A GPU-Based MIS Aggregation Strategy: Algorithms, Comparisons, and Applications Within AMG

The algebraic multigrid (AMG) method is often used as a preconditioner in Krylov subspace solvers such as the conjugate gradient method. An AMG preconditioner hierarchically aggregates the degrees of freedom during the coarsening phase in order to efficiently account for lower-frequency errors. Each degree of freedom in the coarser level corresponds to one of the aggregates in the finer level. The aggregation in each level in the hierarchy has a significant impact on the effectiveness of AMG as a preconditioner. The aggregation can be formulated as a partitioning problem on the graph induced from the matrix representation of a linear system. We present a GPU implementation of a "bottom-up" partitioning scheme based on maximal independent sets (MIS). We also present some novel topology-informed metrics that measure the quality of a partition. To test our implementation and the metrics, we use an existing AMG preconditioned conjugate gradient (PCG-AMG) solver and show that our metrics are correlated with the time and the number of iterations needed for the linear system to converge to a solution. For comparable coarsening ratios, we show that the MIS-based aggregation methods outperform Metis-based "top-down" aggregation method for the PCG-AMG method. Our results also indicate that MIS-based aggregation methods provide aggregates that are evaluated more favorably by our metrics than the aggregates provided by the Metis-based method.