Associate Professor of Mathematics
New York University
Courant Institute
251 Mercer St., #1119
New York, NY 10012

Tandon School of Engineering
2 MetroTech Center, #854
Brooklyn, NY 11201
Sep 2021 - 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
Fast algorithms
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

Computational PDEs and integral equations
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.

Computational statistics
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.

Bahram Khalichi

Graduate Students

Paul Beckman
Paco Rilloraza


Sam Potter (Coreform)
Tristan Goodwill (U. Chicago)
Evan Toler (Argonne)
Dhairya Malhotra (Flatiron Institute)
Yuwei Jiang
Daria Sushnikova (KAUST)
Sunli Tang (Uber)

Open positions

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)

My research has been supported in part by the following awards:

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
Profile on Google Scholar and arXiv.org.

A parametrix method for elliptic surface PDEs
T. Goodwill and M. O'Neil, arXiv.org > math.NA > 2401.12501, 2024.
Fast adaptive high-order integral equation methods for electromagnetic scattering from smooth perfect electric conductors
F. Vico, L. Greengard, M. O'Neil, and M. Rachh, arXiv.org > math.NA > 2306.04473, 2023.
An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces
T. 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 dimensions
D. 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 dimensions
D. 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 expansions
P. 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 quadratures
L. 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 generation
F. 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 fusion
D. 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 Solver
D. 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 objects
J. 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 revolution
C. 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 dimensions
M. O'Neil, Adv. Comput. Math., 44(5): 1385-1409, 2018.
journal (open-access)
A new hybrid integral representation for frequency domain scattering in layered media
J. 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 geometries
M. 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 asymptotics
S. Ament and M. O'Neil, Stat. Comput., 28(1):171-185, 2017.
journal (open-access)
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.
journal arXiv:1602.05301
Robust integral formulations for electromagnetic scattering from three-dimensional cavities
J. 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 applications
S. Ambikasaran, M. O'Neil, and K. R. Singh, technical report, 2016.
Smoothed corners and scattered waves
C. L. Epstein and M. O'Neil, SIAM J. Sci. Comput., 38(5):A2665-A2698, 2016.
journal arXiv:1506.08449
Fast Direct Methods for Gaussian Processes
S. 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 Fields
C. 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 plasmas
A. Cerfon and M. O'Neil, Phys. Plasmas 21, 064501, 2014.
journal arXiv:1406.0481
A generalized Debye source approach to electromagnetic scattering in layered media
M. 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 equation
M. 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 Potentials
A. 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 equation
A. 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 domains
C. 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, II
C. 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 transforms
M. O'Neil, F. Woolfe, and V. Rokhlin, Appl. Comput. Harmon. Anal. 28(2):203-226, 2010.
Slow passage through resonance in Mathieu's equation
L. Ng, R. H. Rand, and M. O'Neil, J. Vib. Control 9(6):685-707, 2003.
Taylor states in stellarators
Using an integral equation formulation for constant-coefficient Beltrami fields, we developed a fast solver for MHD equilibria. See Taylor States in Stellarators above for more information.

Corner and edge rounding
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 Smoothed corners and scattered waves above for more info.

Fast multipole methods
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.
Fast methods for Gaussian processes
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. george is a Python interface for a C++ implementation of the HODLR factorization. See Fast Direct Methods for Gaussian Processes above for more information.
Read the Docs GitHub

Stable density evaluation
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.
Semester Number Title
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