# Fast Algorithms and Integral Equation Methods (CS 598APK) Fall 2019

What Where
Time/place Wed/Fri 2:00pm-3:15pm 1109 Siebel / Catalog
Class URL https://bit.ly/fastalg-f19
Class recordings (also on Echo 360)
Piazza
Calendar

## Final Project Presentations

Date Slot Presenter
Dec 4 1 Guanhua
Dec 4 2 Yashraj
Dec 6 1 Hongliang
Dec 6 2 Vedant
Dec 6 3 Yiming
Dec 11 1 Jonathan
Dec 11 2 Yuchen
• Presentations will take place after Fall Break during the three remaining class periods.
• Presentations will be 22 minutes in length, with three minutes for questions at the end.
• Class attendance during final project presentation is required.

## Why you should take this class

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.

## Instructor

Andreas Kloeckner

(Instructor)

Office: 4318 Siebel

## Course Outline

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

#### Class scribbles (click to view)

CAUTION!

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.

## Computing

We will be using Python with the libraries numpy, scipy and matplotlib for 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.

### Virtual Machine Image

While you are free to install Python and Numpy on your own computer to do homework, the only supported way to do so is using the supplied virtual machine image.

## Books and Papers

### Fast Multipole

Further references:

### Integral Equations/Functional Analysis

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

A third edition is also available.

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

### Background: Numerical Linear Algebra

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: