面向E级计算的并行代数多重网格新型算法研究

Algebraic multigrid(AMG) is one of the most efficient sparse linear solvers for solving large scale linear systems arising from scientific and engineering computing. In the past decades, AMG play an important role in real world numerical simulations. However, with the community of high performance computing (HPC) entering into Petascale computing, the realistic applications become more and more complex. At the same time, the architectures also will become more complicated for the computers of exascale computing, and the number of CPU cores will reach to, may be, hundreds of millions, which is a huge number. Subject to the constraints both the complexity of applications and the complicated architectures, The performance of AMG solver will become a great challenge issue. In order to sustain the performance of AMG with the increasing of CPU cores and solution unknowns, novel algorithms and techniques need to be designed. The purpose of this project is to improve the robustness and the scalability of AMG by designing the components and developing performance optimization which are suite for both complicated architectures and complex applications. The topics in our research plans including, modeling of performance for AMG, high efficient AMG algorithms and optimization techniques with locality property which suite for fine grain parallel computation, robust AMG solver suite for the complex feature of applications, AMG solver development, and finally, the applications of AMG in typical simulations. The main feature of this project is follow the theme of "algorithms designing driven by architectures and applications", and the results of this project, as we hoping , will enhance the more complicated applications on Exascale computers in the future.

代数多重网格（AMG）是高性能科学与工程计算中不可或缺的共性快速算法，在实际应用中发挥重要作用。然而，面向E级计算，受限于"实际应用和体系结构"双重复杂性，AMG计算效率面临"并行度和计算规模扩大1000倍带来的可扩展性问题"以及"复杂体系结构带来的性能优化问题"的新挑战。本项目面向未来E级系统，依托P级和将要出现的百P级系统，立足实际数值模拟应用，研究适应于复杂体系结构和复杂应用特征的AMG新型算法。主要内容包括：适应于体系结构特征和AMG算法特征的性能评价方法和模型；适应细粒度并行、具有良好局部性的AMG算法和结点内性能优化方法；适应复杂应用特征、具备良好并行可扩展的AMG算法；研制AMG解法器；基于典型数值模拟应用，在百P级系统上进行验证。项目成果将为E级系统的实际应用提供高效AMG算法支撑，同时对探索"体系结构和实际应用"驱动的算法设计新模式具有重要意义。

本项目面向E级计算系统，针对AMG算法在超大规模计算下面临的算法与并行可扩展性挑战，研究适应于复杂体系结构和复杂应用特征的AMG新型并行算法，取得如下进展：.（1）针对实际数值模拟应用中大规模方程组序列的求解，提出一种多阶段自适应并行AMG算法。该算法根据序列中相邻方程组性质的差异性与相似性，自适应地选取AMG算法组件和参数，尽可能避免AMG算法中影响并行可扩展性能的因素，从而在保持健壮性的同时提高并行可扩展性。.（2）针对实际应用中一类多尺度稀疏矩阵求解，提出一种基于代数界面粗化和长距离拓展插值的并行AMG算法（ex-AI-AMG）。该算法在不改变并行可扩展性能的前提下，通过代数界面捕捉复杂应用的多尺度特征，并对代数界面采用特殊粗化和插值策略，从而提高算法适应复杂应用特征的能力。.（3）针对ICF数值模拟中辐射扩散耦合方程组求解，提出一类自适应ILU-AMG组合型并行预条件算法。该类算法基于应用中多物理耦合强弱程度和耦合结构特征，给出了AMG和ILU算法的使用条件，根据应用特征自适应地选取算法及其参数，提高计算效率。.（4）针对AMG算法组件在当前复杂体系结构上面临的浮点效率低的问题，提出稀疏矩阵计算性能模型及性能优化方法，改善光滑过程的浮点效率。.（5）设计一个面向E级计算的自动调优多重网格算法框架AutoMG。该框架充分利用应用与体系结构中隐含的各种多级结构（hierarchies），最大限度地权衡算法的收敛性与并行性、扩大算法设计与调优空间，改善多重网格算法对应用特征和超大规模并行计算的适应性。.（6）集成本项目研究成果，研制了并行代数多重网格解法器JXPAMG，支撑激光聚变和工程力学的应用软件在数万核上完成实际模型的大规模计算，显著提升了这些软件的计算效率，取得具有显示度的模拟成果。.本项目紧密围绕应用与体系结构双驱动的算法设计，基于“算法-体系结构-应用特征”的协同设计思路，建立了适应E级计算的多重网格算法框架，全面实现了项目研究目标，为未来E级超大规模计算提供了高效算法基础。