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 will meet during Fall 2019 on Thursdays from
2:00pm - 3:30 in CIWW 705.
Working group for Spring 2019.
Boundary integral equation solvers
FMM-accelerated boundary integral equation solvers in complex geometries in three dimensions for Laplace and Helmholtz problems (and soon Maxwell).
Fast multipole methods
FMMs in three dimensions for Laplace and Helmholtz potentials. Maintained by the Flatiron Institute.
Taylor states in stellarators
High-order boundary integral equation code for computing MHD equilibria in stellarator geometries, see paper Taylor States in Stellerators above.
Quadrature for magnetostatics
High-order quadrature for layer potentials on smooth toroidal surfaces, see paper Efficient high-order singular quadrature schemes in magnetic fusion above.
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.
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
|Title, author, journal||Download|
|Efficient high-order singular quadrature
schemes in magnetic fusion
D. Malhotra, A. J. Cerfon, M. O'Neil, and E. Toler, submitted, 2019.
|Magnetic shear due to localized toroidal flow
shear in tokamaks
J. Lee and A. J. Cerfon, Plasma Physics and Controlled Fusion, 61:105007, 2019.
|Taylor States in Stellarators: A Fast
High-order Boundary Integral Solver
D. Malhotra, A. J. Cerfon, L.-M. Imbert-Gerard, and M. O'Neil,
J. Comput. Phys., 397:108791, 2019.
|An FFT-accelerated direct solver for
electromagnetic scattering from penetrable
J. Lai and M. O'Neil, J. Comput. Phys., 390:152-174, 2019.
|A high-order wideband direct
solver for electromagnetic scattering
from bodies of revolution
C. L. Epstein, L. Greengard, and M. O'Neil, J. Comput. Phys., 387:205-229, 2019.
|A flux-balanced fluid model for
collisional plasma edge turbulence: Model
derivation and basic physical features
A. J. Majda, D. Qi, and A. J. Cerfon, Phys. Plasmas, 25:102307, 2018.
|A flux-balanced fluid model for collisional plasma edge turbulence: Numerical simulations with different aspect ratios
A. J. Majda, D. Qi, and A. J. Cerfon, Phys. Plasmas, 26:082303, 2019.
|Second-kind integral equations for the
Laplace-Beltrami problem on surfaces
in three dimensions
M. O'Neil, Adv. Comput. Math., 44(5): 1385-1409, 2018.
|An integral equation-based numerical
solver for Taylor states in toroidal
M. O'Neil and A. J. Cerfon, J. Comput. Phys., 359:263-282, 2018.
|Pseudo spectral collocation with Maxwell
polynomials for kinetic equations with energy
T. Sanchez-Vizuet and A. J. Cerfon, Plasma Physics and Controlled Fusion, 60:025018, 2018.
|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
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
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.