The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the other relies upon floating-point expansions and, hence, expands the intermediate precision. Instead of converting the entire solver into its ExBLAS-related implementation, we identify those parts that violate reproducibility/non-associativity, secure them, and combine this with the sequential executions. These algorithmic strategies are reinforced with programmability suggestions to assure deterministic executions. Finally, we verify these approaches on two modern HPC systems: both versions deliver reproducible number of iterations, residuals, direct errors, and vector-solutions for the overhead of less than 37.7 % on 768 cores.

中文翻译：

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混合编程环境中并行预处理共轭梯度的再现性

预处理共轭梯度法通常用于求解物理现象数值模拟中出现的线性方程组。在被广泛使用的同时，求解器也因其在计算残差时缺乏准确性而闻名。在本文中，我们提出了两种源自 ExBLAS 项目的算法解决方案，以提高求解器的准确性并确保其在混合 MPI + OpenMP 任务编程环境中的可重复性。一个基于 ExBLAS 并保留每一位信息直到最后舍入，而另一个依赖于浮点扩展，因此扩展了中间精度。我们不是将整个求解器转换为与 ExBLAS 相关的实现，而是识别那些违反再现性/非关联性的部分，保护它们，并将其与顺序执行结合起来。这些算法策略通过可编程性建议得到加强，以确保确定性执行。最后，我们在两个现代 HPC 系统上验证了这些方法：两个版本都提供可重复的迭代次数、残差、直接错误和向量解决方案，在 768 核上的开销低于 37.7%。