GPU accelerated geometric multigrid method: Comparison with preconditioned conjugate gradient

Scientific applications are typically compute intensive, often due to the requirement of solving large sparse linear systems of equations. The geometric multigrid method (GMG) is one of the most efficient algorithms for solving these systems and is well suited for parallelization. Herein we focus on an in-depth analysis of a GPU-based GMG implementation and compare the results against an optimized preconditioned conjugate gradient method. The tests indicate that the smoothing step is the most time consuming operation, and the best performing GMG variant is the V-cycle scheme with 312 smoothing step configuration (3 iterations during restriction, 1 at the coarsest level, and 2 iterations during prolongation). The discretization stencil has a major effect on the runtime and its choice requires a trade-off between execution time performance and numerical accuracy. Overall, the GMG method offers a speed-up of 7.1x-9.2x over the PCG method on the same hardware configuration, while also leading to a smaller average residual.