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#format mathjax
#acl All:read
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= Integral Equations and Fast Methods = = Integral Equations and Fast Algorithms (CS 598AK @ UIUC) =
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|| ''Class Time/Location'' || TBD ||
|| ''Instructor'' || [[http://www.cims.nyu.edu/~kloeckner|Andreas Kloeckner]] ||
|| ''Email'' || kloeckner@cims.nyu.edu ||
|| ''Office'' || TBD ||
|| ''Office Hours'' || TBD ||
|| ''Class Time/Location'' || [[ https://courses.illinois.edu/cisapp/dispatcher/schedule/2013/fall/CS/598|Tuesday/Thursday 2:00pm-3:15pm]] / 1304 [[http://cs.illinois.edu/about-us/directions-siebel-center|Siebel]] ||
|| ''Instructor'' || [[http://www.cs.illinois.edu/~andreask|Andreas Kloeckner]] ||
|| ''Email'' || andreask@illinois.edu ||
|| ''Office'' || Rm. 4318 [[http://cs.illinois.edu/about-us/directions-siebel-center|Siebel]] ||
|| ''Office Hours'' || by appointment ||
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== Lecture Material ==

|| '''Lecture #''' || '''Date''' || '''Topics''' || '''Slides''' || '''Primary Video''' || '''UIUC Video''' || '''Code''' || '''Extra Info''' ||
|| 1 || Aug 27 || Intro || [[attachment:lec1.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec01.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/f2a6ad8e-e011-4e78-bd2e-2f8b0a3e1763|Echo360]] (no sound) || [[https://github.com/inteq13/lec1-demo|Code]] || ||
|| 2 || Aug 29 || Why IEs? Intro Functional Analysis || [[attachment:lec2.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec02.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/61f5224a-0201-4146-b8e9-e2d2085cbf77|Echo360]] || [[https://github.com/inteq13/lec2-demo|Code]] || ||
|| 3 || Sep 3 || Intro Functional Analysis, Intro IEs || [[attachment:lec3.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec03.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/a30797d0-7f67-40e4-a05b-d21f842fe765|Echo 360]] || || ||
|| 4 || Sep 5 || Intro IEs, Neumann series, Compact op. || [[attachment:lec4.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec04.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/435e0e15-d9aa-41f2-bb89-47dd6e9958db|Echo 360]] || || ||
|| 5 || Sep 10 || HW1 discussion, Compact op. || [[attachment:lec5.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec05.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/b069c3e6-635e-4c73-9dea-fcac424daa01|Echo 360]] || || ||
|| 6 || Sep 12 || Compactness of integral operators || [[attachment:lec6.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec06.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/90e9023a-7ed6-41d7-b933-005325985eb2|Echo 360]] (sound dies half-way) || || ||
|| 7 || Sep 17 || Weakly singular i.op., Riesz theory || [[attachment:lec7.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec07.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/666a606a-b2eb-4b32-84fa-6c54ab5294f0|Echo 360]] || || ||
|| 8 || Sep 19 || Riesz theory, Hilbert spaces || [[attachment:lec8.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec08.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/9a822a1a-709f-4f80-8e39-359b81ede178|Echo 360]] || || [[attachment:lec8-estimate.pdf|Notes]] ||
|| 9 || Sep 24 || Fredholm theory, spectral theory || [[attachment:lec9.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec09.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/64104282-18e4-41f4-959c-e87e6f10b0ce|Echo 360]] || || ||
|| 10 || Sep 26 || Spectral theory, potential theory || [[attachment:lec10.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec10.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/505dae51-2eaa-4e18-a31f-4fe38ce5b70a|Echo 360]] || || ||
|| 11 || Oct 1|| PV integrals, Green's thm., formula || [[attachment:lec11.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec11.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/85cf66b6-b110-4007-b02b-685e9add8118|Echo 360]] || || ||
|| 12 || Oct 3 || Jump conditions, ext. domains || [[attachment:lec12.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec12.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/3b21656e-0600-4a14-bd49-7980c0ce8e9b|Echo 360]] || || [[attachment:d-jump-constant.pdf|Notes]] ||
