Adaptive Multilevel Krylov Methods

Abstract

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.

Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

“…They develop a multilevel approach to shift small eigenvalues, leading to a faster convergence of the linear solver [15]. In subsequent work related to multilevel Krylov methods, Kehl, Nabben and Szyld apply preconditioning in a flexible way, via an adaptive number of inner iterations [19]. Baumann and van Gijzen analyze solving shifted linear systems and, by applying flexible preconditioning, also develop nested Krylov solvers [5].…”

Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy

“…They develop a multilevel approach to shift small eigenvalues, leading to a faster convergence of the linear solver. In subsequent work related to multilevel Krylov methods, Kehl et al 16 apply preconditioning in a flexible way, via an adaptive number of inner iterations. Baumann and van Gijzen 17 analyze solving shifted linear systems and, by applying flexible preconditioning, also develop nested Krylov solvers.…”