Parallel Smoothers for Matrix-based Multigrid Methods on Unstructured Meshes Using Multicore CPUs and GPUs

Abstract

Multigrid methods are efficient and fast solvers for problems typically modeled by partial differential equations of elliptic type. For problems with complex geometries and local singularities stencil-type discrete operators on equidistant Cartesian grids need to be replaced by more flexible concepts for unstructured meshes in order to properly resolve all problem-inherent specifics and for maintaining a moderate number of unknowns. However, flexibility in the meshes goes along with severe drawbacks with respect to parallel execution - especially with respect to the definition of adequate smoothers. This point becomes in particular pronounced in the framework of fine-grained parallelism on GPUs with hundreds of execution units. We use the approach of matrix-based multigrid that has high flexibility and adapts well to the exigences of modern computing platforms.

In this work we investigate multi-colored Gauss-Seidel type smoothers, the power(q)-pattern enhanced multi-colored ILU(p) smoothers with fill-ins, and factorized sparse approximate inverse (FSAI) smoothers. These approaches provide efficient smoothers with a high degree of parallelism. In combination with matrix-based multigrid methods on unstructured meshes our smoothers provide powerful solvers that are applicable across a wide range of parallel computing platforms and almost arbitrary geometries. We describe the configuration of our smoothers in the context of the portable lmpLAtoolbox and the HiFlow3 parallel finite element package. In our approach, a single source code can be used across diverse platforms including multicore CPUs and GPUs. Highly optimized implementations are hidden behind a unified user interface. Efficiency and scalability of our multigrid solvers are demonstrated by means of a comprehensive performance analysis on multicore CPUs and GPUs.