求解非对称正定线性方程组的多项式预处理方法

发布时间：2018-05-27 05:45

本文选题：Krylov子空间方法 + 非对称正定矩阵　；参考：《兰州大学》2017年硕士论文

【摘要】：科学计算和工程应用中的许多问题往往需要求解大规模稀疏线性方程组。Krylov子空间方法是求解此类线性方程组的一种重要方法。高效的预处理子能够显著地加快Krylov子空间方法的迭代收敛速度,因此预处理技术方面的研究具有十分重要的理论意义和实际应用价值。在本文中,我们设计了一类多项式预处理子,用于加快Krylov子空间方法求解非对称正定线性方程组时的收敛速度。该类预处理子是基于求解线性方程组的分裂迭代法构造的。在分裂迭代法收敛时,方程组系数矩阵的逆可以用迭代矩阵的级数表示出来,取其部分和后即得到本文的多项式预处理子。我们利用几种不同的分裂迭代格式构造出了相应的m步多项式预处理子。理论分析以及数值实验结果表明,在求解非对称正定线性方程组时,应用该类预处理子的GMRES方法十分有效。
[Abstract]:Many problems in scientific computing and engineering applications often need to solve the large-scale sparse linear equation group.Krylov subspace method is an important method to solve such linear equations. The efficient preprocessor can significantly accelerate the iterative convergence rate of the Krylov subspace method, so the research of preprocessing technology has ten. In this paper, we have designed a class of polynomial preprocessors to speed up the convergence rate of the Krylov subspace method for solving unsymmetric positive definite linear equations. The class preconditioner is based on the split iterative method for solving linear equations. The inverse of the coefficient matrix of the process group can be expressed by the series of iterative matrices, and the polynomial preprocessor of this paper is obtained. We use several different splitting iterations to construct the corresponding M step polynomial preconditioner. The theoretical analysis and numerical experiment results show that the solution of the asymmetric positive definite linear equations is obtained. The GMRES method using this kind of preprocessor is very effective.
【学位授予单位】：兰州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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