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.
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.
Below is an updated list of discussion topics along with any documents that were distributed, or relevant code.
|Jan 29||Electrostatics, magnetostatics||electrostatics.pdf
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
Vico, et. al., 2016
|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||Fast multipole methods|
|Apr 9||Fast multipole methods, butterfly algorithms|
|Apr 16||Fast algorithms for t-SNE (Manas Rachh, Flatiron)||Linderman, et. al., 2017|
|Apr 23||Function extension and Poisson problems|
|Apr 30||Adjoint methods
Fast direct solvers
|MIT Lecture Notes
Gillman, Young, Martinsson, 2012
|May 7||PDEs in corner domains (Kirill Serkh, Yale)||Serkh and Rokhlin, 2016|