空间分数阶微分方程的预处理迭代方法

Fractional calculus have been found very useful in various fields of science and engineering. Fractional differential equations (FDEs) have provided a powerful tool for the description of memory and hereditary properties of different substances. Because most of FDEs cannot be solved analytically, more and more works focus on their numerical solutions. One of the main characteristics of the fractional differential operator is nonlocality, which usually leads to a full coefficient matrix. Therefore the computational cost of the numerical solutions is much more expensive than that for the traditional integer order differential equations. In order to reduce the computational complexity and the huge memory requirement, developing efficient and robust algorithms for solving such kind of problems become more and more important. The purpose of this project is to design fast and reliable preconditioned iterative methods for such kind of large and structured linear systems. More precisely, we will study the following problems: (1) the development of efficient preconditioners for the linear systems arising from the space-fractional diffusion equations; (2) the locally refined composite mesh for the fractional differential equations whose solution may exhibit boundary layer and poor regularity, and the development of fast and faithful preconditioned iterative methods for the resulted linear systems; (3) the preconditioned Newton-Krylov methods for the complex nonlinear systems arising from the fractional nonlinear Schrodinger equations; (4) the preconditioned iterative methods for other related nonlocal models.

分数阶微积分理论和方法被广泛应用于科学和工程中的各个领域。分数阶导数能够比整数阶导数更准确地描述具有记忆和遗传性质的材料与物理过程。分数阶导数的一个主要特征是非局部性，这使得离散后得到的代数方程组通常是稠密的。因此，在数值求解方面，分数阶微分方程所需的运算量比整数阶微分方程要多得多。如何快速高效地求解这些大规模稠密线性方程组是当前急需解决的问题。本项目主要将针对这类结构方程组的高效预处理方法展开研究，具体研究内容如下：（1）针对空间分数阶扩散模型，基于系数矩阵的特殊结构，通过适当的等价变换，讨论和构造高效的预处理方法；（2）考虑带边界层的分数阶扩散方程，讨论和构造实用的网格局部加密方法，以及基于这些网格的离散方程组的预处理方法；（3）分数阶非线性薛定谔方程的预处理方法，构造预处理Newton-Krylov迭代方法；（4）其它相关非局部模型的预处理迭代方法。

近年来，分数阶微分方程被广泛应用于刻画自然界的反常扩散现象，但由于分数阶导数的非局部性质，离散后的代数方程往往是稠密的，这给大规模问题的数值求解带来了很大的困难。（1）我们考虑了带边界层的空间分数阶守恒扩散方程的数值求解，针对一种特殊的网格局部加密方法，提出了快速求解代数方程的预处理方法，并改进了网格加密方法。（2）我们讨论了分数阶非线性耦合薛定谔方程的数值求解，构造了内外迭代求解算法，提出了基于交替方向思想的预处理方法。（3）广义向后差分离散能有效克服传统线性多步法所潜在的稳定性问题及阶障碍，我们考虑了基于广义向后差分离散的非稳态变系数空间分数阶扩散方程的预处理方法，构造了基于Kronecker 积分裂的预处理子和对角补偿预处理子。（4）我们考虑了Levenberg-Marquardt方法的改进，提出了自适应多步Levenberg-Marquardt方法，有效降低Jacobi矩阵的计算次数，提高计算效率。（5）我们考虑了张量Tensor Train分解的加速，提出了σ-重排方法和新的分解策略，在减少计算量的同时也提升了算法的并行性。（6）我们考虑了压缩感知中采样矩阵的构造，提出了三种具有循环矩阵结构的高斯随机采样矩阵，利用快速FFT减少信号重建时间。