Performance tests and analyses are critical to effective high-performance computing software development and are central components in the design and implementation of computational algorithms for achieving faster simulations on existing and future computing architectures for large-scale application problems. In this article, we explore performance and space-time trade-offs for important compute-intensive kernels of large-scale numerical solvers for partial differential equations (PDEs) that govern a wide range of physical applications. We consider a sequence of PDE-motivated bake-off problems designed to establish best practices for efficient high-order simulations across a variety of codes and platforms. We measure peak performance (degrees of freedom per second) on a fixed number of nodes and identify effective code optimization strategies for each architecture. In addition to peak performance, we identify the minimum time to solution at 80% parallel efficiency. The performance analysis is based on spectral and p-type finite elements but is equally applicable to a broad spectrum of numerical PDE discretizations, including finite difference, finite volume, and h-type finite elements.

中文翻译：

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高性能 PDE 求解器的可扩展性

性能测试和分析对于有效的高性能计算软件开发至关重要，并且是设计和实现计算算法的核心组件，以实现对现有和未来计算架构的更快模拟，以解决大规模应用问题。在本文中，我们探讨了用于控制广泛物理应用的偏微分方程 (PDE) 的大规模数值求解器的重要计算密集型内核的性能和时空权衡。我们考虑一系列 PDE 驱动的烘焙问题，旨在为跨各种代码和平台的高效高阶模拟建立最佳实践。我们在固定数量的节点上测量峰值性能（每秒自由度），并为每个架构确定有效的代码优化策略。除了峰值性能之外，我们还确定了在 80% 并行效率下解决问题的最短时间。性能分析基于谱和 p 型有限元，但同样适用于广泛的数值 PDE 离散化，包括有限差分、有限体积和 h 型有限元。