Block-Relaxation Methods for 3D Constant-Coefficient Stencils on GPUs and Multicore CPUs

TL;DR

This paper develops and evaluates highly parallel block relaxation algorithms, specifically block Jacobi and chaotic block Gauss-Seidel, for solving 3D elliptic PDEs efficiently on modern GPU and multicore CPU architectures.

Contribution

It introduces robust, parallel implementations of block relaxation methods with exact block inversion tailored for GPU and multicore systems solving 3D elliptic PDEs.

Findings

Achieved efficient parallel performance on NVIDIA Fermi GPUs.

Demonstrated scalability on AMD multicore systems.

Validated robustness of the algorithms for structured 3D grids.

Abstract

Block iterative methods are extremely important as smoothers for multigrid methods, as preconditioners for Krylov methods, and as solvers for diagonally dominant linear systems. Developing robust and efficient algorithms suitable for current and evolving GPU and multicore CPU systems is a significant challenge. We address this issue in the case of constant-coefficient stencils arising in the solution of elliptic partial differential equations on structured 3D uniform and adaptively refined grids. Robust, highly parallel implementations of block Jacobi and chaotic block Gauss-Seidel algorithms with exact inversion of the blocks are developed using different parallelization techniques. Experimental results for NVIDIA Fermi GPUs and AMD multicore systems are presented.