Some Comments on Multigrid Methods for Computing Propagators

TL;DR

This paper discusses key conceptual principles for multigrid methods in lattice gauge theory, highlighting common violations in existing algorithms and emphasizing the importance of operator stability, symmetry considerations, and the distinction between vector spaces and their duals.

Contribution

It identifies fundamental principles that should guide the development of multigrid algorithms for propagator computations, pointing out violations in current methods.

Findings

Existing algorithms violate key principles of stability and symmetry.

Emphasizes the importance of distinguishing vector space from dual space.

Provides conceptual insights to improve multigrid methods.

Abstract

I make three conceptual points regarding multigrid methods for computing propagators in lattice gauge theory: 1) The class of operators handled by the algorithm must be stable under coarsening. 2) Problems related by symmetry should have solution methods related by symmetry. 3) It is crucial to distinguish the vector space $V$ from its dual space $V^*$. All the existing algorithms violate one or more of these principles.