A Posteriori Error Estimates for Multilevel Methods for Graph Laplacians

Abstract

In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems of graph Laplacians. In earlier works such estimates were computed by solving a perturbed global optimization problem, which could be computationally expensive. We propose a novel strategy to compute these estimates by constructing a Helmholtz decomposition on the graph based on a spanning tree and the corresponding cycle space. To compute the error estimator, we solve efficiently a linear system on the spanning tree and then a least-squares problem on the cycle space. As we show, such estimator has a nearly-linear computational complexity for sparse graphs under certain assumptions. Numerical experiments are presented to demonstrate the efficacy of the proposed method.