- Current
- 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
- 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.

**Integral equations, computational physics, and fast
algorithms**

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.

**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.

**Numerical analysis**

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.

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)

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.

- 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 ]

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

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

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.

CMCL

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

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

Fall 2013

Introductory linear algebra.

Fall 2012 & Spring 2012

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

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