Integral Equations and Fast Algorithms (CS 598AK @ UIUC)
Class Time/Location 

Instructor 

Office 
Rm. 4318 Siebel 
Office Hours 
by appointment 
Class Webpage 

Email Listserv 
Lecture Material
Lecture # 
Date 
Topics 
Slides 
Primary Video 
UIUC Video 
Code 
Extra Info 
1 
Aug 27 
Intro 
Echo360 (no sound) 


2 
Aug 29 
Why IEs? Intro Functional Analysis 


3 
Sep 3 
Intro Functional Analysis, Intro IEs 



4 
Sep 5 
Intro IEs, Neumann series, Compact op. 



5 
Sep 10 
HW1 discussion, Compact op. 



6 
Sep 12 
Compactness of integral operators 
Echo 360 (sound dies halfway) 



7 
Sep 17 
Weakly singular i.op., Riesz theory 



8 
Sep 19 
Riesz theory, Hilbert spaces 


9 
Sep 24 
Fredholm theory, spectral theory 



10 
Sep 26 
Spectral theory, potential theory 



11 
Oct 1 
PV integrals, Green's thm., formula 



12 
Oct 3 
Jump conditions, ext. domains 


13 
Oct 8 
BVPs 



14 
Oct 10 
BVPs, Uniqueness 



15 
Oct 15 
Uniqueness, corners 



16 
Oct 16 
Intro Helmholtz 



17 
Oct 22 
Helmholtz BVP/IE uniqueness 


18 
Oct 24 
Calderón, Intro numerics 
Echo 360 (no sound) 



19 
Oct 29 
HW5, highorder numerics 


20 
Oct 31 
Highorder numerics, IE discretizations 

21 
Nov 5 
Interior Neumann, Nyström 


22 
Nov 7 
Collective compactness, Anselone's thm 



23 
Nov 12 
Anselone's thm, Céa's lemma 


ECE590 
Nov 12 
QBX quadrature 




24 
Nov 14 
Projection error est., Intro quadrature 


25 
Nov 19 
Singular quadrature, Intro fast alg. 


26 
Nov 21 
Fast Multipole Methods 



Nov 26 
Thanksgiving break 





Nov 28 







Dec 3 
Project presentations 





Dec 5 







Dec 10 
No class 





You'll need an uptodate version of Google Chrome to play the videos. You'll also need decent internet bandwidth to do streaming (2 MBit/s should be sufficient). If your internet accesss is too slow, you can always right click and download the video.
As far as the videos are concerned, Internet Explorer and Safari are not supported, because they do not understand the video format we're using.
And Chrome behaves much better than Firefox with the lecture video player, to the point of Firefox not even starting up properly. We're currently investigating. In the meantime, please use Chrome.
If you would still like to use Firefox and the page hangs (keeps displaying the spinny thing), pressing the "seek to start" button will usually return things to working order.
Description
This class will teach you how (and why!) integral equations let you solve many common types of partial differential equations robustly and quickly.
You will also see many fun numerical ideas and algorithms that bring these methods to life on a computer.
What to expect
 A Gentle Intro: Linear Algebra/Numerics/Python warmup
 Some Potential Theory
 The Laplace, Poisson, Helmholtz PDEs, and a few applications
 Integral Equations for these and more PDEs
 Ways to represent potentials
 Quadrature, or: easy ways to compute difficult integrals
 Tree codes and Fast Multipole Methods
 Fun with the FFT
 Linear algebrabased techniques ("Fast direct solvers"if time)
What you should already know
You should have taken some sort of numerical analysis/numerical methods course.
The following questions shouldn't be causing you too much grief:
 What is the divergence theorem? Green's first and second theorem?
 What is Gaussian quadrature?
 Name a numerical method that solves Poisson's equation $\triangle u=f$. (your choice of geometry, boundary conditions and discretization)
 What is the singular value decomposition?
 Name at least three methods for solving a system of linear equations $Ax=b$.
Updates
 August 7, 2013
 Class starts on August 27, 2012, from 23:15pm. We've also been assigned a room. We will be meeting in 1304 Siebel. See you then!
Grading/Evaluation
If you will be taking the class for credit, there will be
 Weekly homework for (a little more than) the first half of the class (60% of your grade)
 A more ambitious final project, which may be inspired by your own research needs (40% of your grade)
If you're planning on auditing or just sitting in, you are more than welcome.
Homework
Homework 1 due: September 5, 2013  out: August 27, 2013 (minor update to problem 1b on Sep 2, some notation fixes on Sep 3)
Homework 2 due: September 17 19, 2013  out: September 6, 2013
Homework 3 due: October 3 4, 2013  out: September 19, 2013
Homework 4 due: October 17, 2013  out: October 4, 2013
Homework 5 due: October 31, 2013  out: October 17, 2013
Homework is due at 11:59pm on the due date, i.e. at the end of the day.
Material
Books
These books cover some of our mathematical needs:
Linear Integral Equations by Kress 

