Homework 7 Task 8: Preconditioning Strategies

April 24, 2019 · View on GitHub

Task: Look for internet sites that include descriptions of preconditioning of systems of equations. Document at least 3 different preconditioning strategies.

Preconditioning

Iterative methods, and the convergence of such methods, are heavily reliant on well-conditioned systems. If a system is poorly conditioned, then the system may not converge to a solution at all (or it will converge to an incorrect system). Thus, a preconditioning matrix can be applied to the original system of equations i.e.:

$\mathbf{\vv{B}}\mathbf{\vv{A}}\mathbf{\vv{x}} = \mathbf{\vv{B}}\mathbf{\vv{b}}$

The condition number for BA may then be smaller than the condition number of A alone^[1]. This also presents the requirement for preconditioners that they must^[2]:

Have better convergence than the original system
Operations with B should be easy to perform

Different preconditioning strategies include:

Jacobi preconditioner^[2]
Gauss-Seidel, SOR, and SSOR^[2]
Incomplete LU factorization^[2]
Multigrid preconditioning^[3]
[Source 1](<https://en.wikipedia.org/wiki/Preconditioner)
Source 2
Source 3