Algebraic multigrid and algebraic multilevel methods: a theoretical comparison

We consider algebraic methods of the two‐level type for the iterative solution of large sparse linear systems. We assume that a fine/coarse partitioning and an algebraic interpolation have been defined in one way or another, and review different schemes that may be built with these ingredients. This includes algebraic multigrid (AMG) schemes, two‐level approximate block factorizations, and several methods that exploit generalized hierarchical bases. We develop their theoretical analysis in a unified way, gathering some known results, rewriting some other and stating some new. This includes lower bounds, that is, we do not only investigate sufficient conditions of convergence, but also look at necessary conditions.