| Description |
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 eciently account for lower-frequency errors. Each degree of freedom in the coarser level corresponds to one of the aggregates in the ner level. The aggregation in each level in the hierarchy has a signicant impact on the eectiveness 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. The contributions of this work are as follows: rst, a GPU implementation of a \bottom-up" partitioning scheme based on maximal independent sets (MIS), including an ecient conditioning scheme for enforcing partition size constraints; second, three novel topological metrics, convexity, eccentricity, and minimum enclosing ball, for measuring partition quality; third, empirical test results comparing our MIS-Based aggregation methods with the MeTis graph partioning library, showing that the metrics correlate more strongly with AMG performance than the commonly used edge-cut metric, and that for ner aggregations, MIS-based aggregation is better suited for AMG coarsening than is the \top down" MeTis graph partitioning library, but that for coarser aggregations, MeTis performs better. |
