Two-stage spectral preconditioners for iterative eigensolvers

In this paper we present preconditioning techniques to accelerate the convergence of Krylov solvers at each step of an Inexact Newton’s method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices arising in large scale scientific computations. We propose a two-stage spectral preconditioning strategy: the first stage produces a very rough approximation of a number of the leftmost eigenpairs. The second stage (Inexact Newton) uses these approximations as starting vectors and also to construct the tuned preconditioner from an initial inverse approximation of the coefficient matrix, as proposed in [1, Martı́nez, Numer. Lin. Alg. Appl., 2016] in the framework of the Implicitly Restarted Lanczos method. The action of this spectral preconditioner results in clustering a number of the eigenvalues of the preconditioned matrices close to one. We also study the combination of this approach with a BFGSstyle updating of the proposed spectral preconditioner as described in [2, Bergamaschi, Martı́nez, Opt. Meth. Softw., 2015]. Extensive numerical testing on a set of representative large SPD matrices gives evidence of the acceleration provided by these spectral preconditioners. Copyright c © 2017 John Wiley & Sons, Ltd.