Optimal interpolation and Compatible Relaxation in Classical Algebraic Multigrid

Abstract

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.

arXiv comment23 pages, submitted to SISC

References in corpus (2)

• Algebraic Multigrid Methods
• An Adaptively Constructed Algebraic Multigrid Preconditioner for Irreducible Markov Chains

Cited by in corpus (2)

• On the Ideal Interpolation Operator in Algebraic Multigrid Methods
• The Role of Energy Minimization in Algebraic Multigrid Interpolation