Integral Equations and Fast Algorithms

Course number: MATH-GA 2011.002
Semester: Fall 2017
Time & Location: Thurs, 1:25pm - 3:15pm in WWH 512
Instructor: Mike O'Neil (oneil@cims.nyu.edu)
Office hours: By appointment
Course description

This course will be an introduction the theory and application of integral equations in classical mathematical physics, as well as the numerical methods required for their efficient and accurate solution. These numerical methods include quadrature for singular functions, analysis-based fast algorithms (e.g. fast multipole methods), iterative and fast-direct solvers (for the resulting dense linear systems). Methods from potential theory, applied analysis, functional analysis, numerical linear algebra, complex analysis, and asymptotic analysis are central to the construction of almost all of these algorithms.

Materials

There is no one textbook for this course. Instead, there will be continually updated lectures notes available. These lecture notes will contain many useful references for each of the topics and algorithms covered in class. As they become relevant, original journal articles and textbooks will be listed below in the table of lecture topics.

Relevant code examples will be posted on gitlab.com/oneilm/integralequations.

Lecture notes, updated throughout the semester, can be downloaded here: int_eq_notes_2017.pdf.

Grading

The grades in the course will be determined by a course project and classroom interactions. A description of the course project can be found here:

Announcements

Important information for the course will appear below as necessary.

  • Class on September 7th is CANCELLED. The first class on September 14
Schedule

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

Date Topics Materials
September 7 No class - cancelled.
September 14 Overview, electrostatics and the Laplace equation Jackson, Classical Electrodynamics
September 21 2D Laplace boundary value problems, layer potentials Colton, Partial Differential Equations
September 28 Fredholm theory Riesz and Nagy, Functional Analysis
Porter and Stirling, Integral equations
October 5 Trapezoidal discretization and quadrature Atkinson, The Numerical Solution of Integral Equations of the 2nd Kind
Kress, 1991
Kapur and Rokhlin, 1997
Alpert, 1999
Hao, et. al., 2014
October 12 Adaptive discretizations, 3D Laplace BVPs
October 19 3D Laplace, surface discretizations Bremer & Gimbutas, 2012
October 26 Barnes-Hut and higher-order tree codes Appel, 1985
Barnes & Hut, 1986
November 2 The 3D Laplace FMM, multipole translations Greengard, 1987
Epton & Dembart, 1995
November 9 Accelerating translation operators
November 16 Integral equations in electromagnetics
November 23 No class - Thanksgiving.
November 30 Butterfly algorithms
December 7 Integral equations in fluid dynamics
December 14 Fast direct solvers