New York University

oneil@cims.nyu.edu

212-998-3125

Courant Institute

251 Mercer St., #1119

New York, NY 10012

Tandon School of Engineering

2 MetroTech Center, #854

Brooklyn, NY 11201

251 Mercer St., #1119

New York, NY 10012

Tandon School of Engineering

2 MetroTech Center, #854

Brooklyn, NY 11201

Sep 2023 - Aug 2024 | Visiting Scholar | CCM, Flatiron Institute |

Sep 2020 - | Associate Professor of Mathematics | Courant Institute, NYU |

Sep 2018 - Aug 2019 | Visiting Assistant Professor of Mathematics | MIT |

Sep 2014 - Aug 2020 | Assistant Professor of Mathematics | Courant Institute, NYU |

Sep 2012 - Aug 2014 | Courant Instructor | Courant Institute, NYU |

Sep 2010 - Aug 2012 | Associate Research Scientist | Courant Institute, NYU |

Aug 2007 - Jul 2020 | Quant Researcher / Assistant Trader | Susquehanna International Group, LLP |

Dec 2007 | PhD Applied Mathematics | Yale University |

May 2003 | AB Mathematics | Cornell University |

Most of my research incorporates the development of fast high-order analysis-based algorithms into problems in computational physics, integral equations, singular quadrature, statistics, and in general, computational science. Almost all problems are rooted in engineering and real-world applications.

Fast Algorithms Research Group at Courant

Almost all partial differential equation occurring in classical mathematical physics can be reformulated as integral equations with an appropriate Green's function. Proper integral formulations are usually very well-conditioned, but result in large dense systems which require fast algorithms to solve. Over the last couple decades, the development of analysis-based algorithms such as fast multipole methods, butterfly algorithms, etc. has enabled these systems to be solved rapidly, usually in near-linear time. I have recently been working on particular problems in electromagnetics, acoustics, and magnetohydrodynamics.

The numerical solution of any of these problems via an integral method requires solving problems in mathematical analysis, numerical analysis (e.g. quadrature for singular integrals), geometry (e.g. well-conditioned triangulations and meshes), fast computational algorithms, and other niches of applied mathematics. The resulting codes are often long and detailed but very efficient.

Recently it has been observed that many of the fast analysis-based algorithms used throughout engineering physics have direct applications in statistics, machine learning, and data analysis. In particular, methods for rapidly inverting structured dense covariance matrices have immediately found applications in Gaussian processes.

Paul Beckman

Annie Liu

Amandin Rabeendran

Paco Rilloraza

Bahram Khalichi (UC Berkeley)

Sam
Potter (Coreform)

Tristan Goodwill (U. Chicago)

Evan Toler (Argonne)

Dhairya
Malhotra (Flatiron Institute)

Yuwei Jiang (Plus)

Daria
Sushnikova (KAUST)

Sunli Tang (Uber)

Please contact me for more info.

Alex
Barnett (Flatiron)

Antoine Cerfon (NYU)

Charlie Epstein (UPenn)

Zydrunas Gimbutas (NIST)

Leslie Greengard (NYU)

Philip Greengard (Columbia)

David W. Hogg (NYU)

Lise-Marie Imbert-Gerard (Arizone)

Andreas Klöckner (UIUC)

Jun Lai
(Zhejiang)

Manas
Rachh (Flatiron)

Felipe
Vico (Valencia)

High-fidelity fast algorithms for inverse problems
and imaging in three dimensions O'Neil (PI) and Borges (PI, UCF), Office of Naval Research Award #N00014-21-1-2383, 9/1/21 - 8/31/24 |

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 |

An integral equation-based solver for the
Laplace-Beltrami operator on triangulated
surfaces O'Neil (PI), Office of Naval Research Award #N00014-15-1-2669, 7/1/2015 - 6/30/2016 |

Fast analysis-based computational methods for
statistics O'Neil (PI), AIG NYU-AIG Partnership on Innovation for Global Resilience, 9/1/2014 - 8/31/2015 |

On the construction of scattering matrices for
irregular or elongated enclosures using Green’s
representation formulaC. Borges, L. Greengard, M. O'Neil, and M. Rachh. arXiv.org > math.NA > 2408.02881, 2024. arXiv:2408.02881 |

A parametrix method for elliptic
surface PDEsT. Goodwill and M. O'Neil, arXiv.org > math.NA > 2401.12501, 2024. arXiv:2401.12501 |

Fast adaptive high-order integral
equation methods
for electromagnetic scattering from smooth
perfect electric
conductorsF. Vico, L. Greengard, M. O'Neil, and M. Rachh, arXiv.org > math.NA > 2306.04473, 2023. arXiv:2306.04473 |

