Multigrid methods despite being known to be asymptotically optimal algorithms, depend on the careful selection of their individual components for efficiency. Also, they are mostly restricted to standard cycle types like V-, F-, and W-cycles. We use grammar rules to generate arbitrary-shaped cycles, wherein the smoothers and their relaxation weights are chosen independently at each step within the cycle. We call this a flexible multigrid cycle. These flexible cycles are used in Algebraic Multigrid (AMG) methods with the help of grammar rules and optimized using genetic programming. The flexible AMG methods are implemented in the software library of hypre, and the programs are optimized separately for two cases: a standalone AMG solver for a 3D anisotropic problem and an AMG preconditioner with conjugate gradient for a multiphysics code. We observe that the optimized flexible cycles provide higher efficiency and better performance than the standard cycle types.

翻译：尽管多重网格方法被公认为渐近最优算法，但其效率依赖于各组成部分的精心选择。此外，这些方法大多局限于V型、F型和W型等标准循环类型。我们利用语法规则生成任意形状的循环结构，其中平滑算子及其松弛权重在循环的每个步骤中独立选择。我们将其称为柔性多重网格循环。这些柔性循环通过语法规则应用于代数多重网格（AMG）方法，并采用遗传编程进行优化。柔性AMG方法已在hypre软件库中实现，并针对两种场景分别优化：针对三维各向异性问题的独立AMG求解器，以及用于多物理场代码的共轭梯度AMG预处理器。实验表明，优化后的柔性循环相较于标准循环类型具有更高的效率和更优的性能。