Volume 51, pp. 512-528, 2019.

Adaptive Multilevel Krylov Methods

René Kehl, Reinhard Nabben, and Daniel B. Szyld


Inexact (variable) preconditioning of Multilevel Krylov methods (MK methods) for the solution of linear systems of equations is considered. MK methods approximate the solution of the local systems on a subspace using a few, but fixed, number of iteration steps of a preconditioned flexible Krylov method. In this paper, using the philosophy of inexact Krylov subspace methods, we use a theoretically-derived criterion to choose the number of iterations needed on each level to achieve a desired tolerance. We use this criterion on one level and obtain an improved MK method. Inspired by these results, a second ad hoc method is also explored. Numerical experiments for the Poisson, Helmholtz, and the convection-diffusion equations illustrate the efficiency and robustness of this adaptive Multilevel Krylov method.

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Key words

Multilevel Krylov methods, flexible GMRES, inexact Krylov subspace methods, inexact preconditioning

AMS subject classifications

65F10, 65F50, 65N22, 65N55

Links to the cited ETNA articles

[9]Vol. 31 (2008), pp. 403-424 Yogi A. Erlangga and Reinhard Nabben: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian

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