An algebraic theory for multigrid methods for variational problems
SIAM Journal on Numerical Analysis
Iterative solution methods
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
SIAM Journal on Scientific Computing
BoomerAMG: a parallel algebraic multigrid solver and preconditioner
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
An Iterative Method with Convergence Rate Chosen a priori
An Iterative Method with Convergence Rate Chosen a priori
Parallel multigrid smoothing: polynomial versus Gauss--Seidel
Journal of Computational Physics
On Generalizing the Algebraic Multigrid Framework
SIAM Journal on Numerical Analysis
Pursuing scalability for hypre's conceptual interfaces
ACM Transactions on Mathematical Software (TOMS) - Special issue on the Advanced CompuTational Software (ACTS) Collection
Reducing Complexity in Parallel Algebraic Multigrid Preconditioners
SIAM Journal on Matrix Analysis and Applications
An Introduction to Algebraic Multigrid
Computing in Science and Engineering
Compatible Relaxation and Coarsening in Algebraic Multigrid
SIAM Journal on Scientific Computing
On the performance of an algebraic multigrid solver on multicore clusters
VECPAR'10 Proceedings of the 9th international conference on High performance computing for computational science
Challenges of Scaling Algebraic Multigrid Across Modern Multicore Architectures
IPDPS '11 Proceedings of the 2011 IEEE International Parallel & Distributed Processing Symposium
Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics
SIAM Journal on Scientific Computing
Architecting the finite element method pipeline for the GPU
Journal of Computational and Applied Mathematics
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This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running on parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [R. D. Falgout and P. S. Vassilevski, SIAM J. Numer. Anal., 42 (2004), pp. 1669-1693] and [R. D. Falgout, P. S. Vassilevski, and L. T. Zikatanov, Numer. Linear Algebra Appl., 12 (2005), pp. 471-494] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several $C$-$F$, polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultraparallel computers. We also introduce the more robust $\ell_1$ smoothers, which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604-623].