Eigenvalue translation based preconditioners for the GMRES(k) method

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Abstract
The paper considers a possible approach to the construction of high‐quality preconditionings for solving large sparse unsymmetric offdiagonally dominant, possibly indefinite linear systems. We are interested in the construction of an efficient iterative method which does not require from the user a prescription of several problem‐dependent parameters to ensure the convergence, which can be used in the case when only a procedure for multiplying the coefficient matrix by a vector is available and which allows for an efficient parallel/vector implementation with only one additional assumption that the most of eigenvalues of the coefficient matrix are condensed in a vicinity of the point 1 of the complex plane. The suggested preconditioning strategy is based on consecutive translations of groups of spread eigenvalues into a vicinity of the point 1. Approximations to eigenvalues to be translated are computed by the Arnoldi procedure at several GMRES(k) iterations. We formulate the optimization problem to find optimal translations, present its suboptimal solution and prove the numerical stability of consecutive translations. The results of numerical experiments with the model CFD problem show the efficiency of the suggested preconditioning strategy.

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