A fundamental tool of numerical computations is the solution of (large) linear systems of equations. Typically, such systems are not solved "directly" (i.e., using Gaussian elimination or one of its many variants) but approximately using an iterative scheme. The most important iterative schemes are presented in the course and their properties will be discussed. Knowledge of a variety of techniques is important to be able to select a good method for a given application.

The course will cover some of the most important techniques for solving iteratively large linear systems of equations. In the first part of the lecture course, rather general methodologies such as the CG- and the GMRES methods will be discussed. The second part of the course will focus on more special methods such as multigrid and domain decomposition methods. These latter methodologies are among the most powerful tools to solve very large systems arising from the discretization of elliptic partial differential equations (e.g., by the FEM). Multigrid, for example, has optimal complexity, i.e., its cost grows linearly with the problem size.

This lecture is part of the AKNUM elective courses.