CAM Colloquium-- Fruzsina Agocs, Flatiron Research Fellow, Center for Computational Mathematics, Flatiron Institute

Description

Title: Wiggles and waves: adaptive high-order methods for oscillatory ODEs and PDEs

Abstract: Oscillatory problems have long posed a challenge to numerical computation, whether they take the form of ordinary or partial differential equations (ODEs or PDEs), yet they are ubiquitous in applications ranging from engineering to astrophysics. In both types of equations, oscillations force algorithms to use more discretization nodes (thus spend more computational effort) as the frequency rises. In PDEs, there are additional difficulties associated with waves interacting with the geometry, e.g. sharp corners, periodic boundaries, cavities. In this talk, I will present two classes of fast, high-order accurate numerical methods to solve oscillatory problems.

For ODEs, I introduce two algorithms that exploit asymptotic expansions in regions of rapid oscillations, but behave as "standard'' solvers otherwise, thus achieving O(1) (frequency-independent) runtime. I will show how they eliminate computational bottlenecks in early-universe astrophysics, quantum physics, and special function evaluation.

Among PDEs, I will focus on two-dimensional acoustic and electromagnetic scattering problems from a nonperiodic source by a periodic boundary. I show how these may be solved using a boundary integral equation (BIE) method, and how the analysis of the solution explains wave-guiding phenomena exploited in applications like antennae, radars, photonic crystals, and acoustic metamaterials.

Bio: I am a postdoctoral researcher at the Center for Computational Mathematics at the Flatiron Institute in New York. I am part of the Numerical Analysis group with research interests in developing fast, adaptive, high-order methods for ODEs, PDEs, and quadrature, particularly oscillatory problems. I also work with physicists on applying these methods in their field, particularly in cosmology and astrophysics, e.g. for speeding up forward modeling related to cosmic inflation, dark matter candidates, beyond-Standard-Model physics. I hold a PhD in Theoretical and Computational Cosmology, and a MSci + BA in Physics from the University of Cambridge (UK).