Scientific Computing II

The module gives deeper insight into two important topics of scientific
computing:
- Iterative solution methods for large, sparse linear systems (relaxation methods, geometric and algebraic multigrid methods, Krylov-subspace methods, preconditioning techniques); intuitive introduction, mathematical performance analysis and sample computations
- Molecular dynamics simulation as a case study for the particle-based simulation approaches in scientific computing (overview; modelling of molecular dynamics; discretization approaches; efficient implementation of all-to-all interaction; techniques for parallelization)

At the end of the module, students are able to remember and identify the main classes of iterative methods to solve large, sparse linear systems. The participants can evaluate the range of application of such methods in standard scenarios and they remember their basic performance features. Students can apply these methods theoretically to a certain problem and also implement them in code. They understand the typical steps of the simulation pipeline from modeling over discretization and numerics to implementation and visualization. For the scenario Molecular Dynamics, they have detailed knowledge of these steps and are able to design and realise suitable simulation software.

Students should have basic knowledge in differential calculus and linear algebra. Knowlegde in numerical programming and scientic computing is recommended (modules IN2005 Einführung in das Wissenschaftliche Rechnen and IN0019 Numerisches Programmieren, e.g.)

This module comprises lectures and accompanying tutorials. The contents of the lectures will be taught by talks and presentations. Students will be encouraged to study literature and to get involved with the topics in depth. In the tutorials, concrete problems will be solved - partially in teamwork - and selected examples will be discussed.

Slides, whiteboard, exercise sheets

- William L. Briggs, Van Emden Henson, Steve F. McCormick. A Multigrid Tutorial. Second Edition. SIAM. 2000.
- J.R. Shewchuk. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Edition 1.25. 1994.
- W. Hackbusch. Iterative Solution of Large Sparse Systems of Equations. Springer, 1993.
- Y. Saad. Iterative Methods for sparse linear systems, SIAM 2003.
- M. Griebel, S. Knapek, G. Zumbusch, and A. Caglar. Numerical Simulation in Molecular Dynamics. Springer, 2007.
- M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, 2003.
- D. Frenkel and B. Smith. Understanding Molecular Simulation from Algorithms to ASpplications. Academic Press (2nd ed.), 2002.
- R. J. Sadus. Molecular Simulation of Fluids; Theory, Algorithms and Object-Orientation. Elsevier, 1999.
- D. Rapaport. The art of molecular dynamics simulation. Camebridge University Press, 1995.

The examination consists of a written exam of 105 minutes in which students show that they are able to find solutions for problems arising in the field of scientific computing in a limited time. By questions on code snippets and given algorithms the participant’s knowledge on iterative methods in general and their implementation is examined. Small example problems test the capability to apply an iterative method to a given problem. The ability to evaluate the eligibility of a certain method is tested by example problems. Questions on performance features of given methods might also arise in this context and test the corresponding knowledge. The understanding of different steps of the simulation pipeline is examined by various tasks such as modelling a given problem, discretizing a given method or questions on numerical properties. Tasks which are used in the context of Molecular Dynamics require a deeper understanding of these steps.