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

PhD Course

Topics in Modern Applied Mathematics

Web page updated May 26, 2005

Misprint in program for Wednesday corrected.

Course plan

The preliminary course plan is as follows. [T] refers to the textbook mentioned below.

Note that there will be short breaks at various points in the program. They are not included below.

Monday, May 15, 9:00 - 16:00
The plan for the day is below. Note that the plan is subject to dynamic modification. If I need to review some background results shifts will occur. Shifts in the other direction will also occur, in case I can be very brief concerning FFT and Fourier Analysis.
Tuesday, May 16, 12:30 - 16:15
The plan for the day is as follows.
Wednesday May 17, 12:30 - 16:15
The plan for the day is as follows.
Thursday May 18, 12:30 - 16:15
The plan for the day is as follows.
Friday May 19, 13:00 - 16:15.
Note that today there is a change in the starting time. This is due to the meeting in the Danish Mathematical Society. All course participants are encoured to attend the lecture by the Abel prize winner Peter Lax. Information on the lecture can be found here. Otherwise, the plan for the day is as follows.
The plan for the second half of the course is as follows. Note the changed dates and times
Monday, May 29, 8:15-12:00
Review of the material covered in [T] during the first half of the course. I have covered most of the material in Chapters 1 to 8, and the first example in Chapter 9. During the review I will give further details on the error estimate for polynomial interpolation in Chapter 5, and on the results in Chapter 4 on regularity.
Monday, May 29, 12:30-16:15
I will start the presentation of the material on pseudospectra. It is based on the copies handed out Friday May 19.
Tuesday, May 30, 12:30-16:15
Further results on pseudospectra. Examples
Wednesday, May 31, 12:30-16:15
Overview of the course. Further examples and applications of both spectral differentiation and pseudspectra.

Course material

The textbook for the course is
[T] 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
[T97] 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.


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 May 8, 2006, by Arne Jensen.