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Preprint 7-2007

Theoretical and Numerical Comparison of Projection Methods derived from Deflation, Domain Decomposition and Multigrid Methods

Source file is available as :   Postscript Document

Author(s) : J.M. Tang , R. Nabben , C. Vuik , Y.A. Erlangga

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 7-2007

MSC 2000

65F10 Iterative methods for linear systems
65F50 Sparse matrices

Abstract :
For various applications, it is well-known that a two-level-preconditioned Krylov method is an efficient method for solving large and sparse linear systems. Beside a traditional preconditioner, like incomplete Cholesky decomposition, a projector is included as preconditioner to get rid of the effect of a number of small or large eigenvalues of the coefficient matrix. In literature, various projection methods are known, coming from the fields of deflation, domain decomposition and multigrid. At a first glance, the projectors seem to be different. However, from an abstract point of view, it can be shown that these methods are closely related. The aim of this paper is to compare these projection methods both theoretically and numerically. We investigate their convergence properties and stability by considering their implementation, effect of rounding-errors, inexact coarse solves, severe termination criteria and perturbed starting vectors. Finally, we end up with a suggestion of the second-level preconditioner, which is as stable as the abstract balancing preconditioner and as cheap and fast as the deflation preconditioner.

Keywords : deflation, domain decomposition, multigrid, conjugate gradients, two-grid schemes, preconditioning, implementation, spd matrices, hybrid methods, coarse grid corrections, projection methods

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