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= Integral Equations and Fast Methods =  = Integral Equations and Fast Algorithms (CS 598AK @ UIUC) = 
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 ''Class Time/Location''  Wednesday 5:107pm, Room 101 Warren Weaver Hall   ''Instructor''  [[http://www.cims.nyu.edu/~kloecknerAndreas Kloeckner]]   ''Email''  kloeckner@cims.nyu.edu   ''Office''  TBD   ''Office Hours''  TBD   ''Class Webpage''  http://bit.ly/ieq13   ''Email Listserv''  [[http://lists.tiker.net/listinfo/ieq13Info page]]  
 ''Class Time/Location''  [[ https://courses.illinois.edu/cisapp/dispatcher/schedule/2013/fall/CS/598Tuesday/Thursday 2:00pm3:15pm]] / 1304 [[http://cs.illinois.edu/aboutus/directionssiebelcenterSiebel]]   ''Instructor''  [[http://www.cs.illinois.edu/~andreaskAndreas Kloeckner]]   ''Email''  andreask@illinois.edu   ''Office''  Rm. 4318 [[http://cs.illinois.edu/aboutus/directionssiebelcenterSiebel]]   ''Office Hours''  by appointment   ''Class Webpage''  http://bit.ly/inteq13   ''Email Listserv''  [[http://lists.tiker.net/listinfo/inteq13Info page]]  
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[[attachment:flyer.pdfClass advertisement]]  ''Email Listserv''  [[http://lists.tiker.net/listinfo/ieq13Info page]]  [[attachment:flyer.pdfClass advertisement]] 
[[attachment:flyer.pdfClass poster]] 
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== Lecture Material ==  '''Lecture #'''  '''Date'''  '''Topics'''  '''Slides'''  '''Primary Video'''  '''UIUC Video'''  '''Code'''  '''Extra Info'''   1  Aug 27  Intro  [[attachment:lec1.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec01.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/f2a6ad8ee0114e78bd2e2f8b0a3e1763Echo360]] (no sound)  [[https://github.com/inteq13/lec1demoCode]]    2  Aug 29  Why IEs? Intro Functional Analysis  [[attachment:lec2.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec02.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/61f5224a02014146b8e9e2d2085cbf77Echo360]]  [[https://github.com/inteq13/lec2demoCode]]    3  Sep 3  Intro Functional Analysis, Intro IEs  [[attachment:lec3.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec03.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/a30797d07f6740e4a05bd21f842fe765Echo 360]]     4  Sep 5  Intro IEs, Neumann series, Compact op.  [[attachment:lec4.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec04.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/435e0e15d9aa41f2bb8947dd6e9958dbEcho 360]]     5  Sep 10  HW1 discussion, Compact op.  [[attachment:lec5.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec05.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/b069c3e6635e4c739deafcac424daa01Echo 360]]     6  Sep 12  Compactness of integral operators  [[attachment:lec6.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec06.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/90e9023a7ed641d7b933005325985eb2Echo 360]] (sound dies halfway)     7  Sep 17  Weakly singular i.op., Riesz theory  [[attachment:lec7.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec07.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/666a606ab2eb4b3284fa6c54ab5294f0Echo 360]]     8  Sep 19  Riesz theory, Hilbert spaces  [[attachment:lec8.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec08.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/9a822a1a709f4f808e39359b81ede178Echo 360]]   [[attachment:lec8estimate.pdfNotes]]   9  Sep 24  Fredholm theory, spectral theory  [[attachment:lec9.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec09.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/6410428218e441f4959ce87e6f10b0ceEcho 360]]     10  Sep 26  Spectral theory, potential theory  [[attachment:lec10.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec10.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/505dae512eaa4e18a31f4fe38ce5b70aEcho 360]]     11  Oct 1 PV integrals, Green's thm., formula  [[attachment:lec11.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec11.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/85cf66b6b1104007b02b685e9add8118Echo 360]]     12  Oct 3  Jump conditions, ext. domains  [[attachment:lec12.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec12.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/3b21656e06004a14bd497980c0ce8e9bEcho 360]]   [[attachment:djumpconstant.pdfNotes]]   13  Oct 8  BVPs  [[attachment:lec13.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec13.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/fda640295c604b3b9260e4f0c3e9f390Echo 360]]     14  Oct 10  BVPs, Uniqueness  [[attachment:lec14.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec14.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/82a617c9f1874b2ea7dbf7df246eba9cEcho 360]]     15  Oct 15  Uniqueness, corners  [[attachment:lec15.