Developing a Multi-GPU-Enabled Preconditioned GMRES with Inexact Triangular Solves for Block Sparse Matrices

Abstract

Solving triangular systems is the building block for preconditioned GMRES algorithm. Inexact preconditioning becomes attractive because of the feature of high parallelism on accelerators. In this paper, we propose and implement an iterative, inexact block triangular solve on multi-GPUs based on PETSc’s framework. In addition, by developing a distributed block sparse matrix-vector multiplication procedure and investigating the optimized vector operations, we form the multi-GPU-enabled preconditioned GMRES with the block Jacobi preconditioner. In the implementation, the GPU-Direct technique is employed to avoid host-device memory copies. The preconditioning step used by PETSc’s structure and the cuSPARSE library are also investigated for performance comparisons. The experiments show that the developed GMRES with inexact preconditioning on 8 GPUs can achieve up to 4.4x speedup over the CPU-only implementation with exact preconditioning using 8 MPI processes.