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

 Class Time/Location Instructor Email 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 half-way) 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, high-order numerics 20 Oct 31 High-order 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 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)

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 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, old-ish 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, old-ish Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems by Nédélec Integral Equations by Tricomi ebook (public) slightly old-fashioned 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

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.

Virtual Machine Image

This information has moved to ComputeVirtualMachineImages.

Teaching/IntegralEquationsFall2013 (last edited 2014-01-23 02:26:25 by AndreasKloeckner)