 250 PhD students perform research related to computational science.

CSE is a multidisciplinary application-driven field that deals with the development and application of computational models and simulations. Often coupled with high-performance computing to solve complex physical problems arising in engineering analysis and design (computational engineering) as well as natural phenomena (computational science). CSE has been described as the "third mode of discovery" (next to theory and experimentation). In many fields, computer simulation, development of problem-solving methodologies and robust numerical tools are integral and therefore essential to business and research. Computer simulations provide the capability to enter fields that are either inaccessible to traditional experimentation or where carrying out traditional empirical inquiries is prohibitively expensive. 


16 juni 2023 12:30 t/m 13:15

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

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.