A Posteriori Error Estimates for Multilevel Methods for Graph Laplacians

07/01/2020
by   Xiaozhe Hu, et al.
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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.

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