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

PhD Course

Topics in Modern Applied Mathematics

Misprint in program for Wednesday corrected.

Course plan

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.

• 9:00 - 9:15. Introduction of lecturer and participants. Coffee and tea will be served.
• 9:15 - 12:00. Formalities concerning the course. After that I start going through Chapter 1 in [T]. I will use Maple to derive the finite difference formulas. The numerical experiments will be done in Matlab. There will be a first introduction to the spectral methods.
• 12:00 - 12:45. Lunch break
• 12:45 - 14:00. Exercises. The participants are expected to bring a laptop with Matlab installed and the files from the book home page downloaded to it. Material for the exercises will appear soon.
• 14:00 - 14:15. Coffee and tea break.
• 14:15 - 16:15. Lectures. I will go through most of Chapter 2 in [T]. I will also cover part of Chapter 3. How far I get depends on how much background material I need to recall. There will also be some numerical experiments. The participants will be asked to do some of these.

Tuesday, May 16, 12:30 - 16:15

The plan for the day is as follows.

• 12:30 - 13:00. Review. Reply to questions on the material from Monday.
• 13:00 - 14:00. I will go through the remaining parts of Chapter 3 in [T].
• 14:00 - 15:00. Exercises. Coffee and tea will be available. We will do some further numerical experiments, based on the .m-files accompanying the book.
• 15:00 - 16:00. I will go through Chapter 4 in [T]. This chapter is somewhat technical. I will try to present the main results. They will be illustrated with numerical experiments.

Wednesday May 17, 12:30 - 16:15

The plan for the day is as follows.

• 12:30 - 13:00. Review. Reply to questions on the material from Tuesday.
• 13:00 - 14:00. I will go through some of the material from Chapter 5 in [T]. The results in Theorem 5 and 6 will be briefly explained.
• 14:00 - 15:00. Exercises. Coffee and tea will be available. We will do some further numerical experiments, based on the .m-files accompanying the book.
• 15:00 - 16:00. We have now come to one of the main topics of the course, the Chebyshev differentiation matrices. I will go through these results from Chapter 6 in [T] in detail.

Thursday May 18, 12:30 - 16:15

The plan for the day is as follows.

• 12:30 - 13:00. Review. Reply to questions on the material from Wednesday.
• 13:00 - 14:00. The use of the Chebyshev differentiation matrices to solve some boundary value problems will be illustrated with several examples from Chapter 7 in [T]. If needed, I will review the basic results concerning the boundary value problems that we solve numerically.
• 14:00 - 15:00. You will now have the opportunity to experiment further with the solution of the boundary value problems. At the end of the period we will have a round of discussions on the conclusions to be drawn from the numerical experiments and their relation to the theory.
• 15:00 - 16:00. If needed, further discussion. I will then give an overview of the FFT approach to spectral differentiation. This will cover part of Chapter 8 in [T].

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.

• 13:00 - 14:00. Overview of the course up to now. Reply to questions.
• 14:00 - 15:00. Discussion of the exercises from the previous days.
• 15:00 - 16:00. Introduction to pseudospectra. I will take this introduction from part of Chapter 9 in [T]. Supplementary material will be handed out.

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

[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.

[T97] Lloyd N. Trefethen: Pseudospectra of linear operators. SIAM Review 39 (1997), 383-406.

Course style

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

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.