Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs

TL;DR

This paper analyzes the convergence and complexity of an algebraic multilevel preconditioner for the graph Laplacian, using matching-based coarse graphs, and demonstrates its effectiveness through theoretical bounds and numerical experiments.

Contribution

It introduces a new algebraic multilevel preconditioner based on matching, with proven convergence rates and complexity estimates, extending to aggregates of multiple vertices.

Findings

Convergence rate approximately (1 - 1/ log n) on structured grids

O(n log n) complexity per cycle

Numerical results confirm theoretical estimates

Abstract

This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level convergence rate where the coarse level graph is obtained by matching. The two-level convergence of the method is then used to establish the convergence of an Algebraic Multilevel Iteration that uses the two-level scheme recursively. On structured grids, the method is proven to have convergence rate $\approx (1-1/\log n)$ and $O(n\log n)$ complexity for each cycle, where $n$ denotes the number of unknowns in the given problem. Numerical results of the algorithm applied to various graph Laplacians are reported. It is also shown that all the theoretical estimates derived for matching can be generalized to the case of aggregates containing more than two vertices.