| 两种正则化方法在求解几类椭圆方程反问题中的应用 |
| Alternative Title | Two regularization methods for solving some inverse problems of elliptic equations
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| 程浩 |
| Thesis Advisor | 傅初黎
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| 2009-05-21
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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 | 不适定性
磨光正则化
滤波法
椭圆方程
Helmholtz方程Cauchy问题
数值解析延拓
误差估计
正则化参数
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| Abstract | 众所周知，磨光正则化和滤波正则化方法是求解不适定问题的两种有效的方法。本文将分别介绍磨光正则化和滤波正则化方法的基本思想，以及如何构造磨光化算子和滤波函数。我们用具有Gaussian核的磨光正则化方法处理了高维变系数椭圆方程的Cauchy问题，用滤波正则化方法处理了带形区域和矩形区域上的Helmholtz方程Cauchy问题以及带形区域上的解析延拓问题。对于上述问题， 我们均得到问题的精确解和正则近似解之间的误差估计，同时数值试验表明了这两种正则化方法是简单易操作的有效方法。 |
| Other Abstract | As we known, the mollification method and the filtering method are two valid regularization methods for solving the ill-posed problems. In this paper, we introduce the basic idea of the mollification method and the filtering method respectively, and describe how to structure the mollification operator and the filtering function. We use the mollification method with Gaussian kernel to deal with the Cauchy problem of an elliptic equation in multi-dimensional case. We use the filtering method to solve the Cauchy problem for the Helmholtz equation in the strip and bounded regions, and the stable analytic continuation problem in the strip. We obtain the error estimates between the exact solutions and their approximations. Moreover, numerical experiments shows that the two methods we proposed work effectively and feasible. |
| 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/262010/224872
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
程浩. 两种正则化方法在求解几类椭圆方程反问题中的应用[D]. 兰州. 兰州大学,2009.
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