Author(s) :
R. Nabben
,
K. Vuik
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 9-2007
MSC 2000
- 65F10 Iterative methods for linear systems
-
65F50 Sparse matrices
Abstract :
In this paper we consider various preconditioners for the cg method to solve
large linear systems of equations with symmetric positive definite system matrix. We continue the comparison between
abstract versions of the deflation, balancing and additive coarse grid correction preconditioning
techniques started by Nabben and Vuik in 2004 and 2006. There the deflation method
is compared with the abstract additive coarse grid
correction preconditioner and the abstract balancing preconditioner.
Here we close the triangle between these three methods.
First of all we show, that a theoretical comparison of the condition
numbers of the abstract additive coarse grid correction and the condition
number of the system preconditioned by the abstract
balancing preconditioner is not possible. We present a counter
example, for which the condition number of the abstract additive coarse
grid correction preconditioned system is below the condition number of the
system preconditioned with the abstract balancing preconditioner.
However, if the cg method is preconditioned by the abstract balancing
preconditioner and is started with a special starting vector, the asymptotic
convergence behavior of the cg method can be described by the so called
effective condition number with respect to the starting vector.
We prove that this effective condition
number of the system preconditioned by the abstract balancing preconditioner
is less or equal to the condition number of the system preconditioned
by the abstract additive coarse grid correction method.
We also provide a short proof of the relation between the effective condition
number and the convergence of CG.
Moreover, we compare the $A$-norm of the errors of the iterates given by the
different preconditioners and we establish the orthogonal invariants of all
three types of preconditioners.