Year: 2015
Numerical Mathematics: Theory, Methods and Applications, Vol. 8 (2015), Iss. 2 : pp. 168–198
Abstract
About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2015.w04si
Numerical Mathematics: Theory, Methods and Applications, Vol. 8 (2015), Iss. 2 : pp. 168–198
Published online: 2015-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 31
-
Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction ($\ell$AIR)
Manteuffel, Thomas A. | Ruge, John | Southworth, Ben S.SIAM Journal on Scientific Computing, Vol. 40 (2018), Iss. 6 P.A4105
https://doi.org/10.1137/17M1144350 [Citations: 35] -
A new algebraic multigrid approach for Stokes problems
Notay, Yvan
Numerische Mathematik, Vol. 132 (2016), Iss. 1 P.51
https://doi.org/10.1007/s00211-015-0710-0 [Citations: 10] -
Algebraic Two-Level Convergence Theory for Singular Systems
Notay, Yvan
SIAM Journal on Matrix Analysis and Applications, Vol. 37 (2016), Iss. 4 P.1419
https://doi.org/10.1137/15M1031539 [Citations: 7] -
Convergence Analysis of Inexact Two-Grid Methods: A Theoretical Framework
Xu, Xuefeng | Zhang, Chen-SongSIAM Journal on Numerical Analysis, Vol. 60 (2022), Iss. 1 P.133
https://doi.org/10.1137/20M1356075 [Citations: 6] -
Symbol based convergence analysis in multigrid methods for saddle point problems
Bolten, Matthias | Donatelli, Marco | Ferrari, Paola | Furci, IsabellaLinear Algebra and its Applications, Vol. 671 (2023), Iss. P.67
https://doi.org/10.1016/j.laa.2023.04.016 [Citations: 2] -
A New Analytical Framework for the Convergence of Inexact Two-Grid Methods
Xu, Xuefeng | Zhang, Chen-SongSIAM Journal on Matrix Analysis and Applications, Vol. 43 (2022), Iss. 1 P.512
https://doi.org/10.1137/21M140448X [Citations: 3] -
Projections, Deflation, and Multigrid for Nonsymmetric Matrices
García Ramos, Luis | Kehl, René | Nabben, ReinhardSIAM Journal on Matrix Analysis and Applications, Vol. 41 (2020), Iss. 1 P.83
https://doi.org/10.1137/18M1180268 [Citations: 5] -
Convergence in Norm of Nonsymmetric Algebraic Multigrid
Manteuffel, Tom | Southworth, Ben S.SIAM Journal on Scientific Computing, Vol. 41 (2019), Iss. 5 P.S269
https://doi.org/10.1137/18M1193773 [Citations: 11] -
On the Ideal Interpolation Operator in Algebraic Multigrid Methods
Xu, Xuefeng | Zhang, Chen-SongSIAM Journal on Numerical Analysis, Vol. 56 (2018), Iss. 3 P.1693
https://doi.org/10.1137/17M1162779 [Citations: 10] -
On the convergence of two‐level Krylov methods for singular symmetric systems
Erlangga, Yogi A. | Nabben, ReinhardNumerical Linear Algebra with Applications, Vol. 24 (2017), Iss. 6
https://doi.org/10.1002/nla.2108 [Citations: 0] -
An Aggregation-Based Two-Grid Method for Multilevel Block Toeplitz Linear Systems
An, Chengtao | Su, YangfengJournal of Scientific Computing, Vol. 98 (2024), Iss. 3
https://doi.org/10.1007/s10915-023-02434-9 [Citations: 0] -
Convergence of VW-cycle and WV-cycle multigrid methods
Xu, Xuefeng
BIT Numerical Mathematics, Vol. 65 (2025), Iss. 1
https://doi.org/10.1007/s10543-024-01042-9 [Citations: 0]