EU Regional School - Voemel Seminar
Dr. Christof Voemel - Efficient Solution of Linear Systems Arising in Finite Discretization Methods
Efficient Solution of Linear Systems Arising in Finite Discretization Methods
The numerical discretization of a physical model by finite differences, volumes, or elements typically leads to large sparse linear systems whose unknowns describe the physical quantities of the underlying problem. In practice, there are two common types of solution methods.
Direct linear solvers allow a numerically stable computation in many relevant cases and can be essentially used as a black box. However, fill-in limits their usefulness as one can incur significant overhead in terms of memory and flops.
Iterative linear solvers offer a more memory-economical, matrix-free solution approach where typically, one only needs the discrete operator as a matrix-vector product. Their main disadvantages are the dependence on preconditioners for fast convergence, and a good choice of parameters such as start vector, basis size, and restart strategy.
This course discusses the theoretical and practical background of the two approaches. The participants are then able to make an informed choice between the various methods that is best suiting their own needs.
The course will cover the following topics.
- Principles of linear systems solving, LU and Cholesky decomposition
- Sparse matrices & sparse direct linear solvers
- Fill-in reducing matrix orderings
- Krylov subspace methods I: GMRES
- Krylov subspace methods II: CG and MinRES