SIAM Journal on Scientific Computing, Ahead of Print.
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 a linear system efficiently on the spanning tree and then a least-squares problem on the cycle space. As we show, such an 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.

中文翻译：

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图拉普拉斯算子的多级方法的后验误差估计

SIAM 科学计算杂志，提前印刷。
在本文中，我们研究了后验误差估计器，它有助于图拉普拉斯算子的线性系统的多级迭代求解器。在早期的工作中，这样的估计是通过解决扰动全局优化问题来计算的，这在计算上可能很昂贵。我们提出了一种新的策略，通过基于生成树和相应的循环空间在图上构建亥姆霍兹分解来计算这些估计值。为了计算误差估计量，我们在生成树上有效地求解线性系统，然后在循环空间上求解最小二乘问题。正如我们所展示的，在某些假设下，这样的估计器对于稀疏图具有近乎线性的计算复杂度。数值实验被提出来证明所提出方法的有效性。