Algebraic Multigrid for Disordered Systems and Lattice Gauge Theories

TL;DR

This paper develops an algebraic multigrid approach for disordered lattice operators, especially in lattice gauge theories, using block LU decomposition and numerical optimization to improve preconditioning and relaxation methods.

Contribution

It introduces a novel algebraic multigrid framework for disordered systems, employing Schur complements and Monte Carlo-like optimization for effective coarse-grid operators.

Findings

Effective coarse-grid operators are constructed via Schur complements.

The method improves preconditioning and relaxation in disordered systems.

Optimized stencils depend on gauge paths and are computationally efficient.

Abstract

The construction of multigrid operators for disordered linear lattice operators, in particular the fermion matrix in lattice gauge theories, by means of algebraic multigrid and block LU decomposition is discussed. In this formalism, the effective coarse-grid operator is obtained as the Schur complement of the original matrix. An optimal approximation to it is found by a numerical optimization procedure akin to Monte Carlo renormalization, resulting in a generalized (gauge-path dependent) stencil that is easily evaluated for a given disorder field. Applications to preconditioning and relaxation methods are investigated.