A comparison of abstract versions of deflation, balancing and additive coarse grid correction preconditioners

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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.