大型稀疏奇异复对称线性系统的高效迭代法研究

Large sparse singular complex symmetric linear systems arise from varieties of complicated problems, such as the problems with Neumann boundary conditions, the PDE problems with periodic boundary conditions, the constrained optimization problems, the least square problems and the computing genetic problems, and so on. Therefore, efficient iterative methods are crucial to solve this kind of linear systems efficiently. However, up to now, very few algorithms can be used to solve large sparse singular complex symmetric linear systems efficiently. In order to solve the singular complex linear systems efficiently, we will first study the efficient iterative methods for nonsingular complex linear systems and the properties of the singular complex symmetric coefficient matrix. Then we will construct new iterative methods by making use of the matrix splitting technique and the extrapolation or parameterized technique. After modifying the splitting matrices from the new iterative methods, we will obtain some new preconditioners more approximate to the original coefficient matrices. By making use of the null space method, the similarity transformation or the orthogonal decomposition method, we will give a detailed analysis for the semi-convergence properties for the new methods and the eigen-properties for the preconditioned matrices. The results obtained by this project not only supply us a class of efficient algorithms, but also propose some strategies to solve the nonsingular complex systems. Furthermore, those new methods may be beneficial to fast solve the associated problems.

在对Neumann边值问题、周期边界PDE问题、约束优化问题、最小二乘以及一些遗传学的计算问题等进行求解时，问题将转化为对大型稀疏奇异复对称线性系统的求解问题。因此，设计大型稀疏奇异复对称线性系统的高效迭代算法对解决这些实际问题起着至关重要的作用。然而，到目前为止，适合这类线性系统求解的高效算法并不多见。为了有效地求解这类问题，本项目拟借鉴非奇异复线性系统的求解思路，通过分析系数矩阵的特殊结构，利用矩阵分裂并结合外推或参数化技术构造新的分裂迭代算法，并把对应的分裂矩阵作为预条件子。通过对该预条件子进行修正，设计出求解这类线性系统的高效迭代算法和预处理技术。结合零空间方法和矩阵正交相似分解等相关理论，给出所提算法的半收敛性和预处理矩阵的谱性质。本项目的研究成果将得到大型稀疏奇异复对称线性系统的高效求解算法，并为非奇异问题的求解提供一定的思路，进而为相关领域实际问题的有效求解提供算法保障。

本研究主要讨论了一类大型稀疏奇异复线性系统的高效求解算法，具体为：.首先，我们通过用复松弛参数外推已经存在的修正的MHSS迭代方法，得到了求解一类奇异复线性系统的复-外推MHSS（CMHSS）迭代算法。接着研究了CMHSS迭代算法的半收敛性以及最优迭代参数。并进一步给出了非奇异复线性系统的收敛性条件。数值实验验证了CMHSS迭代算法在求解非奇异和奇异的复线性系统时的有效性。.其次，对于一类奇异的复线性系统的求解，提出了参数化的广义的MHSS（PGMHSS）迭代方法，通过对谱半径的最小上界进行最优化的办法，研究了PGMHSS算法的半收敛性和最优参数的选取，并用数值实验验证了PGMHSS迭代方法的可行性与有效性。.最后，我们关注了时间-空间分数阶对流扩散方程的快速求解方法。对该问题的研究，首先我们分别利用平移的和不平移的Grünwald-Letnikov（GL）逼近方法对时间和空间分数阶导数项进行离散，进而得到一类大型稠密的线性系统。针对这类系统的系数矩阵的特殊结构，我们提出了带状预条件子，接着分析了该预条件子的相关理论性质。最后用数值实验验证了带状预条件子在合适带宽的条件下的有效性。