An interface formulation of the Laplace-Beltrami
problem on piecewise smooth surfacesT. Goodwill M. O'Neil, SIAM J. Math. Anal.,
55(6): 7575-7615,
2023.journal arXiv:2108.08959 |

FMM-LU: A fast direct solver for
multiscale boundary integral equations in
three
dimensionsD. Sushnikova, L. Greengard, M. O'Neil, and M. Rachh, SIAM Multiscale Model.
Simul., 21(4): 1570-1601, 2023.journal arXiv:2201.07325 |

FMM-accelerated solvers for the
Laplace-Beltrami problem on complex surfaces in three
dimensionsD. Agarwal, M. O'Neil, and M. Rachh, J. Sci. Comput., 97:25, 2023.journal (open-access) |

Efficient reduced-rank methods for
Gaussian processes with eigenfunction
expansionsP. Greengard and M. O'Neil, Stat. Comput., 32:94, 2022.journal (open-access) |

Fast multipole methods for the evaluation
of layer potentials with locally-corrected
quadraturesL. Greengard, M. O'Neil, M. Rachh, and F. Vico, J. Comput. Phys.:
X, 10:100092, 2021.journal (open-access) |

A fast boundary integral method for
high-order multiscale mesh generationF. Vico, L. Greengard, M. O'Neil, and M. Rachh, SIAM
J. Sci. Comput., 42(2):A1380-A1401, 2020.journal arXiv:1909.13356 |

Efficient high-order singular quadrature
schemes in magnetic fusionD. Malhotra, A. J. Cerfon, M. O'Neil, and E. Toler, Plasma
Phys. Control. Fusion, 62(2):024004, 2019.journal arXiv:1909.07417 |

Taylor States in Stellarators: A Fast
High-order Boundary Integral SolverD. Malhotra, A. J. Cerfon, L.-M. Imbert-Gerard, and M. O'Neil, J. Comput. Phys.,
397:108791, 2019.journal arXiv:1902.01205 |

An FFT-accelerated direct solver for
electromagnetic scattering from penetrable
axisymmetric objectsJ. Lai and M. O'Neil, J. Comput. Phys.,
390:152-174, 2019.journal arXiv:1810.07067 |

A high-order wideband direct
solver for electromagnetic scattering
from bodies of revolutionC. L. Epstein, L. Greengard, and M. O'Neil, J. Comput. Phys.,
387:205-229, 2019.journal arXiv:1708.00056 |

Second-kind integral equations for the
Laplace-Beltrami problem on surfaces
in three dimensionsM. O'Neil, Adv. Comput. Math., 44(5):
1385-1409, 2018.journal (open-access) |

A new hybrid integral representation for
frequency domain scattering in layered mediaJ. Lai, L. Greengard, and M. O'Neil, Appl. Comput. Harm. Anal., 45(2):359-378, 2018.journal arXiv:1507.03491 |

An integral equation-based numerical
solver for Taylor states in toroidal
geometriesM. O'Neil and A. Cerfon, J. Comput. Phys., 359:263-282, 2018.journal arXiv:1611.01420 |

Accurate and efficient numerical
calculation of stable densities
via optimized quadrature and asymptoticsS. Ament and M. O'Neil,
Stat. Comput., 28(1):171-185, 2017.journal (open-access) |

Fast algorithms for Quadrature by
Expansion I: Globally valid expansionsM. Rachh, A. Klöckner, and M. O'Neil, J. Comput. Phys.,
345:706-731, 2017.journal arXiv:1602.05301 |

Robust integral formulations for
electromagnetic scattering from three-dimensional
cavitiesJ. Lai, L. Greengard, and M. O'Neil, J. Comput. Phys., 345:1-16, 2017.journal arXiv:1606.03599 |

Fast symmetric factorization of
hierarchical matrices with
applicationsS. Ambikasaran, M. O'Neil, and K. R. Singh, technical report, 2016. arXiv:1405.0223 |

Smoothed corners and scattered
wavesC. L. Epstein and M. O'Neil, SIAM J. Sci. Comput., 38(5):A2665-A2698, 2016.journal arXiv:1506.08449 |

Fast Direct Methods for Gaussian
ProcessesS. Ambikasaran, D. Foreman-Mackey, L. Greengard, D. W. Hogg, and M. O'Neil, IEEE Trans. Pattern
Anal. Mach. Intell., 38(2):252-265, 2016.journal arXiv:1403.6015 |

Debye Sources, Beltrami Fields, and a Complex
Structure on Maxwell FieldsC. L. Epstein, L. Greengard, and M. O'Neil, Comm. Pure Appl. Math.
68(12):2237-2280, 2015.journal arXiv:1308.5425 |

