Adaptive AMG with coarsening based on compatible weighted matching

We introduce a new composite adaptive Algebraic Multigrid (composite?AMG) method to solves ystems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of ?AMG is a composite solver built through a bootstrapstrategyaimedtoobtainadesiredconvergencerate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on weighted matching in a graph, successfully exploited for reordering algorithms in sparse direct methods to enhance diagonal dominance, and compatible relaxation. The proposed compatible matching process replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs).