Fast Iterative Solution Methods

Iterative methods are among the most efficient and prominent methods for solving large scale linear and non-linear systems of equations. In addition to classical Krylov-space methods such as cg or GMRES, also multigrid and domain decomposition methods are nowadays widely used, as they are solution approaches with optimal complexity.

In this course, we will treat
- Krylov space methods
- multigrid methods
- domain decomposition methods.

In extension of the established theory, we will not only treat the linear elliptic case, but we will also consider multigrid and domain decomposition methods for constrained, coupled, and non-linear problems.

Requirements
The students are expected to be familiar with main concepts from numerical analysis, functional analysis (Banach spaces, Hilbert spaces), and to have basic knowledge in partial differential equations and finite element methods.

REFERENCES

• Wolfgang Hackbusch. Multi-Grid Methods and Applications. Springer Verlag, Berlin, Heidelberg, New York, 1985.
• P. Oswald. On function spaces related to finite element approximation theory. Zeitschrift für Analysis und ihre Anwendungen, 9:43--64, 1990.
• Saad Iterative Methods for Sparse Linear Systems http://www-users.cs.umn.edu/~saad/IterMethBook_2ndEd.pdf
• The Mathematical Theory of Finite Element Methods. Brenner, Susanne; Scott, Ridgway; Springer
• Optimization and Nonsmooth Analysis. Frank H. Clarke
• Numerical Optimization" Jorge Nocedal, Stephen J. Wright; Springer