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.
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:
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.
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.
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. |
arXiv:1705.00069 |
Fast symmetric factorization of
hierarchical matrices with
applications S. Ambikasaran, M. O'Neil, and K. R. Singh, submitted. |
arXiv:1405.0223 |
An integral equation-based numerical
solver for Taylor states in toroidal
geometries M. O'Neil and A. Cerfon, submitted. |
arXiv:1611.01420 |
Robust integral formulations for
electromagnetic scattering from three-dimensional
cavities J. Lai, L. Greengard, and M. O'Neil, to appear, J. Comput. Phys., 2017. |
journal arXiv:1606.03599 |
Fast algorithms for Quadrature by
Expansion I: Globally valid expansions M. Rachh, A. Klöckner, and M. O'Neil, to appear J. Comput. Phys., 2017. |
journal arXiv:1602.05301 |
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. |
journal arXiv:1507.03491 |
Accurate and efficient numerical
calculation of stable densities
via optimized quadrature and asymptotics S. Ament and M. O'Neil, to appear Stat. Comput., 2017. |
journal open-access arXiv:1607.04247 |
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. |
journal |
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. |
journal |
View all my code on GitLab and GitHub
Corner and edge roundingElliptic 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 methodsTwo-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 processesThe 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 |