Non-Galerkin Multigrid Based on Sparsified Smoothed Aggregation

Algebraic multigrid (AMG) methods are known to be efficient in solving linear systems arising from the discretization of partial differential equations and other related problems. These methods employ a hierarchy of representations of the problem on successively coarser meshes. The coarse-grid operators are usually defined by (Petrov--)Galerkin coarsening, which is a projection of the original operator using the restriction and prolongation transfer operators. Therefore, these transfer operators determine the sparsity pattern and operator complexity of the multigrid hierarchy. In many scenarios the multigrid operators tend to become much denser as the coarsening progresses. Such behavior is especially problematic in parallel AMG computations, where it imposes an expensive communication overhead. In this work we present a new algebraic technique for controlling the sparsity pattern of the operators in the AMG hierarchy, independently of the choice of the restriction and prolongation. Our method is based on...