| 两个不适定数学物理问题的最优正则化逼近 |
| Alternative Title | Optimal Regularization Approximation for two Ill-posed Mathematical Physics Problems
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| 李洪芳 |
| Thesis Advisor | 傅初黎
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| 2002-05-12
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 不适定数学物理问题
最优正则化方法
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| Abstract | 本文研究了两类不适定数学物理问题的最优正则化逼近方法: 第一类是一维非标准型逆热传导问题，即含有对流项的抛物方程侧边值问题；第二类是特殊形式的椭圆方程的Cauchy问题。对于这两类问题，本文给出了由相应的扰动数据来确定未知解的最优误差界，同时给出了三种基于谱分析理论的广义正则化方法: 广义Tikhonov正则化方法和两类广义奇异值分解法，并且证明了当正则化参数按给定规则选取时可实现Holder型或对数型最优误差估计。对于第一类问题本文还考虑了最优滤波方法，证明了当滤波函数适当选取时可得到Holder型最优性结果。
文中第3.4节还就第一类问题给出数值试验，结果表明本文所给出的正则化方法是有效可行的。 |
| Other Abstract | In this paper we consider the optimal regularization approximation for two ill-posed mathematical physics problems: one dimension non-standard inverse heat conduction problem, i.e., a sideways parabolic equation with convection, and Cauchy problems for a special elliptic equation. For these problems, we give the optimal error bounds for identifying unknown solution from the noisy data. At the same time, we give three generalized regularization methods based on spectral representation: the generalized Tikhonov regularization method and the methods of generalized singular value decomposition. We obtain the Holder stability and logarithmic stability optimal error estimation respectively, when the regularization parameter is chosen optimally. Furthermore, we also give an optimal filtering method for the first problem, and give the Holder stability optimal error estimation when the filter function is chosen properly.
In the last of the section 3.4 we give a numerical experimentation for the first problem which prove that the above methods are viable. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224918
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
李洪芳. 两个不适定数学物理问题的最优正则化逼近[D]. 兰州. 兰州大学,2002.
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