Mike O'Neil

Assistant 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 2014 - Assistant Professor Courant Institute, NYU
Sep 2018 - Aug 2019 Visiting Assistant Professor Mathematics, MIT
Sep 2012 - Aug 2014 Courant Instructor Courant Institute, NYU
Sep 2010 - Aug 2012 Associate Research Scientist Courant Institute, NYU
Aug 2007 - Jul 2010 Quant Researcher, Assistant Trader Susquehanna International Group, LLP
Dec 2007 Ph.D. Applied Mathematics Yale University
May 2003 A.B. 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.

Research group: 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 complicated but very efficient.

Complementary to solving PDEs or integral equations, algorithms which stably and rapidly compute special functions, invert matrices, apply operators, etc. must be developed. These schemes fall broadly under numerical analysis, and constitute the components that go into necessary software toolboxes for applied mathematics.

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.

Alex Barnett (Flatiron)
Antoine Cerfon (NYU)
Charlie Epstein (UPenn)
Zydrunas Gimbutas (NIST)
Leslie Greengard (NYU)
David W. Hogg (NYU)
Lise-Marie Imbert-Gerard (UMD)
Andreas Klöckner (UIUC)
Jun Lai (Zhejiang)
Manas Rachh (Flatiron)


Dhairya Malhotra
Daria Sushnikova

Graduate Students

Tristan Goodwill
Evan Toler


Yuwei Jiang
Sunli Tang (Uber)

Open positions

Please contact me for more info.


Profile on Google Scholar and arXiv.org.

Title, author, journal Download
Fast multipole methods for the evaluation of layer potentials with locally-corrected quadratures
L. Greengard, M. O'Neil, M. Rachh, and F. Vico, arXiv:2006.02545, 2020.
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.
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.
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 axisymmetric objects
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Exact axisymmetric Taylor states for shaped plasmas
A. Cerfon and M. O'Neil, Phys. Plasmas 21, 064501, 2014.
A generalized Debye source approach to electromagnetic scattering in layered media
M. O'Neil, J. Math. Phys. 55, 012901, 2014.
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.
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.
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.
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.
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.
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.

Title Semester Number
Fast solvers Fall 2020 MATH-GA 2011.003
Theory of probability Fall 2020 MATH-UA 233
Numerical analysis Spring 2020 MA-UY 4424
Ordinary differential equations Fall 2019 MATH-UA 262
Computational methods for integral equations Spring 2019 MATH-GA 2840.001
Numerical analysis Spring 2018 MATH-UA 252
Integral equations and fast algorithms Fall 2017 MATH-GA 2011.002
Linear algebra and differential equations Fall 2016 MA-UY 2034
Introductory numerical analysis Spring 2016 MA-UY 4423
Fast analysis-based algorithms Fall 2015 MATH-GA 2830.002
Introductory numerical analysis Spring 2015 MA-UY 4423
Capstone project in Data Science Fall 2014 DS-GA 1006
Mathematical statistics Spring 2014 MATH-UA 234
Data Science Projects Fall 2013 MATH-GA 2011.001
Mathematical statistics Spring 2013 MATH-UA 234
Linear Algebra Fall 2012 MATH-UA 140
Linear Algebra Spring 2012 MATH-UA 140