|| 13 || Oct 8 || BVPs || [[attachment:lec13.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec13.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/fda64029-5c60-4b3b-9260-e4f0c3e9f390|Echo 360]] || || ||
|| 14 || Oct 10 || BVPs, Uniqueness || [[attachment:lec14.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec14.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/82a617c9-f187-4b2e-a7db-f7df246eba9c|Echo 360]] || || ||
|| 15 || Oct 15 || Uniqueness, corners || [[attachment:lec15.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec15.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/f55ba3d4-002a-4adb-8391-ff711f50c4bb|Echo 360]] || || ||
|| 16 || Oct 16 || Intro Helmholtz || [[attachment:lec16.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec16.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/7d87067e-68f3-42cb-94c4-e176d5fe0960 |Echo 360]] || || ||
|| 17 || Oct 22 || Helmholtz BVP/IE uniqueness || [[attachment:lec17.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec17.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/80d13a31-42de-479b-9d59-381e309dc22c|Echo 360]] || || [[attachment:brakhage-werner.pdf|Notes]] ||
|| 18 || Oct 24 || Calderón, Intro numerics || [[attachment:lec18.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec18.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/babc0f9e-5533-4826-a6e0-b83d8b685bef|Echo 360]] (no sound) || || ||
|| 19 || Oct 29 || HW5, high-order numerics || [[attachment:lec19.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec19.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/08d25fb6-cbf8-4816-8f5f-186c0aa39b6b|Echo 360]] || [[https://github.com/inteq13/lec19-demo|Code]] || ||
|| 20 || Oct 31 || High-order numerics, IE discretizations || [[attachment:lec20.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec20.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/0b3a2c70-e3f2-4d26-8d58-44bdd6566860|Echo 360]] || [[https://github.com/inteq13/lec19-demo|Code]] || [[attachment:vt-notes.pdf|Notes]] ||
|| 21 || Nov 5 || Interior Neumann, Nyström || [[attachment:lec21.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec21.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/a4182585-3887-486a-aa99-a67d1c345519|Echo 360]] || || [[attachment:interior-neumann.pdf|Notes]] ||
|| 22 || Nov 7 || Collective compactness, Anselone's thm || [[attachment:lec22.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec22.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/befcf9a3-bef2-479d-afdc-c5a9e4310667|Echo 360]] || || ||
|| 23 || Nov 12 || Anselone's thm, Céa's lemma || [[attachment:lec23.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec23.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/fdf675b2-43be-4509-88cf-d3e89bc330fb|Echo 360]] || || [[attachment:lec23-notes.pdf|Notes]] ||
|| ECE590 || Nov 12 || QBX quadrature || [[attachment:ece590.pdf|Slides]] || || || || [[http://arxiv.org/abs/1207.4461|Paper]] ||
|| 24 || Nov 14 || Projection error est., Intro quadrature || [[attachment:lec24.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec24.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/90705dc7-8d5d-4e1c-b7b8-9f2b8d5d8f59|Echo 360]] || [[https://github.com/inteq13/lec24-demo|Code]] || ||
|| 25 || Nov 19 || Singular quadrature, Intro fast alg. || [[attachment:lec25.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec25.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/10bb843c-66f7-43a1-b32b-9a6f9a612738|Echo 360]] || [[https://github.com/inteq13/lec26-demo|Code]] || ||
|| 26 || Nov 21 || Fast Multipole Methods || [[attachment:lec26.pdf|Slides]] || [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec26.json|Video]] || [[http://recordings.engineering.illinois.edu/ess/echo/presentation/2a826fe8-4a22-4494-8c05-08217b69c68e|Echo 360]] || [[https://github.com/inteq13/lec26-demo|Code]] || ||
||<|2> || Nov 26 ||<|2> Thanksgiving break || || || || || ||
||Nov 28 || || || || || ||
||<|2> || Dec 3 ||<|2> Project presentations || || || || || ||
|| Dec 5 || || || || || ||
|| || Dec 10 || No class || || || || || ||
 
You'll need an up-to-date version of [[http://google.com/chrome|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.

