Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying problem and its discretization, algebraic multigrid aims to be less intrusive, requiring less knowledge about the origin of the linear system. A key step in algebraic multigrid is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In work by MacLachlan and Saad, a constrained combinatorial optimization problem is used to define the "best" coarse grid within the setting of a two-level reduction-based algebraic multigrid method and is shown to be NP-complete. Here, we develop a new coarsening algorithm based on simulated annealing to approximate solutions to this problem, which yields improved results over the greedy algorithm developed previously. We present numerical results for test problems on both structured and unstructured meshes, demonstrating the ability to exploit knowledge about the underlying grid structure if it is available.

翻译：多层次技术是解决因离散部分差异方程式和其他问题产生的大型线性系统的有效方法。虽然几何多格格需要详细了解根本问题及其离散化,但代数多格旨在减少侵扰性,较少了解线性系统的来源。代数多格中的一个关键步骤是选择粗略/细微分解,目的是平衡迭代与成本的趋同。在MacLachlan和Saad的工作中,一个有限的组合优化问题被用来界定基于两级降级的代数多格方法设置中的“最佳”粗略格格格格,并显示其为NP的完整。在这里,我们开发了一种新的基于模拟线性算法的粗略法算法,以近似这一问题的解决办法,从而取得比以前开发的贪婪算法更好的结果。我们提出了在结构化和无结构的介质上测试问题的数字结果,表明如果具备的话,将利用关于底基网格结构的知识的能力。