# PhD Course

## Topics in Modern Applied Mathematics

Web page updated May 30, 2005 The course plan was updated. Two sets of notes have been added to the Course files. m-files concerning pseudospectra have been added.

### Course files

The course files can be found here.

### Course plan

The preliminary course plan is here.

#### Note

In order to do the exercises in the course you need to have access to a computer with Matlab installed. Preferably a laptop that you bring with you to the lectures.

I use the computer algebra system Maple to illustrate various things in the course. If you wish to experiment with my files, you need to have Maple installed. You can get a copy of Maple here. Note that Aalborg University has a site license for Maple. Follow the instructions concerning installation give in the link above, in particular with respect to the license file.

## Course description

The official description of the course etc. can be found here.

A more precise description follows below.

### Topics to be covered

Pseudospectral methods
Pseudospectral methods (also called collocation methods) are important fairly recent methods for solving differential equations numerically. We will describe them in detail and compare with other methods, such as finite difference and finite element methods. Depending on the background of the participants, a number of important results and techniques will be reviewed, including results from approximation theory, Fourier analysis, and differential equations. A number of examples of applications will be given.
Pseudospectra of linear operators
Although the name sounds like the previous topic, it is actually something quite different. Given a matrix, its eigenvalues and (generalized) eigenvectors characterize it completely. This is sometimes called the Jordan normal form of a matrix. As is well known, the set of eigenvalues is called the spectrum. If the matrix is nice (for example hermitean (also called selfadjoint), the spectrum gives valuable information on the matrix. But if the matrix is far from being hermitean, this information may be very misleading. To remedy this problem, the pseudospectrum has been introduced. I will introduce the pseudospectrum, show some of its properties, and describe the kind of information it contains. A number of computational examples will be given. They can be chosen to fit interests of the participants.

### Course material

The textbook for the course is
Lloyd N. Trefethen: Spectral Methods in Matlab, SIAM 2000. ISBN 0-89871-465-6. A link to the SIAM homepage is here. A link to the author's page for the book is here. Note that this page also contains a number of matlab m-files we will use during the course.
This book covers both topics. There will also be some papers about the pseudospectrum that will be handed out during the course. An overview of the psesudospectrum can be found in
Lloyd N. Trefethen: Pseudospectra of linear operators. SIAM Review 39 (1997), 383-406.
You can access this paper electronically through the library.

### Course style

The course will be taught in a style, where I emphasize computation in Matlab with the algorithms and methods presented in the course. This is viewed as an aid to understanding the contents of the formal results. Formal mathematical proofs of the results presented will rarely be given, since most of them require prerequisites that few PhD students have.

There will be a number of Matlab exercises posed during the course. You will be reguired to hand in solutions of some of them, as part of the evaluation of the course.

### Prerequisites

The usual mathematics background obtained during studies for the MSc in engineering or natural science at Aalborg University will suffice. If in doubt, please contact me.

Beyond this some familiarity with Matlab is required. You should know the basic matrix operations, and how to write and edit an .m file, carrying out a sequence of computations.

Written by Arne Jensen.