两类不适定问题的正则化方法研究 Alternative Title The study of the regularization methods for two classes of ill-posed problems 邓志亮 Thesis Advisor 傅初黎 2010-05-27 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 不适定问题 数值解析延拓 Tikhonov正则化 磨光正则化 非线性不适定问题 对数型源条件 边界识别 Abstract 本论文主要讨论了两类不适定问题的正则化方法。一类是复分析中的数值解析延拓问题，这是一个线性的严重不适定问题。 多年来研究进展不大，而且早期的结果基本上 集中在条件稳定性上，近年来蓬勃发展的正则化理论及算法在这方面的应用则很少。本文讨论了一个带型区域上的数值解析延拓问题，我们用 Fourier变换将其转化为一个线性算子方程，分析了该算子方程的不适定性，进而讨论了正则化方法的最优误差界。在此基础上，为了稳定地计算它，分别提出了 一个修改的Tikhonov正则化方法和一个磨光正则化方法，并分别在先验与后验参数选取规则下给出误差估计，最后用数值例子验证了方法的有效性。 另一类是对非线性不适定算子方程 一般理论中的有关问题进行了研究，在对非线性条件的某种限制下，针对经典的Tikhonov正则化方法，在对数源条件下分别给出了先验参数选取和两个后验参数选取规则下的 收敛速率，并对一个有重要应用背景的未知边界识别问题用所得结果进行了数值模拟，数值结果也体现了方法的有效性。 Other Abstract This thesis investigates the regularization methods for two classes of ill-posed problems. The first one is the problem of numerical analytic continuation in complex analysis,which is a linear and severely ill-posed problem.Over the years, advances in this research are not remarkable and the earlier results mostly focus on the conditional stability. However, it seems that there are few applications of modern theory of regularization methods which have been developed intensively in the last few decades. In this thesis, we consider the problem of numerical analytic continuation on a strip domain.Firstly, we transform it into a linear operator equation by using Fourier transform and analyze the ill-posedness. Moreover, the optimal error bound of regularization method is also given. Based on this, for stably computing it, we provide a modified Tikhonov regularization method and a mollification regularization method,and give error estimate under a-priori and a-posteriori parameter choice rules,respectively.Several numerical examples are provided,which show the two methods work effectively.The other kind is that some related problems in the general theory on a nonlinear ill-posed operator equation are considered.Imposing some nonlinear conditions on the nonlinear operator, we provide a-priori and two a-posteriori parameter choice rules which are specific to Tikhonov regularization method under logarithmic-type source condition and give the convergence rates, respectively. In addition, an unknown boundary identification problem, which has important application background, is also discussed. Numerical experiments support our theoretical results and show the effectiveness of the methods. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224910 Collection 数学与统计学院 Recommended CitationGB/T 7714 邓志亮. 两类不适定问题的正则化方法研究[D]. 兰州. 兰州大学,2010.
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