Fast algebraic multigrid for block-structured dense and Toeplitz-like-plus-Cross systems arising from nonlocal diffusion problems

TL;DR

This paper introduces a fast algebraic multigrid method tailored for block-structured dense and Toeplitz-like-plus-Cross systems from nonlocal diffusion problems, demonstrating convergence and efficiency with $ ext{O}(N ext{log} N)$ complexity.

Contribution

It provides the first detailed convergence proof and efficient solution approach for Toeplitz-like-plus-Cross systems using a simple AMG framework.

Findings

Convergence of the two-grid method is proven for these systems.

Numerical experiments confirm the method's efficiency and convergence.

The approach achieves $ ext{O}(N ext{log} N)$ computational complexity.

Abstract

Algebraic multigrid (AMG) is one of the most efficient iterative methods for solving large sparse system of equations. However, how to build/check restriction and prolongation operators in practical of AMG methods for nonsymmetric {\em sparse} systems is still an interesting open question [Brezina, Manteuffel, McCormick, Runge, and Sanders, SIAM J. Sci. Comput. (2010); Manteuffel and Southworth, SIAM J. Sci. Comput. (2019)]. This paper deals with the block-structured dense and Toeplitz-like-plus-Cross systems, including {\em nonsymmetric} indefinite, symmetric positive definite (SPD), arising from nonlocal diffusion problem and peridynamic problem. The simple (traditional) restriction operator and prolongation operator are employed in order to handle such block-structured dense and Toeplitz-like-plus-Cross systems, which is convenient and efficient when employing a fast AMG. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such SPD situations. The numerical experiments are performed in order to verify the convergence with a computational cost of only $\mathcal{O}(N \mbox{log} N)$ arithmetic operations, by using few fast Fourier transforms, where $N$ is the number of the grid points. To the best of our knowledge, this is the first contribution regarding Toeplitz-like-plus-Cross linear systems solved by means of a fast AMG.