预条件迭代法求解矩阵方程

发布时间：2018-04-18 16:51

本文选题：矩阵方程 + 预条件处理　；参考：《华东理工大学》2017年硕士论文

【摘要】：在工程应用与科学计算的诸多领域中,矩阵计算都有着十分广泛的应用:比如Lyapunov方程、Sylvester方程等矩阵方程的数值求解问题,在控制系统的设计和分析、偏微分方程边值问题、图像识别以及大规模线性动力系统、模型降价等领域中都扮演着十分重要的角色;因此,研究矩阵方程的一类数值迭代求解算法具有非常重要的实用价值。在矩阵方程的实际求解问题中,其系数矩阵往往是大型稀疏矩阵,这使得迭代法成为研究该类问题求解的最常用方法。但随着矩阵方程系数矩阵规模的增大,采用经典迭代法进行求解就会出现计算量巨大、收敛速度十分缓慢的情况,预条件技术则能够很好地处理好这样一类问题。而本文正是基于预条件处理思想,提出了一类预条件平方Smith迭代算法数值求解矩阵方程,具体内容如下:利用交替方向隐式法即ADI迭代法构造预条件算子并处理矩阵方程,将该原矩阵方程转化为谱性质更好的方程,然后利用平方Smith法迭代产生Krylov子空间中的低秩逼近数值解形式,同时给出预条件平方Smith算法以及误差和残量的估计;最后,在数值实验部分对工程领域中最常见的两类矩阵方程Lyapunov方程和Sylvester方程进行数值编程实现,用实验结果验证本文提出的一类预条件迭代法比传统的Jacobi迭代法、Global-Krylov子空间法、Block-Krylov子空间法等迭代法具有更好的收敛效果,说明本文提出的预条件平方Smith算法是十分有效的。
[Abstract]:In many fields of engineering application and scientific calculation, matrix calculation has been widely used: for example, the numerical solution of matrix equations such as Lyapunov equation and Sylvester equation, the design and analysis of control system, the boundary value problem of partial differential equation, etc.Image recognition, large-scale linear dynamical systems and model reduction play a very important role, so it is very important to study a kind of numerical iterative algorithm for solving matrix equations.In the practical solving problem of matrix equation, the coefficient matrix is usually a large sparse matrix, which makes iterative method the most common method to solve this kind of problem.However, with the increase of matrix size of coefficient matrix of matrix equation, using classical iterative method to solve the problem will result in a huge amount of computation and a very slow convergence rate. The preconditioning technique can deal with this kind of problem well.Based on the idea of preconditioned treatment, a class of preconditioned square Smith iterative algorithm is proposed to solve the matrix equation numerically. The main contents are as follows: the alternating direction implicit method, I. E. the ADI iterative method, is used to construct the preconditioned operator and to deal with the matrix equation.The original matrix equation is transformed into a better spectral equation, and then the numerical solution of low rank approximation in Krylov subspace is generated iteratively by using the square Smith method. The preconditioned square Smith algorithm and the estimation of error and residual amount are also given.In the part of numerical experiment, two kinds of matrix equations, Lyapunov equation and Sylvester equation, are numerically programmed in the field of engineering.The experimental results show that the proposed preconditioned Smith method is more effective than the traditional Block-Krylov subspace method, which shows that the preconditioned square Smith method proposed in this paper is very effective.
【学位授予单位】：华东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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