Generalizing Reduction-Based Algebraic Multigrid

12/16/2022
by   Tareq Zaman, et al.
0

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.

READ FULL TEXT

page 1

page 13

page 18

research
02/12/2020

Algebraic multigrid block preconditioning for multi-group radiation diffusion equations

The paper focuses on developing and studying efficient block preconditio...
research
01/06/2022

Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Domain decomposition methods are among the most efficient for solving sp...
research
03/12/2020

Learning Algebraic Multigrid Using Graph Neural Networks

Efficient numerical solvers for sparse linear systems are crucial in sci...
research
05/27/2021

Coarse-Grid Selection Using Simulated Annealing

Multilevel techniques are efficient approaches for solving the large lin...
research
07/01/2023

Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems

This paper focuses on developing a reduction-based algebraic multigrid m...
research
05/10/2023

Algebraic multigrid methods for metric-perturbed coupled problems

We develop multilevel methods for interface-driven multiphysics problems...
research
03/01/2023

Robust and Practical Solution of Laplacian Equations by Approximate Elimination

We introduce a new algorithm and software for solving linear equations i...

Please sign up or login with your details

Forgot password? Click here to reset