For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson’s equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.

对于由偏微分方程离散化产生的许多线性方程组，构建高效的多重网格求解器具有挑战性。在这里，我们介绍了 EvoStencils，这是一种使用语法引导的遗传编程优化几何多重网格方法的新方法，这是一种受自然进化原理启发的随机程序优化技术。多重网格求解器表示为我们根据形式语法生成的数学表达式树。通过使用代码生成自动生成优化的实现，然后在目标平台上执行以测量所有相关的性能指标，从收敛和计算性能方面评估每个求解器的质量。基于此，使用基于非支配排序的选择执行多目标优化。为了并行评估大量求解器，将它们分布到多个计算节点。我们通过构建几何多重网格求解器来证明我们的实现的有效性，这些求解器能够在具有 Ivy Bridge 和 Broadwell 微体系结构的多核处理器上优于泊松方程和线性弹性边界值问题的手工方法，该问题具有多达 1600 万个未知数。