Fast Algorithms at Courant

Large-scale linear systems arise in a myriad of areas in physics, mathematics, and statistics. Often when directly discretizing PDEs, the resulting linear systems are sparse and efficient methods for applying the discretized operators are straightforward, and relatively efficient methods for computing their inverse has been developed. However, linear systems arising from the discretization of integral formulations of the very same PDEs are dense, and somewhat more sophisticated algorithms must be developed in order to apply and invert there operators with near optimal computational complexity. Often these algorithms are based on the underlying physics of the problem, and are therefore commonly referred to as analysis-based fast algorithms.

The most well-known analysis-based fast algorithm algorithm is the fast multipole method, which allows for various N-body calculations to be performed in O(N) time, often resulting in speedups over the direct calculation of 1000x or more. Most of the work of the Fast Algorithms group is focused on designing novel computational schemes for solving PDE and integral equations arising in electromagnetics, fluid dynamics, quantum electrodynamics, heat flow, and magnetohydrodynamics. Various aspects of the work include numerical quadrature design, special function evaluation, applied analysis (e.g. deriving novel integral equations), computational geometry (e.g. surface meshing), optimization, and high-performance computing.

The group is supported in part by the following sources:

Hidden Symmetries and Fusion Energy
A. Bhattacharjee (PI), Cerfon (Co-I), O'Neil (Co-I), et. al., Simons Foundation, 6/1/18 - 5/31/22

Multi-level randomized algorithms for high-frequency wave propagation
Greengard (PI) and O'Neil (Co-I), Office of Naval Research Award #N00014-18-1-2307, 6/1/18 - 5/31/22

Toward real-time electromagnetic design: Fast, accurate, and robust integral equation-based solvers
O'Neil (PI), Office of Naval Research Award #N00014-17-1-2451, 6/1/17 - 5/31/20

Fast high-order CAD-compatible Nystrom methods for frequency domain electromagnetics
O'Neil (PI), Office of Naval Research Award #N00014-17-1-2059, 1/1/17 - 12/31/19

Plasma Properties (Task III)
H. Weitzner (PI) and Cerfon (Co-I), US Department of Energy, Office of Science, Fusion Energy Sciences DE-FG02-86ER53223, 5/1/16 - 4/30/19

High Performance Equilibrium Solvers for Integrated Magnetic Fusion Simulations
Cerfon (PI), US Department of Energy, Office of Science, Fusion Energy Sciences DE-SC0012398, 7/15/14 - 7/14/19