Fast Algorithms at Courant

Eigenmodes of Beltrami
					    fields in a tokamak
						 scattering from a cylinder
Parabolic antenna mesh


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.


Dhairya Malhotra
Kirill Serkh
Denis Silantyev
Grad students
Yuwei Jiang
Jason Kaye
Sunli Tang


Seminar schedules and materials will be posted here beginning in the fall.


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

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Publications and Technical Reports

Title, author, journal Download
An integral equation-based numerical solver for Taylor states in toroidal geometries
M. O'Neil and A. Cerfon, J. Comput. Phys., 359:263-282, 2018.
Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion
T. Sanchez-Vizuet and A. J. Cerfon, Plasma Physics and Controlled Fusion, 60:025018, 2018.
A high-order wideband direct solver for electromagnetic scattering from bodies of revolution
C. L. Epstein, L. Greengard, and M. O'Neil, technical report, 2017.
An adaptive fast multipole accelerated Poisson solver for complex geometries
T. Askham and A. J. Cerfon, J. Comput. Phys., 344:1, 2017.
Sparse grid techniques for particle-in-cell schemes
L. F. Ricketson and A. J. Cerfon, Plasma Physics and Controlled Fusion, 59:024002, 2017.
Fast algorithms for Quadrature by Expansion I: Globally valid expansions
M. Rachh, A. Klöckner, and M. O'Neil, J. Comput. Phys., 345:706-731, 2017.
Accurate Derivative Evaluation for any Grad-Shafranov Solver
L. F. Ricketson, A. J. Cerfon, M. Racch, and J. P. Freidberg, J. Comput. Phys., 305:744, 2016.
ECOM: A fast and accurate solver for toroidal axisymmetric MHD equilibria
J. P. Lee and A. J. Cerfon, Comput. Phys. Commun., 190:72, 2015.


Fast multipole methods
Two-dimensional and three-dimensional fast multipole codes developed by Leslie Greengard and Zydrunas Gimbutas for Laplace, Helmholtz, elastostatic, and Maxwell potentials can be downloaded on the CMCL webpage. Code source on CMCL.

Plasma physics
Various simulators for plasma equilibria in axisymmetric geometries (e.g. tokamaks) have been developed by members of the group. Links to individual codes can be found here.