{{{#!wiki note
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 [[https://bugzilla.mozilla.org/show_bug.cgi?id=795686|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.
}}}
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This class will teach you how how (and why!) integral equations let you solve many common types of partial differential equations robustly and quickly. This class will teach you how (and why!) integral equations let you solve many common types of partial differential equations robustly and quickly.
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 * A Gentle Intro: Calculus/Linear Algebra/Python warm-up  * A Gentle Intro: Linear Algebra/Numerics/Python warm-up
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 * Fast Randomized Linear Algebra (if time)  * 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?
 * 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 $Ax=b$.
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 Month 1, 2013:: Class starts on MONTH DAY, 2012, from XX-XXpm. We've also been assigned a room. We will be meeting in XXX. See you then!  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!
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{{{#!wiki note
This isn't final yet.
}}}
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 * Weekly homework for (roughly) the first half of the class (50% of your grade)
 * A more ambitious final project, which may be inspired by your own research needs (50% of your grade) (also see [[/ProjectGuidelines]])
 * 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)
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 * [[attachment:hw1.pdf|Homework 1]] due: September 5, 2013 - out: August 27, 2013 (minor update to problem 1b on Sep 2, some notation fixes on Sep 3)
 * [[attachment:hw2.pdf|Homework 2]] due: September --(17)-- 19, 2013 - out: September 6, 2013
 * [[attachment:hw3.pdf|Homework 3]] due: October --(3)-- 4, 2013 - out: September 19, 2013
 * [[attachment:hw4.pdf|Homework 4]] due: October 17, 2013 - out: October 4, 2013
 * [[attachment:hw5.pdf|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.
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|| ''Linear Integral Equations'' by Kress || [[http://books.google.com/books?id=R3BIOfKssQ4C|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_7387527|UIUC library]] || || probably the book with the best overall coverage, but little on numerics and algorithms ||
|| ''Integral equation methods in scattering theory'' by Colton and Kress || [[http://books.google.com/books?isbn=047186420X|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=047186420X|UIUC library]] || [[http://link.springer.com/book/10.1007/978-1-4614-4942-3/page/1|ebook]] || a more advanced book, builds on Kress LIE ||
|| ''Partial Differential Equations of Mathematical Physics and Integral Equations'' by Guenther and Lee || [[http://books.google.com/books?isbn=0486688895|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=0486688895|UIUC library]] || || Good for potential theory, little FA, no numerics, old-ish ||
|| ''Partial Differential Equations: An Introduction'' by Colton || [[http://books.google.com/books?isbn=0486138437|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=0486138437|UIUC library]] (not available) || ||
|| ''Integral Equations'' by Hackbusch || [[http://books.google.com/books?isbn=3764328711|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=3764328711|UIUC library]] || || comprehensive, but different emphasis than our class ||
|| ''Foundations of Potential Theory'' by Kellogg || [[http://books.google.com/books?isbn=0486601447|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_136521|UIUC library]] || [[http://hdl.handle.net/2027/mdp.39015000990989|ebook]] (public) || a potential theory book, little FA, no numerics, old-ish ||
|| ''Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems'' by Nédélec || [[http://books.google.com/books?isbn=9780387951553|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=9780387951553|UIUC library]] || || ||
|| ''Integral Equations'' by Tricomi || [[http://books.google.com/books?id=FYs8ua1X6xIC|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Search/Home?lookfor=0486648281|UIUC library]] || [[http://hdl.handle.net/2027/mdp.39015000996259|ebook]] (public) || slightly old-fashioned IE theory book, little numerics, no FA ||
|| ''Linear Operator Theory in Engineering and Science'' by Naylor and Sell || [[http://books.google.com/books?id=t3SXs4-KrE0C|Google Books]] || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_58667|UIUC library]] || || good for FA, uses int.eq. examples (recommended by Steven Dalton) ||
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 * [[http://books.google.com/books?isbn=0486601447|Foundations of Potential Theory]] by Kellogg
 * [[http://books.google.com/books?isbn=9780387951553|Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems]] by Nédélec
 * [[http://books.google.com/books?isbn=3764328711|Integral Equations]] by Hackbusch
(FA=functional analysis)
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==== UIUC ebooks ====

|| ''The fast solution of boundary integral equations'' by Rjasanow || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_5346726|ebook]] ||
|| ''Linear Integral Equations'' by Kanwal || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_7044974|ebook]] ||
|| ''Linear and Nonlinear Integral Equations'' by Wazwaz || [[http://vufind.carli.illinois.edu/vf-uiu/Record/uiu_6696792|ebook]] ||
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==== Some possible articles for a final project ====

