Mike O'Neil

Assistant Professor of Mathematics
New York University

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

Tandon School of Engineering
6 Metrotech Center, #321F
Brooklyn, NY 11201
2014 - Current Assistant Professor Courant Institute, NYU
2012 - 2014 Courant Instructor Courant Institute, NYU
2010 - 2012 Associate Research Scientist Courant Institute, NYU
2007 - 2010 Quant Researcher, Assistant Trader Susquehanna International Group, LLP
2007 Ph.D. Applied Mathematics Yale University
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.

PDEs, 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.

Related research groups:

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 (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)
Nick Knight (NYU)
Jun Lai (Zhejiang)
Manas Rachh (Yale)
Jon Wilkening (Berkeley)

Graduate Students
Sunli Tang (NYU)

Open positions
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 Tech Reports

My profile on Google Scholar and arXiv.org.

Title, author, journal Download
Second-kind integral equations for the Laplace-Beltrami problem on surfaces in three dimensions
M. O'Neil, submitted.
Fast symmetric factorization of hierarchical matrices with applications
S. Ambikasaran, M. O'Neil, and K. R. Singh, submitted.
An integral equation-based numerical solver for Taylor states in toroidal geometries
M. O'Neil and A. Cerfon, submitted.
Robust integral formulations for electromagnetic scattering from three-dimensional cavities
J. Lai, L. Greengard, and M. O'Neil, to appear, J. Comput. Phys., 2017.
Fast algorithms for Quadrature by Expansion I: Globally valid expansions
M. Rachh, A. Klöckner, and M. O'Neil, to appear J. Comput. Phys., 2017.
A new hybrid integral representation for frequency domain scattering in layered media
J. Lai, L. Greengard, and M. O'Neil, to appear Appl. Comput. Harm. Anal., 2017.
Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics
S. Ament and M. O'Neil, to appear Stat. Comput., 2017.
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.

View all my code on GitLab and 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. GitLab repo: 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. Code source on CMCL.

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. Get the software on GitHub: george.

Title Semester Number
Integral equations and fast algorithms Fall 2017
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