High Performance AMG Solvers on GPU-Architecture

Multigrid methods are the most effective methods for solving positive definite linear systems in scientific computing fields. The systems can be discretized from a pressure equation or a compositional model from a black oil... [ view full abstract ]

Multigrid methods are the most effective methods for solving positive definite linear systems in scientific computing fields. The systems can be discretized from a pressure equation or a compositional model from a black oil model in reservoir simulation. The multigrid methods have the optimal convergence rate. Our efforts are devoted into developing new generation parallel AMG solvers on GPU-architecture to accelerate the solution. Because GPUs have different architecture from CPUs, it is necessary to develop new implementation algorithms to make maximal use of the parallel capability provided by GPUs.

This work consists of a classical AMG solver and a smoothed aggregation multigrid solver. An AMG algorithm has a setup phase and a solution phase. The setup phase is responsible for building the coarser grids, smoothers, restriction operators and prolongation operations. We implemented two coarsening strategies (the Ruge-Stuben method and the CLJP method), several interpolation operators (standard interpolation, direct interpolation and multi-pass interpolation), a series of smoothers (Jacobi, damped Jacobi, weighted Jacobi, block Jacobi, Gauss-Seidel, etc.). The solution phase is responsible for solving the multi-level system in which we implemented the V-cycle, F-cycle and W-cycle. The setup phase is completed on the host (CPU). The solution phase is completed on the device (GPU).

The Intel Xeon X5570 with NVIDIA Tesla C2050/2070 is used as the testing platform. Some structural matrices and non-structural matrices are tested. The result shows that our AMG solver can be accelerated up to over ten times faster on a GPU than only running sequentially on a CPU.