Preconditioner for linear solvers (markdown test)
When solving a system of equations
iteratively (using, e.g., the conjugate gradient method), the convergene speed is usually determined by the condition number , where
Thus, if , which happends for example when solving a Poison problem using a finite element approximation with meshsize , iterative solvers have a very hard time to solve the system efficiently.
In order to archieve robustness w.r.t. , one strategy is to apply preconditioning to the linear system. This means that instead of solving , we now solve
where is a positive-definite matrix. This system has the same solution but we choose in such a way, that the preconditioned system matrix has a . Also should be easy-to-compute.
Preconditoning is very important. Wikipedia, e.g., states for the CG method:
In most cases, preconditioning is necessary to ensure fast convergence of the conjugate gradient method.
In the best case, one can show that the is uniformly bounded in , i.e.
where the constant is independent of . In this case, we would have archieved full robustness w.r.t. .2023-01-06