Proseminar / Seminar Krylov Methods

Topic

Linear systems of equations of the form $$Ax = b$$ play a key role in many areas of mathematics, natural sciences, economics and engineering. In particular, they are a fundamental tool in many numerical methods. For example complex numerical methods for the simulation of an aircraft wing or weather forecasts ultimately rely on the solution of linear systems.

While simple Gaussian elimination is theoretically sufficient to solve linear systems, the aforementioned applications easily lead to systems with millions of degrees of freedom. Such cases are entirely out of reach of the more simple methods, and state-of-the-art solvers become a necessity.

This seminar is about Krylov methods, a very efficient and widely used class of iterative solvers. We will get familiar with the methods themselves, prove their convergence to the solution and investigate their efficiency.

Organization

The seminar will take place Tuesdays, 11:00-13:00. For dates and topics, refer to the google doc.
Since we use MÜSLI for email communication, please register at: https://muesli.mathi.uni-heidelberg.de/lecture/view/1329

Contact

Main contact:

Linus Seelinger: linus.seelinger at iwr.uni-heidelberg.de

Secondary contacts:

Chupeng Ma: chupeng.ma at uni-heidelberg.de

Robert Scheichl: r.scheichl at uni-heidelberg.de

Literature

During the seminar, we will be following these books:

Y. Saad, Iterative methods for sparse linear systems, 2. Edition, Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics, 2007.
J. Liesen and Z. Strakos, Krylov Subspace Methods, Principles and Analysis, Oxford University Press, 2013.

Structure

As part of this seminar, you will

develop a solid understanding of an individual topic,
give a talk on that topic to the other students,
write a report on that topic,
and participate in peer assessment of other presentations.

We will hold a combined Proseminar and Seminar, and assign topics according to your level.

Talks

The length of each talk is 35 min. + 10 min. for questions.

Proseminar participants may choose between German and English.

Before your talk you should meet with one of us to discuss your presentation.

Reports

Reports will be due 4 weeks after your seminar date. The page limit for the report is around 10 pages.

Proseminar participants may choose between German and English, according to the language of your talk.

Your report should have a clear structure, and you should correctly reference sources used.

By default, we will assume that the workload was shared evenly, and will assign a single mark for all group members. If you believe that the workload wasn’t shared evenly, then in your summary you can designate how the workload was split (as a percentage).

Peer assessment

You will be responsible for assessing one other presentation using the same criteria we apply (Assessment guide), and your assessment will enter in the presenters’ grade. The quality of your assessment will in turn be part of your own grade.

Hint: Asking questions is good!

Grades

Your seminar contribution will be graded based on

talk (60%), depending on content, structure and quality of presentation,
report (30%), depending on content, structure and quality of presentation,
peer assessment (10%), depending on consistency and quality of your critique / feedback.

For peer assessment, please use the assessment form. You can directly fill in the PDF form, this is supported by most PDF readers. Use the assessment guide as a reference for how many points to award.

Talk guidelines

Give a general overview of the topic, focus on one or two specific points in detail.
Make the topic accessible to others (e.g. carefully introduce notation, do not assume too much prior knowledge).
Show key theoretical aspects and possibly conduct numerical experiments (focus may vary depending on topic and preference).
Keep a clear structure, carefully select what to show and what to omit. The talk should have an introduction with an overview of the talk, a central part containing the content, and some conclusions at the end. The central part should in itself have a clear structure.
Make the talk interesting to hear (i.e. “tell a story”, show to the audience why your topic is interesting and useful).
Give an easy to follow presentation (e.g. do not overload slides, do not talk too slowly or too fast, give conclusions to sections of your talk to wrap them up).

Report guidelines

Cover the same content as in your talk. The points above mostly apply to the report as well.
You should go into more detail. For example, omitting technical details may be fine in a talk for clarity of presentation; the report however should be technically correct.
Properly cite sources you use.
Write in a concise, yet accessible style. Relevant points should be written out in an easy to follow way. On the other hand redundancy should be avoided since it distracts from the content.
As with the talk your report should have a clear structure. Do not forget to add conclusions.
There is no formal page requirement or limit, but around 10 pages (without appendices, e.g., containing code) should be okay. Of course that depends on how many figures you add to your report.