Seminar
Domain decomposition and multilevel methods analysis for scalable parallel computing

Tutors:	D. Arndt, P. Lucero
Class data:	LSF,
First meeting:	Mon, October 16, 2017, 14h, Mathematikon (INF 205), SR 11
Seminar:	Mon 14-16h, Mathematikon (INF 205), SR 11

Summary

The evolution of computers towards high parallelism has led to a large amount of research on numerical methods that would be suited for such architectures, with the goal of finding scalability. Regardless of the hardware capabilities, if the methods used do not acomplish a slow growth in cost with respect to the size of the problem, the method is doomed to reach a limit on applicability that can be significantly low. Domain decomposition methods have proven to satisfy these requirements at least for suitably well-behaved problems. In particular, multilevel methods are the state-of-the-art technique for solving linear equation systems resulting from the discretization of partial differential equations with a cost scaling linearly. In this seminar we aim to an introduction to these methods and their implementation. We focus on their asymptotic analysis through an abstract framework with implementation examples and we intend to achieve a general intuition on the behaviour and domain of applicability of these techniques.

Target group

Students of mathematics (BSc and MSc) as well as students of scientific computing (MSc).
The seminar is in particular recommended for those students that have some experience with discretization techniques (such as finite elements or finite differences) for partial differential equations. On the other hand, the initial talks don't require anything more than basic knowledge in linear algebra. Experience with software for dealing with numerical linear algebra is desirable but not crucial.

Participation

For participation in the seminar you have to register via Müsli (binding registration).

Additionally, please write us a mail with your subject of study, semester, BSc / MSc, ... . Please also mention related lectures you have already attended, courses you will take in the winter term and, if you already have ideas, the area where you would like to write your thesis.

Preparation

The topics will be assigned starting from now. Interested students can prepare their presentations already in the semester break (we do recommend that). Latest assignment is in the beginning of the summer term. Generally, before presenting we recommend to discuss the structure of your talk with us.

Topics

The seminar covers the following topics:

1. Iterative methods: Richardson, Jacobi and Gauß-Seidel (Meister "Numerik linearer Gleichungssysteme": p. 23-118) Proseminar
2. Coarse grid correction (Toselli, Widlund; Smith)
3. Two-level preconditioners (Toselli, Widlund; Smith)
4. Multilevel methods (Toselli, Widlund; Smith; Meister)
5. Schwarz methods (Quarteroni; Toselli, Widlund; Smith)

Schedule

t.b.a

Report

• The report should be submitted in electronical form preferrably using Latex.
• It should serve as a summary of your presentation containing between 5 and 10 pages. In particular, make sure to write in the way you would write this yourself and don't copy & paste from the reference.
• Don't forget to include citations to results you are using and don't prove.

Literature

• Meister, Andreas. "Numerik linearer Gleichungssysteme". Springer Spektrum, 2011.
• Quarteroni, Alfio and Valli, Alberto. Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, 1999.
• Barry Francis Smith, Petter E. Bjørstad, and William D. Gropp. "Domain decomposition : parallel multilevel methods for elliptic partial differential equations". Cambridge University Press, Cambridge, 1996.
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• Meister, Andreas. "Numerik linearer Gleichungssysteme". Springer Spektrum, 2011.
• Andrea Toselli and Olof B. Widlund. Domain decomposition methods : algorithms and theory. Springer series in computational mathematics. Springer, Berlin, 2005.