In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems with graph Laplacians. In earlier works such estimates were computed by solving global optimization problems, 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 the linear system on the spanning tree, and then we solve approximately 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.
翻译:在本文中,我们研究一个事后误差估计器,该测算器有助于用图解拉平面图为线性系统提供多层迭代解答器。在早期的工程中,这种估计是用解决全球优化问题来计算的,而这些问题在计算上可能非常昂贵。我们提出了一个新颖的战略来计算这些估计数,方法是在基于横贯树和相应的循环空间的图表上构造一个赫姆霍尔茨分解法。为了计算误差估计器,我们有效地解决了横贯树上的线性系统,然后我们解决了循环空间上一个大约最少的方块问题。正如我们所显示的那样,这样的测算器在某些假设下对稀有的图表具有近线性计算复杂性。做了数字实验,以证明拟议方法的有效性。