An Adaptive Multigrid Method Based on Path Cover

TL;DR

This paper introduces PC-αAMG, an adaptive multigrid method that efficiently solves linear systems from graph Laplacians and PDE discretizations by using path covers to form aggregations, resulting in near-optimal performance.

Contribution

The paper presents a novel adaptive multigrid approach based on path cover aggregation, improving efficiency and robustness for solving weighted graph Laplacian systems.

Findings

Achieves nearly optimal V-cycle performance for graph Laplacian systems.

Demonstrates robustness on ill-conditioned graph problems.

Provides a low-cost multilevel hierarchy rebuilding mechanism.

Abstract

We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic partial differential equations. The PC-$\alpha$AMG is based on unsmoothed aggregation AMG (UA-AMG). To preserve the structure of smooth error down to the coarse levels, we approximate the level sets of the smooth error by first forming vertex-disjoint path cover with paths following the level sets. The aggregations are then formed by matching along the paths in the path cover. In such manner, we are able to build a multilevel structure at a low computational cost. The proposed PC-$\alpha$AMG provides a mechanism to efficiently re-build the multilevel hierarchy during the iterations and leads to a fast nonlinear multilevel algorithm. Traditionally, UA-AMG requires more sophisticated cycling techniques, such as AMLI-cycle or K-cycle, but as our numerical results show, the PC-$\alpha$AMG proposed here leads to nearly optimal standard V-cycle algorithm for solving linear systems with weighted graph Laplacians. Numerical experiments for some real world graph problems also demonstrate PC-$\alpha$AMG's effectiveness and robustness, especially for ill-conditioned graphs.