Although multigrid is asymptotically optimal for solving many important partial differential equations, its efficiency relies heavily on the careful selection of the individual algorithmic components. In contrast to recent approaches that can optimize certain multigrid components using deep learning techniques, we adopt a complementary strategy, employing evolutionary algorithms to construct efficient multigrid cycles from proven algorithmic building blocks. Here, we will present its application to generate efficient algebraic multigrid methods with so-called \emph{flexible cycling}, that is, level-specific smoothing sequences and non-recursive cycling patterns. The search space with such non-standard cycles is intractable to navigate manually, and is generated using genetic programming (GP) guided by context-free grammars. Numerical experiments with the linear algebra library, \emph{hypre}, demonstrate the potential of these non-standard GP cycles to improve multigrid performance both as a solver and a preconditioner.

翻译：尽管多重网格方法在求解许多重要偏微分方程时具有渐近最优性，但其效率高度依赖于算法各组件的精细选择。与近期采用深度学习技术优化特定多重网格组件的方法不同，我们采用互补策略——利用进化算法从经过验证的算法构建模块中构造高效的多重网格循环。本文将此方法应用于生成具有所谓"灵活循环"（即层级特异性平滑序列与非递归循环模式）的高效代数多重网格方法。包含此类非标准循环的搜索空间难以通过人工方式导航，我们采用基于上下文无关文法的遗传编程（GP）进行引导生成。基于线性代数库hypre的数值实验表明，这些非标准GP循环在提升多重网格作为求解器与预处理器的性能方面具有显著潜力。