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Recursive Krylov‐based multigrid cycles
Author(s) -
Notay Yvan,
Vassilevski Panayot S.
Publication year - 2008
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.542
Subject(s) - krylov subspace , multigrid method , preconditioner , mathematics , conjugate gradient method , positive definite matrix , solver , linear system , convergence (economics) , bounded function , iterative method , relaxation (psychology) , conjugate residual method , grid , generalized minimal residual method , mathematical optimization , computer science , eigenvalues and eigenvectors , partial differential equation , mathematical analysis , psychology , social psychology , physics , gradient descent , quantum mechanics , machine learning , artificial neural network , economics , economic growth , geometry
We consider multigrid (MG) cycles based on the recursive use of a two‐grid method, in which the coarse‐grid system is solved by µ⩾1 steps of a Krylov subspace iterative method. The approach is further extended by allowing such inner iterations only at the levels of given multiplicity, whereas V‐cycle formulation is used at all other levels. For symmetric positive definite systems and symmetric MG schemes, we consider a flexible (or generalized) conjugate gradient method as Krylov subspace solver for both inner and outer iterations. Then, based on some algebraic (block matrix) properties of the V‐cycle MG viewed as a preconditioner, we show that the method can have optimal convergence properties if µ is chosen to be sufficiently large. We also formulate conditions that guarantee both, optimal complexity and convergence, bounded independently of the number of levels. Our analysis shows that the method is, at least, as effective as the standard W‐cycle, whereas numerical results illustrate that it can be much faster than the latter, and actually more robust than predicted by the theory. Copyright © 2007 John Wiley & Sons, Ltd.