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec15.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/f55ba3d4002a4adb8391ff711f50c4bbEcho 360]]     16  Oct 16  Intro Helmholtz  [[attachment:lec16.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec16.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/7d87067e68f342cb94c4e176d5fe0960 Echo 360]]     17  Oct 22  Helmholtz BVP/IE uniqueness  [[attachment:lec17.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec17.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/80d13a3142de479b9d59381e309dc22cEcho 360]]   [[attachment:brakhagewerner.pdfNotes]]   18  Oct 24  Calderón, Intro numerics  [[attachment:lec18.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec18.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/babc0f9e55334826a6e0b83d8b685befEcho 360]] (no sound)     19  Oct 29  HW5, highorder numerics  [[attachment:lec19.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec19.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/08d25fb6cbf848168f5f186c0aa39b6bEcho 360]]  [[https://github.com/inteq13/lec19demoCode]]    20  Oct 31  Highorder numerics, IE discretizations  [[attachment:lec20.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec20.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/0b3a2c70e3f24d268d5844bdd6566860Echo 360]]  [[https://github.com/inteq13/lec19demoCode]]  [[attachment:vtnotes.pdfNotes]]   21  Nov 5  Interior Neumann, Nyström  [[attachment:lec21.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec21.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/a41825853887486aaa99a67d1c345519Echo 360]]   [[attachment:interiorneumann.pdfNotes]]   22  Nov 7  Collective compactness, Anselone's thm  [[attachment:lec22.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec22.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/befcf9a3bef2479dafdcc5a9e4310667Echo 360]]     23  Nov 12  Anselone's thm, Céa's lemma  [[attachment:lec23.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec23.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/fdf675b243be450988cfd3e89bc330fbEcho 360]]   [[attachment:lec23notes.pdfNotes]]   ECE590  Nov 12  QBX quadrature  [[attachment:ece590.pdfSlides]]     [[http://arxiv.org/abs/1207.4461Paper]]   24  Nov 14  Projection error est., Intro quadrature  [[attachment:lec24.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec24.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/90705dc78d5d4e1cb7b89f2b8d5d8f59Echo 360]]  [[https://github.com/inteq13/lec24demoCode]]    25  Nov 19  Singular quadrature, Intro fast alg.  [[attachment:lec25.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec25.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/10bb843c66f743a1b32b9a6f9a612738Echo 360]]  [[https://github.com/inteq13/lec26demoCode]]    26  Nov 21  Fast Multipole Methods  [[attachment:lec26.pdfSlides]]  [[http://tiker.net/inteq13/html/player.html?descriptor=metadata/lec26.jsonVideo]]  [[http://recordings.engineering.illinois.edu/ess/echo/presentation/2a826fe84a2244948c0508217b69c68eEcho 360]]  [[https://github.com/inteq13/lec26demoCode]]   <2>  Nov 26 <2> Thanksgiving break       Nov 28       <2>  Dec 3 <2> Project presentations        Dec 5         Dec 10  No class       You'll need an uptodate version of [[http://google.com/chromeGoogle 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=795686investigating]]. 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 (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 $Ax=b$. 

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Month 1, 2013:: Class starts on XXXX 5, 2012, from XXXXpm. 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 23:15pm. 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 (40% of your grade) (also see [[/ProjectSubmissionGuidelines]]) 
* 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.pdfHomework 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.pdfHomework 2]] due: September (17) 19, 2013  out: September 6, 2013 * [[attachment:hw3.pdfHomework 3]] due: October (3) 4, 2013  out: September 19, 2013 * [[attachment:hw4.pdfHomework 4]] due: October 17, 2013  out: October 4, 2013 * [[attachment:hw5.pdfHomework 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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*  These books cover some of our mathematical needs:  ''Linear Integral Equations'' by Kress  [[http://books.google.com/books?id=R3BIOfKssQ4CGoogle Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=0387987002UIUC library]]  [[http://link.springer.com/book/10.1007/9781461205593/page/1ebook]] (not free at UIUC)  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=047186420XGoogle Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=047186420XUIUC library]]  [[http://link.springer.com/book/10.1007/9781461449423/page/1ebook]]  