Computational Methods for Integral Equations

Course number: MATH-GA 2840.001
Semester: Spring 2019
Time & Location: Tues, 3:20pm - 5:10pm in CIWW 517
Instructor: Antoine Cerfon (cerfon@cims.nyu.edu)
Leslie Greengard (greengard@cims.nyu.edu)
Mike O'Neil (oneil@cims.nyu.edu)
Course description

This course will be run as a working group consisting of faculty, post-docs, and graduate students. The topics that will be covered include: integral equation formulations of PDEs, numerical methods for singular quadrature, fast algorithms, and optimization methods. The material will range from standard/introductory level to recent research results, and will be presented via paper discussion, lectures by faculty, and reports on recent research progress by graduate students and post-docs (roughly 1/3 time spent on each). Application areas will include fluid mechanics, magnetofluid dynamics (plasma physics), and electromagnetics. The working group will be advertised and open to all graduate students, pending approval of the faculty organizers.

Materials

There is no textbook for this course. Supplemental references will be suggested as necessary.

Relevant code examples will be posted on gitlab.com/oneilm/sp19-math2840.

Announcements
  • The first class will meet on January 29, 2019.
Schedule

Below is an updated list of discussion topics along with any documents that were distributed, or relevant code.

Date Topics Materials
Jan 29 Electrostatics, magnetostatics electrostatics.pdf
magnetostatics.pdf
Feb 5 Electromagnetics
Layer potential jump conditions
Schrodinger equation (Jason Kaye, Courant)
 
Chapter 3 of http://bit.ly/2TC77zt
 
Feb 12 Nystrom discretization, quadrature
Int Eqs for the Heat Equation and Beyond (Jun Wang, Flatiron)
Feb 19 Discretization and quadrature

FFT-accelerated Poisson solver
Hao, et. al., 2014
Alpert, 1999
Vico, et. al., 2016
Feb 26
Mar 5 Conditioning of integral equations (Euan Spence, Bath UK)
Mar 12 Magnetically confined plasma equilibria
Equilibria in stellarators (Dhairya Malhotra, Courant, NYU)
Mar 19 No class, Spring Recess
Mar 26 Eigenvalues and PDEs (Travis Askham, NJIT)
Apr 2 Int eqs in domains with corners and edges (Kirill Serkh, Yale)
Apr 9
Apr 16 Fast algorithms for t-SNE (Manas Rachh, Flatiron) Linderman, et. al., 2017
Apr 23 Fast direct solvers (Adrianna Gillman, Rice U)
Apr 30
May 7