Multigrid Smoothers for Ultraparallel Computing

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].