Michael O'Neil

Assistant Professor of Mathematics
New York University

Courant Institute of Mathematical Sciences
251 Mercer St., #1122
New York, NY 10012

Tandon School of Engineering
6 Metrotech Center, #321F
Brooklyn, NY 11201


Short bio

Assistant Professor, Courant Institute and Tandon School of Engineering, NYU
2012 - 2014
Courant Instructor, Courant Institute, NYU
2010 - 2012
Associate Research Scientist, Courant Institute, NYU
2007 - 2010
Quantitative researcher & assistant trader, Susquehanna International Group, LLP
Ph.D. Applied Mathematics, Yale University
A.B. Mathematics, Cornell University


Sep 10, 2015 I'm co-organizer of a NIPS Workshop this December in Montreal, send in your abstracts!


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. For more information, check out my research page.

Integral equations, computational physics, fast algorithms, and numerical analysis

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 stable, 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.

Also visit the Courant Mathematics and Computing Laboratory and the Greengard research group page for related research.

Alex Barnett (Dartmouth)
Antoine Cerfon (NYU)
Charlie Epstein (UPenn)
Zydrunas Gimbutas (NIST)
Leslie Greengard (NYU)
David W. Hogg (NYU)
Lise-Marie Imbert-Gerard (NYU)
Andreas Klöckner (UIUC)
Jun Lai (NYU)
Manas Rachh (Yale)
Jon Wilkening (Berkeley)

Graduate Students

Sunli Tang (NYU)

Please contact me if you are a graduate student interested in computational science and looking for an advisor or a post-doc position.

Publications and pre-prints [Google Scholar] [arXiv.org]

2017 An integral equation-based numerical solver for Taylor states in toroidal geometries
(with A. Cerfon), submitted.
Robust integral formulations for electromagnetic scattering from three-dimensional cavities
(with J. Lai and L. Greengard), submitted.
Fast algorithms for Quadrature by Expansion I: Globally valid expansions
(with M. Rachh and A. Klöckner), submitted.
A new hybrid integral representation for frequency domain scattering in layered media
(with J. Lai and L. Greengard), to appear Appl. Comput. Harm. Anal.
journal arXiv:1507.03491
Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics
(with S. Ament), to appear Stat. Comput.
2016 Smoothed corners and scattered waves
(with C. L. Epstein), SIAM J. Sci. Comput., 38(5):A2665-A2698, 2016.
journal arXiv:1506.08449
Fast Direct Methods for Gaussian Processes
(with S. Ambikasaran, D. Foreman-Mackey, L. Greengard, and D. W. Hogg),
IEEE Trans. Pattern Anal. Mach. Intell., 38(2):252-265, 2016.
journal arXiv:1403.6015
2015 Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
(with C. L. Epstein and L. Greengard), Comm. Pure Appl. Math. 68(12):2237-2280, 2015.
journal arXiv:1308.5425
Fast symmetric factorization of hierarchical matrices with applications
(with S. Ambikasaran).
2014 Exact axisymmetric Taylor states for shaped plasmas
(with A. Cerfon), Phys. Plasmas 21, 064501, 2014.
journal arXiv:1406.0481
A generalized Debye source approach to electromagnetic scattering in layered media
J. Math. Phys. 55, 012901, 2014.
journal arXiv:1310.4241
On the efficient representation of the impedance Green's function for the Helmholtz equation
(with L. Greengard and A. Pataki), Wave Motion 51(1):1-13, 2014.
journal arXiv:1109.6708
2013 Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials
(with A. Klöckner, A. Barnett, and L. Greengard), J. Comput. Phys. 252:332-349, 2013.
journal arXiv:1207.4461
A fast, high-order solver for the Grad-Shafranov equation
(with A. Pataki, A. J. Cerfon, J. P. Freidberg, and L. Greengard), J. Comput. Phys. 243:28-45, 2013.
journal arXiv:1210.2113
A consistency condition for the vector potential in multiply-connected domains
(with C. L. Epstein, Z. Gimbutas, L. Greengard, and A. Klöckner),
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
(with C. L. Epstein and L. Greengard), Comm. Pure Appl. Math. 66(5):753-789, 2013.
journal arXiv:1105.3217
2010 An algorithm for the rapid evaluation of special function transforms
(with F. Woolfe and V. Rokhlin), Appl. Comput. Harmon. Anal. 28(2):203-226, 2010.
2003 Slow passage through resonance in Mathieu's equation
(with L. Ng and R. Rand), J. Vib. Control 9(6):685-707, 2003.

Software [GitLab] [GitHub]

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.
Corner rounding

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.

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 systematic structure that can be exploited. george is a Python interface for a C++ implementation of the HODLR factorization. An optimized Fortran version is currently in development.
george - HODLR


Linear Algebra and Differential Equations
MA-UY 2034 @ NYU SoE

Combination linear algebra and ordinary differential equations course.

Introductory Numerical Analysis
MA-UY 4423 @ NYU SoE

Introductory numerical analysis intended for undergraduate and graduate students covering fundamental topics such as floating-point arithmetic, numerical integration, interpolation, linear algebra, solution of ODEs, etc.
Spring 2016
Spring 2015

Data Science Projects
MATH-GA 2011.001 @ NYU Courant

A projects mentoring course for students concentrating on a data science track within the computational science masters program at the Courant Institute.
Fall 2013

Linear algebra

Introductory linear algebra.
Fall 2012
Spring 2012

Fast analysis-based algorithms
MATH-GA 2830.002 @ NYU Courant

An introduction to several numerical methods known as fast analysis-based algorithms, including fast multiple methods, butterfly algorithms, hierarchical matrix compression and fast direct solvers.
Fall 2015

Capstone Project in Data Science
DS-GA 1006 @ NYU Courant

This is the crowning project course for students enrolled in the Data Science masters program at NYU through the Center for Data Science. Working with industry and/or faculty mentors, students complete and present a thorough treatment of a real-world data science problem.
Fall 2014

Mathematical Statistics
MATH-UA 234 @ NYU Courant

This course is a junior/senior level introduction to the mathematical theory of statistics to be taken after a similarly focused course on the theory of probability has been taken.
Spring 2014
Spring 2013