Michael O'Neil

Assistant Professor of Mathematics

Contact Information

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

Polytechnic School of Engineering
New York University
6 Metrotech Center, #321F
Brooklyn, NY 11201


Short bio
Assistant Professor, Courant Institute and Polytechnic 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


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)
Andreas Klöckner (UIUC)
Jon Wilkening (Berkeley)

Siva Ambikasaran (NYU)

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 - Google Scholar Profile - arXiv Profile
Fast symmetric factorization of hierarchical matrices with applications
(with S. Ambikasaran), submitted.
[ arXiv:1405.0223 ]

Fast Direct Methods for Gaussian Processes
(with S. Ambikasaran, D. Foreman-Mackey, L. Greengard, and D. W. Hogg), submitted.
[ arXiv:1403.6015 ]

Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
(with C. L. Epstein and L. Greengard), to appear, Comm. Pure Appl. Math..
[ pdf ] [ arXiv:1308.5425 ]

Exact axisymmetric Taylor states for shaped plasmas
(with A. Cerfon), Phys. Plasmas 21, 064501, 2014.
[ pdf ] [ arXiv:1406.0481 ]

A generalized Debye source approach to electromagnetic scattering in layered media
J. Math. Phys. 55, 012901, 2014.
[ pdf ] [ 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.
[ pdf ] [ arXiv:1109.6708 ]

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.
[ pdf ] [ 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.
[ pdf ] [ 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.
[ pdf ] [ 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.
[ pdf ] [ arXiv:1105.3217 ]

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.
[ pdf ]

Slow passage through resonance in Mathieu's equation
(with L. Ng and R. Rand), J. Vib. Control 9(6):685-707, 2003.
[ pdf ]

Selected talks

Below are slides from a few selected talks that I've given over the years.

Software - Bitbucket - GitHub
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

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.

Integral equations MA-UY xxxx @ NYU SoE

Spring 2016

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 2015

Data Science Projects MATH-GA 2011, CSCI-GA 2945 @ 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 MATH-UA 140 @ NYU

Introductory linear algebra.
Fall 2012 & Spring 2012

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

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