低算子复杂度的高效并行AMG法及其在两类PDEs中的应用

It is an essential computational bottleneck to acquire the solution of large-scale linear systems arising from discretization of partial differential equations in science and engineering. Parallel algebraic multigrid (AMG) is one of the most common but popular solvers for these systems. The efficient parallel AMG method with low operator complexity is a hotspot but also a difficulty in AMG field..This project firstly develops an efficient parallel AMG method with low operator complexity and its theoretical analysis for high-dimensional complicated Poisson-like equation, which is a fundamental issue to AMG. Secondly, efficient parallel AMG methods with low operator complexity are constructed for the strongly multi-physical coupled and multi-scaled high-dimensional multi-group radiation diffusion equations and linear systems from finite element discretization of high-dimensional multi-term time-space fractional differential equation, and persuasive supports for the proposed algorithms are also provided. At last, we build the aforementioned efficient parallel AMG solvers under some famous computing platforms both at home and abroad. Here are numerous innovations and difficulties, such as serious impacts to aforementioned AMG algorithms from the non-Galerkin coarse grid, complicated multi-physical couplings, strongly multiple scales and nonlocality of the time-space fractional differential operator. We need to explore and present some novel ways and new techniques to dispose of these difficulties. The achievements of the project will play a positive role in enriching and promoting AMG's parallel algorithms, theoretical analyses, software developments and practical applications.

偏微分方程组大规模离散化系统的求解是科学工程领域的计算瓶颈之一，并行代数多重网格（AMG）法是解决该瓶颈问题的一条有效途径，低算子复杂度的高效并行AMG法是目前AMG领域的研究热点。.本项目首先针对高维复杂Poisson类方程的离散化系统，研究具有低算子复杂度的高效并行AMG法及相关理论，这属于AMG领域的基础性问题；接着针对一类具有强耦合性、强多尺度性的高维多群辐射扩散方程组和高维时间多项时空分数阶微分方程的有限元代数系统，设计具有低算子复杂度的高效并行AMG法，并提供理论支撑；最后基于新算法和国内外相关计算平台，研发高效并行AMG解法器。这些研究涉及许多难点，如非Galerkin型、多物理量的强耦合性、强多尺度性、时空分数阶微分算子的非局部性等对高效并行AMG法带来的影响，因此需要探索和发展一些新方法、新技术、新技巧。所获成果将对AMG的并行算法、理论、软件及实际应用起着积极作用。

低算子复杂度的高效并行AMG法是科学与工程计算领域的重要研究课题。本项目首先针对两种复杂Poisson类方程的时空有限元格式，理论上获得其系数矩阵的条件数估计式，并利用快速Fourier变换，构造和分析了一种低复杂度（甚至渐近最优）的自适应AMG法，数值实验结果表明新算法具有一致收敛性和高效性；接着，针对一种多介质轴对称二维三温辐射扩散问题的保对称有限体元格式，设计了两种具有简单粗空间的非重叠型区域分解预条件子，并在一定的数学假设下获得预条件系统的渐近最优条件数估计式，数值实验结果验证了理论估计式的正确性及并行解法器的良好算法与并行可扩展性；最后，针对二维空间分数阶扩散问题的时空有限元格式，发展了一种基于时间依赖的时间传播算子的时间维度上多重网格规约（MGRIT）算法，在时间传播算子的稳定性与同时可对角化的假设下，给出其两水平收敛理论，并通过数值实验验证理论的正确性、MGRIT相对parareal与串行时间步进算法的可观加速比。该工作在一定程度上对并行多层迭代算法及其理论进行完善。

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