Sep 10, 2015 | I'm co-organizer of a NIPS Workshop this December in Montreal, send in your abstracts! |
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.
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.
2016 | An integral equation-based numerical
solver for Taylor states in toroidal
geometries (with A. Cerfon), submitted. |
arXiv:1611.01420 | |
A new hybrid integral representation for
frequency domain scattering in layered media (with J. Lai and L. Greengard), to appear Appl. Comput. Harm. Anal. |
journal | arXiv:1507.03491 | |
Accurate and efficient numerical
calculation of stable densities
via optimized quadrature and asymptotics (with S. Ament), submitted. |
arXiv:1607.04247 | ||
Robust integral formulations for
electromagnetic scattering from three-dimensional
cavities (with J. Lai and L. Greengard), submitted. |
arXiv:1606.03599 | ||
Fast algorithms for Quadrature by
Expansion I: Globally valid expansions (with M. Rachh and A. Klöckner), submitted. |
arXiv:1602.05301 | ||
Smoothed corners and scattered
waves (with C. L. Epstein), SIAM J. Sci. Comput., 38(5):A2665-A2698, 2016. |
journal | arXiv:1506.08449 | |
Fast Direct Methods for Gaussian
Processes (with S. Ambikasaran, D. Foreman-Mackey, L. Greengard, and D. W. Hogg), IEEE Trans. Pattern Anal. Mach. Intell., 38(2):252-265, 2016. |
journal | arXiv:1403.6015 | |
2015 | Debye Sources, Beltrami Fields, and a Complex
Structure on Maxwell Fields (with C. L. Epstein and L. Greengard), Comm. Pure Appl. Math. 68(12):2237-2280, 2015. |
journal | arXiv:1308.5425 |
Fast symmetric factorization of
hierarchical matrices with
applications (with S. Ambikasaran). | arXiv:1405.0223 | ||
2014 | Exact axisymmetric Taylor states for
shaped plasmas (with A. Cerfon), Phys. Plasmas 21, 064501, 2014. |
journal | arXiv:1406.0481 |
A generalized Debye source approach to electromagnetic
scattering in layered
media J. Math. Phys. 55, 012901, 2014. |
journal | 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. |
journal | arXiv:1109.6708 | |
2013 | 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. |
journal | 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. |
journal | 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. |
journal | 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. |
journal | arXiv:1105.3217 | |
2010 | 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. |
journal | |
2003 | Slow passage through resonance in Mathieu's
equation (with L. Ng and R. Rand), J. Vib. Control 9(6):685-707, 2003. |
journal |
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.
Corner rounding
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
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
Combination linear algebra and ordinary differential equations course.
Fall 2016
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 2016
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
An introduction to several numerical methods known
as fast analysis-based algorithms, including fast
multiple methods, butterfly algorithms, hierarchical matrix
compression and fast direct solvers.
Fall 2015
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