An algebraic multigrid solver (AMG) is an iterative method for solving symmetric positive definite linear systems arising from elliptic partial differential equations. When solving linear systems, we know that oscillatory errors converge quickly on a fine grid, but smooth errors are less likely to converge. The AMG solver changes the fine grid to a coarser grid, and smooth errors on a fine grid are converted to oscillatory errors on the coarser grid. In this way, smooth errors are easier to converge. By repeating this process, an AMG solver with a hierarchical structure can be assembled. Compared to normal Krylov solvers that use only one grid, an AMG solver is more complicated. However, it is much more efficient when solving symmetric positive definite linear systems. It is also scalable, which is attractive for applications on parallel computers. An efficient implementation of an AMG solver is essential to many scientific applications.

The authors present work on developing a parallel AMG solver on clusters of central processing units (CPUs) and graphical processing units (GPUs). GPUs are specially designed to accelerate the manipulation of images by processing large amounts of data in parallel. The architecture of GPUs is highly parallel, unlike CPUs. For floating-point computation, GPUs are around ten times faster than CPUs. By using GPUs, scientific applications can be highly accelerated. The authors develop a solver package, Parallel Toolbox, in which a conjugate gradient solver and an AMG preconditioner are implemented. To implement an AMG solver/preconditioner, it is necessary to calculate coarser grids, interpolation operators, and coarser matrices. These components have a significant influence on the efficiency of the AMG.

In this paper, the local Ruge-Stuben algorithm is chosen as the grid coarsening strategy. During the coarsening process, there is no communication among the processors. A simple interpolation operator is employed, and it has no communication either. The coarsening strategy and interpolation operator used in this paper are simple, but they are effective. When implemented on a GPU cluster, the speed-up is very large, which the authors demonstrate with numerical experiments, achieving a peak speed-up factor of 34. Compared with other papers, the results presented here are much better. I think the paper would be a good reference for researchers who work on GPU computing, parallel computing, and AMGs.

The authors present the first AMG solver on GPUs for fully unstructured discretizations of elliptic differential equations. The strategies used in this paper are effective for GPU clusters. The proposed package, Parallel Toolbox, is available online, and I believe this package and its design will be helpful to other researchers.