A Theory of Relaxation-Based Algebraic Multigrid

TL;DR

This paper introduces a relaxation-centric theoretical framework for algebraic multigrid methods, deriving exact coarse-level equations and transfer operators by modeling relaxation as a dynamical system.

Contribution

It presents a novel approach that unifies and extends existing AMG theories by focusing on relaxation dynamics and deriving key multigrid components analytically.

Findings

Recovered known results for non-symmetric systems

Derived exact expressions for coarse-level operators

Identified dynamical corrections to improve AMG performance

Abstract

Algebraic multigrid (AMG) methods derive their optimal efficiency from the interplay between a relaxation process and a corresponding coarse grid correction. In many standard formulations, relaxation and coarse-graining are analyzed and treated as largely separate of one another. Here we propose an alternative theoretical approach centered entirely on the relaxation process, which exposes its fundamental role in the coarse-graining of the fine-scale problem. By treating the relaxation of the error as a dynamical system and applying a dimensional-reduction procedure analogous to the Mori-Zwanzig-Nakajima formalism, we derive exact expressions for the coarse-level equations and the interpolation operations, as well as a natural way of computing complementary transfer operators. We illustrate the unifying nature of this framework by recovering several well-known results for general non-symmetric systems, including ideal and optimal restriction and interpolation, as well as the limiting case of exact elimination. We further emphasize the pivotal importance of compatible-relaxation and identify dynamical corrections that naturally arise in our theory, which have the potential to enhance the convergence, robustness, and adaptivity of future algebraic multigrid methods.