Faculty of Engineering and Science
The International Doctoral School of Technology and Science
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.