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=0486688895Google Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=0486688895UIUC library]]   Good for potential theory, little FA, no numerics, oldish   ''Partial Differential Equations: An Introduction'' by Colton  [[http://books.google.com/books?isbn=0486138437Google Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=0486138437UIUC library]] (not available)    ''Integral Equations'' by Hackbusch  [[http://books.google.com/books?isbn=3764328711Google Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=3764328711UIUC library]]   comprehensive, but different emphasis than our class   ''Foundations of Potential Theory'' by Kellogg  [[http://books.google.com/books?isbn=0486601447Google Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Record/uiu_136521UIUC library]]  [[http://hdl.handle.net/2027/mdp.39015000990989ebook]] (public)  a potential theory book, little FA, no numerics, oldish   ''Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems'' by Nédélec  [[http://books.google.com/books?isbn=9780387951553Google Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=9780387951553UIUC library]]     ''Integral Equations'' by Tricomi  [[http://books.google.com/books?id=FYs8ua1X6xICGoogle Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Search/Home?lookfor=0486648281UIUC library]]  [[http://hdl.handle.net/2027/mdp.39015000996259ebook]] (public)  slightly oldfashioned IE theory book, little numerics, no FA   ''Linear Operator Theory in Engineering and Science'' by Naylor and Sell  [[http://books.google.com/books?id=t3SXs4KrE0CGoogle Books]]  [[http://vufind.carli.illinois.edu/vfuiu/Record/uiu_58667UIUC 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  [[http://vufind.carli.illinois.edu/vfuiu/Record/uiu_5346726ebook]]   ''Linear Integral Equations'' by Kanwal  [[http://vufind.carli.illinois.edu/vfuiu/Record/uiu_7044974ebook]]   ''Linear and Nonlinear Integral Equations'' by Wazwaz  [[http://vufind.carli.illinois.edu/vfuiu/Record/uiu_6696792ebook]]  === 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 ==== * [[http://dx.doi.org/10.1137/0912004The fast Gauss Transform]] by Greengard and Strain * Highfrequency FMMs: [[http://scholar.google.com/scholar?cluster=9876726404460117961&hl=en&as_sdt=0,14&sciodt=0,141]] [[http://dx.doi.org/10.1016/j.jcp.2005.12.0012]] * Distributedmemory parallelization of FMMs: (still looking for a good intro article, possibly [[http://dx.doi.org/10.1177/1094342011429952A 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/S003614450343200XAccelerating the Nonuniform Fast Fourier Transform]] by Greengard and Lee * [[http://dx.doi.org/10.1016/j.acha.2009.08.005An algorithm for the rapid evaluation of special function transforms]] by O'Neil et al. * [[http://dx.doi.org/10.1155/2013/938167Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial]] by Johan Helsing * [[http://dx.doi.org/10.1016/j.jcp.2003.08.011A 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.2004.10.033A 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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* [Shidong Jiang](http://web.njit.edu/~jiang/math707.html) * [Alex Barnett](http://www.math.dartmouth.edu/~m126w12/) * [Shravan Veerapaneni](https://sites.google.com/a/umich.edu/math671fastalgorithms/) * Leslie Greengard * [Gunnar Martinsson](http://amath.colorado.edu/faculty/martinss/main_teaching.html) * [Jianlin Xia](http://www.math.purdue.edu/~xiaj/teaching/692.11s/) * [Francesco Andriulli](http://public.telecombretagne.eu/~fandriul/teaching.html) 
* [[http://web.njit.edu/~jiang/math707.htmlShidong Jiang]] (NJIT) * [[http://www.math.dartmouth.edu/~m126w12/Alex Barnett]] (Dartmouth) * [[https://sites.google.com/a/umich.edu/math671fastalgorithmsShravan Veerapaneni]] (Michigan) * [[https://cs.nyu.edu/courses/spring12/CSCIGA.2945001/index.htmlLeslie Greengard]] (NYU) * [[http://amath.colorado.edu/faculty/martinss/Teaching/APPM5720_2011s/index.htmlGunnar Martinsson]] (UC Boulder) * [[http://www.math.purdue.edu/~xiaj/teaching/692.11s/Jianlin Xia]] (Purdue) * [[http://public.telecombretagne.eu/~fandriul/teaching.htmlFrancesco Andriulli]] (ENS TELECOM Bretagne) * [[http://cims.nyu.edu/%7Etygert/gradcourse/survey.pdfMark Tygert]] (NYU, now Yale) === Online resources === ==== Math ==== * The [[http://dlmf.nist.govDigital Library of Mathematical Functions]] * [[http://people.math.sfu.ca/~cbm/aands/Abramowitz and Stegun: Handbook of Mathematical Functions]] ==== Python ==== * [[http://scipylectures.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.pdfThe 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. 
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 
ebook (not free at UIUC) 
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
This information has moved to ComputeVirtualMachineImages.