求解线性离散不适定问题的改进Tikhonov算法

发布时间：2018-07-24 17:25

【摘要】：随着科学技术和生产水平的发展,不适定问题广泛出现在地球物理、自动控制等多种领域。正则化方法是求解此类问题近似解的有效算法。本文研究求解线性离散不适定问题的正则化算法。论文主要工作包括:首先,基于求解线性离散不适定问题的Tikhonov正则化方法和TSVD正则化方法,分析两者的优缺点,提出了一种求解线性离散不适定问题的混合Tikhonov正则化算法。其次,将Fractional Tikhonov正则化算法应用于投影算法,提出了求解大规模线性离散不适定问题的Arnoldi-Fractional Tikhonov正则化算法。再次,进一步提出了广义Arnoldi-Fractional Tikhonov正则化算法和限制值域的Arnoldi-Fractional Tikhonov正则化算法。而且,论文对所提出的算法都编写程序实现算法,针对经典算例,进行了数值试验和比较。数值试验结果表明了新算法是有效且具有优势的。
[Abstract]:With the development of science and technology and production level, ill-posed problems are widely used in many fields such as geophysics, automatic control and so on. Regularization method is an effective algorithm for solving approximate solutions of this kind of problems. In this paper, a regularization algorithm for solving linear discrete ill-posed problems is studied. The main work includes: firstly, based on the Tikhonov regularization method and TSVD regularization method for linear discrete ill-posed problems, a hybrid Tikhonov regularization algorithm is proposed to solve the linear discrete ill-posed problems by analyzing their advantages and disadvantages. Secondly, the Fractional Tikhonov regularization algorithm is applied to the projection algorithm, and a Arnoldi-Fractional Tikhonov regularization algorithm is proposed for solving large scale linear discrete ill-posed problems. Thirdly, the generalized Arnoldi-Fractional Tikhonov regularization algorithm and the Arnoldi-Fractional Tikhonov regularization algorithm are proposed. In addition, the algorithms are programmed to realize the algorithms, and numerical experiments and comparisons are carried out for classical examples. Numerical results show that the new algorithm is effective and has advantages.
【学位授予单位】：南京航空航天大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241

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