Eigenvalue translation based preconditioners for the GMRES(k) method

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. KEY WORDS Arnoldi procedure Spread and condensed eigenvalues Eigenvalue translations

On the Construction of Deflation-Based Preconditioners

“…In this case, deflation is not applied as a preconditioner, but the deflation vectors are augmented with the Krylov subspace, and the minimization property of GMRES ensures that the deflation subspace is projected out of the residual. For more discussion on deflation methods for nonsymmetric systems, see [15,8,6,21,5,2]. Other authors have attempted to choose a subspace a priori that effectively represents the slowest modes.…”

An Efficient Preconditioned CG Method for the Solution of a Class of Layered Problems with Extreme Contrasts in the Coefficients

“…Deflation is also used in iterative methods for non-symmetric systems of equations [5,9,10,13,24,27,33]. In these papers the smallest eigenvalues have been shifted away from the origin.…”

Adaptively Accelerated Gmres Fast Fourier Transform Method for Electromagnetic Scattering

“…A disadvantage of this approach is that the convergence behavior in many situations seems to depend quite critically on the value of m. Even in situations in which satisfactory convergence takes place, the convergence is less than optimal, since the history is thrown away so that potential superlinear convergence behavior is inhibited [10]. There are many acceleration techniques that attempt to mimic the convergence of full GMRES more closely, or to accelerate the convergence of the regular GMRES by retaining some historical information at the time of restart [11][12][13][14][15]. Deflation methods are a main class of acceleration techniques for GMRES.…”

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