第一类Fredholm积分方程的正则化解 Alternative Title Regularized solution to the Fredholm integral equation of the first kind 温瑾 Thesis Advisor 魏婷 2008-05-17 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 第一类Fredholm积分方程 再生核Hilbert空间 正则化方法 正则化参数 先验选取方法 收敛率 Abstract 在本文中，在再生核空间的框架下，我们使用了一种正则化方法去解第一类Fredholm积分方程。在测量数据带有确定性误差以及被误差水平所控制的假设下，得出了误差估计。在正则化参数及测量点数的适当选择条件下可以得到收敛率。几个数值例子表明我们的方法是有效并且是稳定的。 本文共分四个部分，第二章和第三章是本文的主体。 在第二章中，我们给出了一些预备知识，分别为积分方程的分类，第一类Fredholm积分方程的不适定性，经典的Tikhonov正则化方法，以及再生核Hilbert空间与再生核。 在第三章，我们首先给出本文的研究问题及方法，即基于再生核空间框架下的特殊的正则化方法，并给出了正则化解的表达式以及两种证明方法，其次给出几个假设和一个引理，然后，在确定性误差的假设以及正则化参数的适当选取下，我们讨论了正则化解的收敛性估计，最后给出几个数值例子，表明我们的方法是有效和稳定的。 第四章为总结。 Other Abstract In this thesis, based on the reproducing kernel spaces, we use a regularization method to solve the Fredholm integral equation of the first kind. Under the assumption that measured data are contaminated with deterministic errors whose root mean square value is bounded by a noise level, we obtain two error estimates. The convergence rates can be obtained under the suitable choices of regularization parameters and the number of measured points. Some numerical experiments show that the proposed method is effective and stable. This thesis can be divided into four chapters; Chapter 2 and Chapter 3 are the main body of the thesis. As some preliminaries, we give some important knowledge in Chapter 2 as follows: the classification of the integral equations, ill-posedness of Fredholm integral equations of the first kind, the classical Tikhonov regularization method, reproducing kernel Hilbert spaces and reproducing kernels. In Chapter 3, at first, we propose the problem of this thesis and its regularization method, i.e. a special regularization method in the framework of the reproducing kernel spaces, and deduce the regularized solution by two methods. Secondly, we give some assumptions and a lemma. Thirdly, we discuss the convergence rates of the regularized solution under the assumption of deterministic errors and suitable choices of the regularization parameters. At last, some numerical results show that our method is effective and stable. In Chapter 4, we give a conclusion. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225522 Collection 数学与统计学院 Recommended CitationGB/T 7714 温瑾. 第一类Fredholm积分方程的正则化解[D]. 兰州. 兰州大学,2008.
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