Optimal interpolation and Compatible Relaxation in Classical Algebraic Multigrid

TL;DR

This paper introduces a new optimal algebraic multigrid interpolation method that minimizes convergence rate, compares it with ideal forms, and demonstrates superior performance in scalar diffusion problems with variable coefficients.

Contribution

It develops a sharp measure for coarse grid quality, derives a lower bound on convergence, and proposes a bootstrap AMG setup for sparse optimal interpolation.

Findings

Sparse optimal interpolation outperforms dense ideal interpolation in numerical tests.

The new measure guarantees a lower bound on two-grid convergence rate.

Bootstrap AMG effectively computes sparse approximations for multilevel methods.

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.