 * [[http://dx.doi.org/10.1137/0912004|The fast Gauss Transform]] by Greengard and Strain
 * High-frequency FMMs: [[http://scholar.google.com/scholar?cluster=9876726404460117961&hl=en&as_sdt=0,14&sciodt=0,14|1]] [[http://dx.doi.org/10.1016/j.jcp.2005.12.001|2]]
 * Distributed-memory parallelization of FMMs: (still looking for a good intro article, possibly [[http://dx.doi.org/10.1177/1094342011429952|A tuned and scalable fast multipole method as a preeminent algorithm for exascale systems]] by Yokota and Barba and references therein)
 * [[http://dx.doi.org/10.1137/S003614450343200X|Accelerating the Nonuniform Fast Fourier Transform]] by Greengard and Lee
 * [[http://dx.doi.org/10.1016/j.acha.2009.08.005|An algorithm for the rapid evaluation of special function transforms]] by O'Neil et al.
 * [[http://dx.doi.org/10.1155/2013/938167|Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial]] by Johan Helsing
 * [[http://dx.doi.org/10.1016/j.jcp.2003.08.011|A fast solver for the Stokes equations with distributed forces in complex geometries]] by Biros, Ying, and Zorin
 * [[http://dx.doi.org/10.1016/j.jcp.2004.10.033|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.
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 * [[http://web.njit.edu/~jiang/math707.html|Shidong Jiang]]
 * [[http://www.math.dartmouth.edu/~m126w12/|Alex Barnett]]
 * [[https://sites.google.com/a/umich.edu/math-671-fast-algorithms|Shravan Veerapaneni]]
 * Leslie Greengard
 * [[http://amath.colorado.edu/faculty/martinss/Teaching/APPM5720_2011s/index.html|Gunnar Martinsson]]
 * [[http://www.math.purdue.edu/~xiaj/teaching/692.11s/|Jianlin Xia]]
 * [[http://public.telecom-bretagne.eu/~fandriul/teaching.html|Francesco Andriulli]]
 * [[http://cims.nyu.edu/%7Etygert/gradcourse/survey.pdf|Mark Tygert]]
 * [[http://web.njit.edu/~jiang/math707.html|Shidong Jiang]] (NJIT)
 * [[http://www.math.dartmouth.edu/~m126w12/|Alex Barnett]] (Dartmouth)
 * [[https://sites.google.com/a/umich.edu/math-671-fast-algorithms|Shravan Veerapaneni]] (Michigan)
 * [[https://cs.nyu.edu/courses/spring12/CSCI-GA.2945-001/index.html|Leslie Greengard]] (NYU)
 * [[http://amath.colorado.edu/faculty/martinss/Teaching/APPM5720_2011s/index.html|Gunnar Martinsson]] (UC Boulder)
 * [[http://www.math.purdue.edu/~xiaj/teaching/692.11s/|Jianlin Xia]] (Purdue)
 * [[http://public.telecom-bretagne.eu/~fandriul/teaching.html|Francesco Andriulli]] (ENS TELECOM Bretagne)
 * [[http://cims.nyu.edu/%7Etygert/gradcourse/survey.pdf|Mark Tygert]] (NYU, now Yale)
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==== Math ====
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==== Python ====

 * [[http://scipy-lectures.github.io/|The SciPy lectures]]
 * [[http://mentat.za.net/numpy/numpy_advanced_slides/|The Numpy MedKit]] by Stéfan van der Walt
 * [[http://web.mit.edu/dvp/Public/numpybook.pdf|The Numpy User Guide]] by Travis Oliphant
 * [[http://docs.scipy.org/doc/|Numpy/Scipy documentation]]
 * [[http://www.reddit.com/r/Python/comments/1lgxbf/best_tutorial_to_learn_numpy/|More in this reddit thread]]
 * [[https://code.google.com/p/spyderlib/|Spyder]] (a Python IDE, like Matlab) is installed in the virtual machine. (Applications Menu > Development > Spyder)

=== Virtual Machine Image ===

This information has moved to ComputeVirtualMachineImages.

Flyer thumbnail

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

Class Time/Location

Tuesday/Thursday 2:00pm-3:15pm / 1304 Siebel

Instructor

Andreas Kloeckner

Email

andreask@illinois.edu

Office

Rm. 4318 Siebel

Office Hours

by appointment

Class Webpage

http://bit.ly/inteq13

Email Listserv

Info page

Class poster

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?
  • 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 2-3:15pm. 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

Google Books

UIUC library

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

Integral equation methods in scattering theory by Colton and Kress

Google Books

UIUC library

ebook

a more advanced book, builds on Kress LIE

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

Google Books

UIUC library

Good for potential theory, little FA, no numerics, old-ish

Partial Differential Equations: An Introduction by Colton

Google Books

UIUC library (not available)

Integral Equations by Hackbusch

Google Books

UIUC library

comprehensive, but different emphasis than our class

Foundations of Potential Theory by Kellogg

Google Books

UIUC library

ebook (public)

a potential theory book, little FA, no numerics, old-ish

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)

slightly old-fashioned IE theory book, little numerics, no FA

Linear Operator Theory in Engineering and Science by Naylor and Sell

Google Books

UIUC library

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

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

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.

Online resources

Math

Python

Virtual Machine Image

This information has moved to ComputeVirtualMachineImages.

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