In this paper, we consider a classical form of optimal algebraic multigrid (AMG) interpolation that directly minimizes the two-grid convergence rate and compare it with the so-called ideal form that minimizes a certain weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap algebraic multigrid setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the BAMG method with sparse interpolation matrix (and spanning multiple levels) outperforms the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.
@article{arxiv.1703.10240,
title = {Optimal interpolation and Compatible Relaxation in Classical Algebraic Multigrid},
author = {James Brannick and Fei Cao and Karsten Kahl and Rob Falgout and Xiaozhe Hu},
journal= {arXiv preprint arXiv:1703.10240},
year = {2017}
}