非线性不光滑问题的邻近修正Landweber迭代法研究

发布时间：2018-01-01 12:11

本文关键词：非线性不光滑问题的邻近修正Landweber迭代法研究　出处：《大连海事大学》2017年硕士论文　论文类型：学位论文

【摘要】：本文主要研究了关于非线性不光滑问题的邻近修正Landweber迭代法。在许多实际应用中都会涉及到反问题,反问题最大的特点就是不适定性。由于反问题的不适定性,所以需要用正则化方法来得到真实解的稳定近似解。在大规模非线性反问题中,通常采用迭代正则化方法,我们采用的迭代正则化方法是修正Landweber迭代法。然而,L~2范数罚函数可能会引起解过度光滑,解的过度光滑会导致与实际情况出现较大的偏差,为了克服这一困难,我们引入了稀疏正则化方法,增加一项L~1范数罚项,但是,这样会导致不光滑的情况。为了解决这一问题,我们引入了邻近算子,将邻近算子与修正Landweber迭代法相结合,用于解决非线性不光滑反问题。本文首先对邻近算子和修正Landweber迭代法的定义及其相关性质进行了回顾。在此基础上,将邻近算子与修正Landweber迭代法相结合,得到了最终的邻近修正Landweber迭代法。经过分析,我们证明了该算法的收敛性,并得到了相应的收敛率。最后,用该算法求解了参数识别问题,进行了一维、二维数值实验,验证了算法的有效性。理论分析和数值实验的结果表明:邻近修正Landweber迭代法可以有效地解决非线性不光滑参数识别问题。
[Abstract]:In this paper, the neighborhood modified Landweber iterative method for nonlinear nonsmooth problems is studied. The inverse problem is involved in many practical applications. The biggest characteristic of inverse problem is ill-posed. Due to the ill-posed of inverse problem, we need to use regularization method to obtain the stable approximate solution of real solution. In large-scale nonlinear inverse problem. The iterative regularization method is usually used, and the modified Landweber iterative method is used. However, the penalty function of Ln 2 norm may cause the solution to be excessively smooth. In order to overcome this difficulty, we introduce a sparse regularization method to add a penalty term of L ~ (1) norm, but in order to overcome the problem, the excessive smoothness of the solution will lead to a large deviation from the actual situation. In order to solve this problem, we introduce the adjacent operator and combine the adjacent operator with the modified Landweber iterative method. In this paper, the definitions of adjacent operators and modified Landweber iterative methods and their related properties are reviewed. By combining the adjacent operator with the modified Landweber iterative method, the final neighborhood modified Landweber iterative method is obtained. After analysis, we prove the convergence of the algorithm. The corresponding convergence rate is obtained. Finally, the algorithm is used to solve the parameter identification problem, and the one-dimensional and two-dimensional numerical experiments are carried out. The results of theoretical analysis and numerical experiments show that the neighborhood modified Landweber iteration method can effectively solve the problem of nonlinear nonsmooth parameter identification.
【学位授予单位】：大连海事大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.6

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