Exact axisymmetric Taylor states for
shaped plasmasA. Cerfon and M. O'Neil, Phys. Plasmas 21, 064501, 2014.journal arXiv:1406.0481 |

A generalized Debye source approach to electromagnetic
scattering in layered
mediaM. O'Neil, J. Math. Phys. 55, 012901,
2014.journal arXiv:1310.4241 |

On the efficient representation of the
impedance Green's function for the Helmholtz
equationM. O'Neil, L. Greengard, and A. Pataki, Wave Motion 51(1):1-13, 2014.journal arXiv:1109.6708 |

Quadrature by Expansion: A New Method for
the Evaluation of Layer
PotentialsA. Klöckner, A. Barnett, L. Greengard, and M. O'Neil, J. Comput. Phys. 252:332-349, 2013.journal arXiv:1207.4461 |

A fast, high-order solver for the
Grad-Shafranov equationA. Pataki, A. J. Cerfon, J. P. Freidberg, L. Greengard, and M. O'Neil, J. Comput. Phys. 243:28-45, 2013.journal arXiv:1210.2113 |

A consistency condition for the vector potential in
multiply-connected domainsC. L. Epstein, Z. Gimbutas, L. Greengard, A. Klöckner, and M. O'Neil, IEEE
Trans. Magn. 49(3):1072-1076, 2013.journal arXiv:1203.3993 |

Debye sources and the numerical solution of the time
harmonic Maxwell equations, IIC. L. Epstein, L. Greengard, and M. O'Neil, Comm. Pure Appl. Math. 66(5):753-789, 2013.journal arXiv:1105.3217 |

An algorithm for the rapid evaluation of special
function transformsM. O'Neil, F. Woolfe, and V. Rokhlin, Appl. Comput. Harmon. Anal.
28(2):203-226, 2010.journal |

Slow passage through resonance in Mathieu's
equationL. Ng, R. H. Rand, and M. O'Neil, J. Vib. Control 9(6):685-707, 2003.journal |

Using an integral equation formulation for constant-coefficient Beltrami fields, we developed a fast solver for MHD equilibria. See

GitHub

Elliptic PDEs in singular geometries are often computaitonally more expensive to solve than those in nearby regularized geometries. We have released preliminary Matlab code for regularizing polygons in 2D and polyhedra in 3D. See

GitHub

Three-dimensional fast multipole codes Laplace, Helmholtz, and Maxwell potentials can be downloaded from GitHub, and are supported by the Flatiron Institute. This is a collaborative effort between many researchers.

GitHub

The largest computational task encountered when modeling using Gaussian processes is the inversion of a (dense) covariance matrix. Often, these matrices have a hierarchical structure that can be exploited.

Read the Docs GitHub

Random variables whose distribution family is closed under addition are known as stable distributions, including normal and Cauchy distributions. In general, there are no closed form expressions for their evaluation. Using custom designed Generalized Gaussian Quadrature rules and asymptotics, the density function can be evaluated for a wide range of stable distributions.

GitLab

Semester | Number | Title |
---|---|---|

Fa 2024 | MA 4204 | Ordinary Differential Equations |

Sp 2023 | MATH 396 | Honors numerical analysis |

Fa 2022 | MATH 2011 | Computational electromagnetics |

Sp 2022 | MATH 2012 | Randomized numerical linear algebra |

Sp 2022 | MATH 396 | Honors numerical analysis |

Fa 2021 | MA 6973 | Computational statistics |

Sp 2021 | MA 6963 | Statistics |

Fa 2020 | MATH 2011 | Fast solvers |

Fa 2020 | MATH 233 | Theory of probability |

Sp 2020 | MA 4424 | Numerical analysis |

Fa 2019 | MATH 262 | Ordinary differential equations |

Sp 2019 | MATH 2840 | Computational methods for integral equations |

Sp 2018 | MATH 252 | Numerical analysis |

Fa 2017 | MATH 2011 | Integral equations and fast algorithms |

Fa 2016 | MA 2034 | Linear algebra and differential equations |

Sp 2016 | MA 4423 | Introductory numerical analysis |

Fa 2015 | MATH 2830 | Fast analysis-based algorithms |

Sp 2015 | MA 4423 | Introductory numerical analysis |

Fa 2014 | DS 1006 | Capstone project in Data Science |

Sp 2014 | MATH 234 | Mathematical statistics |

Fa 2013 | MATH 2011 | Data Science Projects |

Sp 2013 | MATH 234 | Mathematical statistics |

Fa 2012 | MATH 140 | Linear Algebra |

Sp 2012 | MATH 140 | Linear Algebra |