高阶广义k-对角线性系统的异构协同并行求解算法研究与探索

Today high performance and low energy consumption advantages of heterogeneous computing systems are approved increasingly. Large-scale heterogeneous computing systems developed by our country have a leading position in the world. But the parallel numerical simulation application softwares processed on these systems are still relatively lack. The core problem of numerical simulation is solving the linear systems. The linear systems arising from chemical process, nuclear physics, heat conduction, and circuit simulation have generalized k -diagonal features. So, we will study an efficient parallel solving algorithm for these higher-order generalized k–diagonal linear systems on the large-scale heterogeneous computing systems. Firstly, the preconditioned techniques will be studied for generalized diagonal features of liear systems, and the parallel data model will be formed according to the architecture of the heterogeneous computing systems. Then the hybrid solving algorithms on the heterogeneous computing systems are designed, which can be executed in multi-level parallel, furthermore, the key factors affecting the performance of solving higher-order generalized k –diagonal linear systems will be analyzed, and the optimization tactics of the data access and communications will be explored. Lastly in order to improve the development efficiency and performance of numerical simulation applications, the common function library will be developed, which can shield the details of cooperative programming on the heterogeneous computing systems. The numerical simulation applications using our algorithms will be tested on “TianHe1A-HN” supercomputer. The research results of the project can improve the service ability of the large-scale heterogeneous computing systems developed by our country and promote sofrware development for scientific computing and engineering application on these heterogeneous platforms.

目前异构计算系统的高性能和低能耗优势日渐得到业界认可，我国国产大规模异构计算机系统在国际上已处于领先地位，但与之相适应的并行数值模拟应用软件尚较为缺乏。数值模拟计算中的核心问题是线性系统求解，而在化学过程、热传导、电路模拟以及核物理模拟等许多领域的数值模拟中需要求解的线性系统具有广义k-对角特征。为此，本项目针对高阶广义k-对角线性系统在大规模异构计算系统上的高效求解展开研究。首先针对广义对角特征和异构计算系统的体系结构研究矩阵的预处理技术，构建并行的数据模型；然后设计适合异构处理器多层次协同并行的求解算法，并探索影响求解性能的关键因素—数据访问和通信的优化策略；最后力争形成可屏蔽异构协同编程细节的共性函数库，以提高数值模拟应用的开发效率和求解性能。本项目研究成果将在“天河一号”超级计算机上进行实验验证，这将对提高国产大规模异构计算系统在科学计算和工程应用方面的服务水平起到一定的推动作用.

随着数值模拟在科学计算和工程应用中的地位突出及普遍使用，许多行业及领域对数值模拟的软件应用和开发都产生了强烈的需求，而线性系统的求解方法是数值模拟的核心。由于不同的问题得到特征各异的线性系统，在化学过程、热传导、电路模拟以及核物理模拟等许多领域产生的线性系统大多具有稀疏特征。但是随着数据网格的复杂化和问题域边界的不规则性，有时候产生的线性系统的稀疏模式较为多样。随着问题规模的扩大和模拟的精细化，要求解的线性系统的阶数越来越高，有的甚至达到百亿以上，对求解的性能提出了更高的要求。但是目前通用的求解算法很难满足如此高阶的稀疏线性系统的求解性能要求。.目前大规模异构计算系统由于其经济和高效性，已逐步成为各种数值模拟计算普遍选用的高性能计算平台。针对高阶稀疏线性系统的多样性，研究大规模稀疏矩阵的压缩存储方法和其核心运算SpMV在CPU+GPU异构平台上的快速并行算法，以及针对在科学计算和工程应用中广泛存在的准三对角和块三角稀疏线性系统，研究一种高性能并且具有较好鲁棒性的混合并行求解算法，并针对异构体系结构的特征，从数据划分、任务分配和协同编程等方面研究求解算法的异构并行化策略，以充分发挥异构计算系统的计算效率。

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