# Integral Equations and Fast Algorithms (CS 598AK @ UIUC)

What Value
Class Time/Location Tuesday/Thursday 2:00pm-3:15pm / 1304 Siebel
Instructor Andreas Kloeckner
Office Rm. 4318 Siebel
Office Hours by appointment
Class Webpage http://bit.ly/inteq13
Email Listserv Info page

## Lecture Material

Lecture # Date Topics Slides Primary Video UIUC Video Code Extra Info
1 Aug 27 Intro Slides Video Echo360 (no sound) Code
2 Aug 29 Why IEs? Intro Functional Analysis Slides Video Echo360 Code
3 Sep 3 Intro Functional Analysis, Intro IEs Slides Video Echo 360
4 Sep 5 Intro IEs, Neumann series, Compact op. Slides Video Echo 360
5 Sep 10 HW1 discussion, Compact op. Slides Video Echo 360
6 Sep 12 Compactness of integral operators Slides Video Echo 360 (sound dies half-way)
7 Sep 17 Weakly singular i.op., Riesz theory Slides Video Echo 360
8 Sep 19 Riesz theory, Hilbert spaces Slides Video Echo 360 Notes
9 Sep 24 Fredholm theory, spectral theory Slides Video Echo 360
10 Sep 26 Spectral theory, potential theory Slides Video Echo 360
11 Oct 1 PV integrals, Green's thm., formula Slides Video Echo 360
12 Oct 3 Jump conditions, ext. domains Slides Video Echo 360 Notes
13 Oct 8 BVPs Slides Video Echo 360
14 Oct 10 BVPs, Uniqueness Slides Video Echo 360
15 Oct 15 Uniqueness, corners Slides Video Echo 360
16 Oct 16 Intro Helmholtz Slides Video Echo 360
17 Oct 22 Helmholtz BVP/IE uniqueness Slides Video Echo 360 Notes
18 Oct 24 Calderón, Intro numerics Slides Video Echo 360 (no sound)
19 Oct 29 HW5, high-order numerics Slides Video Echo 360 Code
20 Oct 31 High-order numerics, IE discretizations Slides Video Echo 360 Code Notes
21 Nov 5 Interior Neumann, Nyström Slides Video Echo 360 Notes
22 Nov 7 Collective compactness, Anselone's thm Slides Video Echo 360
23 Nov 12 Anselone's thm, Céa's lemma Slides Video Echo 360 Notes
ECE590 Nov 12 QBX quadrature Slides Paper
24 Nov 14 Projection error est., Intro quadrature Slides Video Echo 360 Code
25 Nov 19 Singular quadrature, Intro fast alg. Slides Video Echo 360 Code
26 Nov 21 Fast Multipole Methods Slides Video Echo 360 Code
Nov 26 Thanksgiving break
Nov 28
Dec 3 Project presentations
Dec 5
Dec 10 No class

You'll need an up-to-date 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 warm-up
• 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 algebra-based 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?
• 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 $Ax=b$.

August 7, 2013 : Class starts on August 27, 2012, from 2-3:15pm. We've also been assigned a room. We will be meeting in 1304 Siebel. See you then!

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 Google Books UIUC library
Integral equation methods in scattering theory by Colton and Kress Google Books UIUC library ebook
Partial Differential Equations of Mathematical Physics and Integral Equations by Guenther and Lee Google Books UIUC library
Partial Differential Equations: An Introduction by Colton Google Books UIUC library (not available)
Integral Equations by Hackbusch Google Books UIUC library
Foundations of Potential Theory by Kellogg Google Books UIUC library ebook (public)
Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems by Nédélec Google Books UIUC library
Integral Equations by Tricomi Google Books UIUC library ebook (public)
Linear Operator Theory in Engineering and Science by Naylor and Sell Google Books UIUC library

(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

[[!table header="no" class="mointable" data=""" The fast solution of boundary integral equations by Rjasanow | ebook Linear Integral Equations by Kanwal | ebook Linear and Nonlinear Integral Equations by Wazwaz | ebook """]]

### 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

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.

### Virtual Machine Image

This information has moved to ComputeVirtualMachineImages.