大规模分数阶微分系统的高性能并行算法研究

Large-scale fractional differential systems arise from various applications, including the simulation of integrated circuits with superconductor material, anomalous diffusion of ions in nerve cell and micro-electromechanical systems. Because of the nonlocal property of the fractional differential operator, it might not achieve the application goal by using the traditional computational methods such as finite difference method to solve large-scale or super-large scale fractional differential systems. The main reason is that the computing time is too long and the accumulated errors are too large. One possible way to address the issue is to study high-performance parallel algorithms in terms of high speed parallel computers. In this project, we firstly give a new waveform relaxation method according to the nonlocal property of the fractional differential operator, and analyze the convergence rate of the proposed method and the computation cost; Secondly, we study the parareal method of fractional differential equations, and analyze the convergence, the stability and the parallel efficiency of the method; Third, we study the parareal waveform relaxation method of fractional functional differential equations and then discuss the convergence of the method and the impact of the functional parts on the errors; Finally, we establish experiment platform of large-scale fractional differential systems to make a lot of numerical experiments and improve the performance of the proposed methods. We hope that some new ideas and new methods will have been put forward on scientific calculation of fractional differential equations through this research.

大规模分数阶微分系统的来源非常广泛，例如含有超导材料的集成电路模拟，神经细胞中离子的反常扩散过程模拟和微电子力学系统等。由于分数阶微分算子是非局部算子，使得利用有限差分法等传统的数值方法求解大规模或超大规模分数阶微分系统时，都因计算时间过长、计算误差积累过大，难以达到实际应用的目的。解决该困难的一个方向是研究适合高速并行计算机的高性能并行算法。本课题拟根据分数阶微分算子的非局部性，首先给出一种新的波形松弛格式，分析算法的收敛速度及计算成本；其次，研究分数阶微分方程的parareal方法，分析算法的收敛性、稳定性及并行效率；再次，研究分数阶泛函微分方程的parareal波形松弛方法，分析算法的收敛性及泛函项对计算误差的影响；最后，建立大规模分数阶微分系统的数值测试平台，进行大量数值模拟，验证并改进所提各种新算法的有效性。通过本课题的研究，有望为分数阶微分方程的科学计算提供新思路和新方法。

大规模分数阶微分系统的来源非常广泛，例如含有超导材料的集成电路模拟，神经细胞中离子的反常扩散过程模拟和微电子力学系统等。由于分数阶微分算子是非局部算子，使得利用有限差分法等传统的数值方法求解大规模或超大规模分数阶微分系统时，都因计算时间过长、计算误差积累过大，难以达到实际应用的目的。解决该困难的一个主要方向是研究适合高速并行计算机的高性能并行算法。本课题根据分数阶微分算子的非局部性，首先给出了一种新的波形松弛算法，即基于分数阶算子的窗口波形松弛方法，分析了算法的收敛速度及计算成本；其次，研究了分数阶泛函微分方程的波形松弛算法，分析算法的收敛性及泛函项对计算误差的影响；最后，建立大规模分数阶微分系统的数值测试平台，进行大量数值模拟，验证并改进所提各种新算法的有效性。我们希望本课题的研究为分数阶微分方程的科学计算提供新思路和新方法。