Generalizing Reduction-Based Algebraic Multigrid

TL;DR

This paper enhances reduction-based algebraic multigrid methods to improve robustness and performance on non-diagonally dominant matrices, using strength of connection, SPAI, and interpolation techniques, with demonstrated success on diffusion problems.

Contribution

The authors introduce modifications to classical AMGr algorithms, improving their robustness and efficiency for non-diagonally dominant matrices through new techniques and analysis.

Findings

Improved robustness on non-diagonally dominant matrices.

Effective performance on anisotropic diffusion problems.

Maintained control of computational costs.

Abstract

Algebraic Multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.