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Fast Algorithms and Integral Equation Methods (CS 598APK) Fall 2024

What Where
Time/place Tue/Thu 11:00am-12:15pm 1302 Siebel / Catalog
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Project Presentations

Tuesday May 7

Thursday May 9

Why you should take this class

The numerical solution of a 3D scattering problem

A quadtree as used in a Fast Multipole Method

Many of the algorithms of introductory scientific computing have super-linear runtime scaling. Gaussian elimination or LU decomposition are good examples, their runtime scales as $O(n^3)$ with the number of unknowns $n$. Even simple matrix-vector multiplication exhibits quadratic scaling. Problems in scientific computing, especially those arising from science and engineering questions, often demand large-scale computation in order to achieve acceptable fidelity, and such computations will not tolerate super-linear, let alone quadratic or cubic scaling.

This class will teach you a set of techniques and ideas that successfully reduce this asymptotic cost for an important set of operations. This leads to these techniques being called "fast algorithms". We will begin by examining some of these ideas from a linear-algebraic perspective, where for matrices with special structure, large cost gains can be achieved. We will then specialize to PDE boundary value problems, which give rise to many of the largest-scale computations. We will see that integral equations are the natural generalization of the linear-algebraic tools encountered earlier, and we will understand the mathematical and algorithmic foundations that make them powerful tools for computation. All throughout, we will pay much attention to the idea of representation–i.e. the choice of what the numerical unknowns of the system to be solved should be.


Andreas Kloeckner

Andreas Kloeckner



Office: 4318 Siebel

Course Outline

Note: the section headings in this tree are clickable to reveal more detail.


These scribbled PDFs are an unedited reflection of what we wrote during class. They need to be viewed in the context of the class discussion that led to them. See the lecture videos for that.

If you would like actual, self-contained class notes, look in the outline above.

These scribbles are provided here to provide a record of our class discussion, to be used in perhaps the following ways:

  • as a way to cross-check your own notes
  • to look up a formula that you know was shown in a certain class
  • to remind yourself of what exactly was covered on a given day

By continuing to read them, you acknowledge that these files are provided as supplementary material on an as-is basis.


We will be using Python with the libraries numpy, scipy and matplotlib for in-class work and assignments. No other languages are permitted. Python has a very gentle learning curve, so you should feel at home even if you've never done any work in Python.

Running Code on your Own Computer

While running code in this online system should technically suffice to do your work for this class, you may find it useful to also install Python on your own computer.

The recommended way of doing so involves downloading the Anaconda Python distribution. Note that this is a commercial product (even if it is free of charge), and this is not intended as an endorsement of the company or the product. Note that we cannot promise to provide technical support for this installation.

Download Anaconda Python »

Another way to run Python code is through an online JupyterLab available through the course. Go to get started. NOTE that this environment runs entirely in your browser. If you clear your browser data, any work 'saved' there will be irretrievably lost.

Books and Papers

Randomized Linear Algebra

Fast Multipole

Further references:

Integral Equations/Functional Analysis

Linear Integral Equations
Rainer Kress, Linear integral equations. (second edition) The references in the notes are for the second edition.

A third edition is also available.

Inverse Acoustic and Electromagnetic Scattering Theory
David Colton and Rainer Kress, Inverse Acoustic and Electromagnetic Scattering Theory. (3rd edition)

Background: Numerical Linear Algebra

Scientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey / E-Book (accessible free of charge from campus network/VPN)

Michael T. Heath, Revised Second Edition, Society for Industrial and Applied Mathematics

Further references:

Previous editions of this class

Many of these have videos available.

Related Classes Elsewhere

Python Help

Numpy Help

Grading Policies

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