Aggregation-based aggressive coarsening with polynomial smoothing

TL;DR

This paper introduces a scalable algebraic multigrid preconditioner for graph Laplacian systems, combining aggressive aggregation coarsening with polynomial smoothing to improve efficiency in solving discretized Poisson problems.

Contribution

It presents a novel combination of aggressive coarsening and polynomial smoothing within an algebraic multilevel framework for graph Laplacian systems.

Findings

The method is scalable and effective for large graph Laplacian systems.

Numerical results demonstrate improved convergence rates.

Applicable to finite element discretizations of Poisson problems.

Abstract

This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient iteration in the solve phase. We show that by combining these techniques it is possible to design a simple and scalable algorithm. Results of the algorithm applied to graph Laplacian systems arising from the standard linear finite element discretization of the scalar Poisson problem are reported.