probably the book with the best overall coverage, but little on numerics and algorithms 

Integral equation methods in scattering theory by Colton and Kress 
a more advanced book, builds on Kress LIE 

Partial Differential Equations of Mathematical Physics and Integral Equations by Guenther and Lee 

Good for potential theory, little FA, no numerics, oldish 

Partial Differential Equations: An Introduction by Colton 
UIUC library (not available) 


Integral Equations by Hackbusch 

comprehensive, but different emphasis than our class 

Foundations of Potential Theory by Kellogg 
ebook (public) 
a potential theory book, little FA, no numerics, oldish 

Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems by Nédélec 



Integral Equations by Tricomi 
ebook (public) 
slightly oldfashioned IE theory book, little numerics, no FA 

Linear Operator Theory in Engineering and Science by Naylor and Sell 

good for FA, uses int.eq. examples (recommended by Steven Dalton) 
(FA=functional analysis)
There really aren't any books to cover our numerical and algorithmic needs.
So, unfortunately, there isn't one book that covers the entire class, or even a reasonable subset. It will occasionally be useful to refer to these books, but I would not recommend you go out and buy them just for this course. I will make sure they are available in the library for you to refer to.
UIUC ebooks
The fast solution of boundary integral equations by Rjasanow 

Linear Integral Equations by Kanwal 

Linear and Nonlinear Integral Equations by Wazwaz 
Source articles
Because of the (no)book situation (see above), I will post links to the research articles underlying the class here.
Some possible articles for a final project
The fast Gauss Transform by Greengard and Strain
Distributedmemory parallelization of FMMs: (still looking for a good intro article, possibly A tuned and scalable fast multipole method as a preeminent algorithm for exascale systems by Yokota and Barba and references therein)
Accelerating the Nonuniform Fast Fourier Transform by Greengard and Lee
An algorithm for the rapid evaluation of special function transforms by O'Neil et al.
Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial by Johan Helsing
A fast solver for the Stokes equations with distributed forces in complex geometries by Biros, Ying, and Zorin
A fast direct solver for boundary integral equations in two dimensions by Martinsson and Rokhlin (or one of the many variants thereof)
I'll probably be adding more things to this list over time. Also note that this list is not intended to be exhaustive. If you've got an article that you like or want to read, let's talk.
Some of these require quite a bit of machinery in order to do meaningful implementation work. We'll have to do some negotiating on existing software you can use as a starting point.
Related classes elsewhere
Shidong Jiang (NJIT)
Alex Barnett (Dartmouth)
Shravan Veerapaneni (Michigan)
Leslie Greengard (NYU)
Gunnar Martinsson (UC Boulder)
Jianlin Xia (Purdue)
Francesco Andriulli (ENS TELECOM Bretagne)
Mark Tygert (NYU, now Yale)
Online resources
Math
Python
The Numpy MedKit by Stéfan van der Walt
The Numpy User Guide by Travis Oliphant
Spyder (a Python IDE, like Matlab) is installed in the virtual machine. (Applications Menu > Development > Spyder)
Virtual Machine Image
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