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
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.
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.
Dhairya
Malhotra
Daria Sushnikova
Tristan Goodwill
Evan Toler
Yuwei Jiang
Sunli Tang (Uber)
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. |
arXiv:2006.02545 |
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. |
journal arXiv:1909.13356 |
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. |
journal arXiv:1909.07417 |
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. |
journal arXiv:1902.01205 |
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. |
journal arXiv:1810.07067 |
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. |
journal arXiv:1708.00056 |
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. |
journal (open-access) |
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. |
journal arXiv:1507.03491 |
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. |
journal arXiv:1611.01420 |
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. |
open-access |
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. |
journal arXiv:1602.05301 |
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. |
journal arXiv:1606.03599 |
Fast symmetric factorization of
hierarchical matrices with
applications S. Ambikasaran, M. O'Neil, and K. R. Singh, technical report, 2016. |
arXiv:1405.0223 |
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 |
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
GitHub
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.
GitHub
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.
GitHub
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
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.
GitLab
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 |