分数阶微分方程并行算法研究

Because of the non- locality of space and long tail effect of time on fractional differential operator, the computational complexity of the fractional differential operator’s numerical approximation has higher order than that of the traditional integer differential operator. This presents a new challenge for the fast numerical approximation of fractional differential equations (FDE). So there is a need for the study of parallel algorithms for fractional operators. This project tries to do research on the coarse-grained and fine-grained parallel algorithms of FDE, combining with the architectural features of parallel computer systems. Specific studies include the following three aspects: parallel algorithms and optimization of fractional ordinary differential equations, parallel algorithms of unbalanced space fractional and efficient implicit parallel iterative algorithm of FDE. In order to achieve scalable, coarse-grained task-level and fine-grained data-level parallel computing for the numerical approximation of FDE, the task allocation, load balancing, communication and memory access methods are designed and optimized for the corresponding parallel algorithms from the algorithm and implementation levels. The purpose of this project is to investigate the above parallel algorithms, which can effectively improve simulation speed and size of fractional differential problem. This will promote the development of theoretical and applied research in the relevant fields of fractional differential operator.

由于分数阶微分算子空间上的非局部性和时间上的长尾效应，与传统整数阶微分算子相比，其数值近似的时间复杂度有阶级的提高。这对分数阶微分方程的数值近似计算提出了新的挑战，需要研究针对分数阶算子的并行算法。本项目结合并行计算机体系结构特征，展开分数阶微分方程粗细粒度并行算法研究。具体研究内容包括以下三个方面：分数阶常微分方程并行算法及优化、非均衡空间分数阶并行算法和分数阶微分方程高效隐式并行迭代算法。从算法层和实现层对相应并行算法的任务分配、负载平衡、通信和访存方法进行设计和优化，实现分数阶微分方程数值近似的可扩展粗粒度任务级和细粒度数据级并行计算。本项目的目的是通过对上述并行算法的研究，能有效提高分数阶微分问题的数值模拟速度和求解规模，推动分数阶微分算子在各相关领域的理论和应用研究的发展。

项目完成分数阶常微分方程并行算法及其优化方法，非均衡空间分数阶微分方程并行算法和分数阶微分方程高效隐式并行迭代算法外，还设计并实现了粒子输运并行算法、流体力学新型并行算法和基于笛卡尔背景网格的快速网格变形算法，及申请了流体并行算法方面的专利。学术成果24项，其中期刊论文14篇，其中SCI检索9篇，一作或者通信作者7篇；会议论文5篇，第一发明人专利5个。在人才培养方面，指导本科毕业4名，协助指导博士毕业2名，硕士毕业5名，工程师晋升高级工程师2名，高级工程师晋升研究员1名。在学术交流方面，参加了HPC China 2016/2017、IHPCF 2017和Corphy 2016等会议和学术交流。