My question is around the efficiency of AMG. In which case AMG can perform better,as solver or as a preconditioner(for example a Krylov space method as CG)? Assume the case of elliptic pdes.
Most people use (algebraic as well as geometric) multigrid as preconditioners these days. It's an empirical observation that that leads to faster convergence in terms of iterations, given that in a typical CG iteration, applying a multigrid preconditioner takes up the vast majority of the effort compared to all other operations. In other words, you get the effect of the CG iteration basically for free, and it turns out that that is apparently beneficial because it covers components of the solution for which multigrid is slow to converge.