| 具奇异或退化性质的二阶抛物型方程的系数反演问题 |
| Alternative Title | Inverse coefficient problems for second order parabolic equations with singular or degenerate properties
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| 杨柳 |
| Thesis Advisor | 魏婷
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| 2016-05-29
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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 | 博士
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| Keyword | 反问题
辐射系数
初值函数
源项函数
奇异
退化
二阶抛物型方程
最优控制方法
存在性
唯一性
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| Abstract | 本文主要考虑具奇异或退化性质的二阶抛物型方程的系数反演问题，研究在适当的附加条件下解的唯一性和条件稳定性，正则化问题的解的存在性，唯一性，稳定性，收敛性，以及有效的数值重构方法。第一章，首先介绍了偏微分方程系数反问题的研究背景，其后引入了本文的数学模型，并详细阐述了
研究动机和研究的主要困难。第二章，介绍了一些函数空间和相应的积分嵌入理论，以及二阶抛物型方程的适定性结果，这些结果在后面章节的证明中起到了重要作用。第三章，研究了一个利用终端观测值确定二阶抛物型方程的辐射系数的反问题。与通常的终端控制问题
不同，这里的观测数据仅在某个固定方向上给出，而不是整个区域，这会导致抛物型方程的共轭理论在此并不适用。第四章，研究了一个利用附加条件同时重构二阶退化抛物型方程的初值和源项系数的反问题。第五章，讨论了前一章中提出的反问题的数值重构。我们利用Landweber迭代算法来求反问题的数值解，其中的关键是求出正问题算子的共轭算子的具体形式。然而，由于两个未知函数的相互耦合，我们很难直接看出共轭算子的结构。为此，我们采用算子分解方法，通过将正问题算子分解为四个独立的算子，并分别求出对应的共轭算子，最后再组合在一起而得到了正问题算子的共轭算子。我们还进行了数值实验，并给出了典型的具体算例。数值实验表明我们的算法是稳定而有效的，两个未知函数都重构得很好。 |
| Other Abstract | This thesis mainly considers some inverse coefficients problems of second order partial differential
equations (PDEs) with singular or degenerate properties. Under some appropriate additional conditions,
we will investigate the uniqueness of the solution, the existence, uniqueness, stability and convergence
of the solution for the corresponding regularized problem, and effective numerical reconstruction methods for the solution of the inverse problem.
In Chapter 1, we firstly introduce the background of inverse coefficients problems of PDEs, and the mathematical models arising in the thesis. Then, we elaborate the motivation and main
difficulties of the research.
In Chapter 2, we introduce some function spaces and the
corresponding integral imbedding theory, and some well-posedness
results of the second order parabolic equations which are quite important in the
later proofs.In Chapter 3, we consider an inverse problem of identifying the radiative coefficient in a second order parabolic equation utilizing
the terminal observations.In Chapter 4, we investigate an inverse problem of simultaneously
reconstructing the initial value and source coefficient in a second
order degenerate parabolic equation using some additional
conditions.In Chapter 5, we discuss the numerical reconstruction of the inverse problem proposed in the
previous chapter. We utilize the Landweber iteration algorithm to
obtain the numerical solution of the inverse problem, where the key
point is to solve the specific form of the adjoint operator for
the forward problem. However, due to the coupling of the two unknown
functions, it is quite difficult to directly find the structure of the adjoint operator. The operator decomposition method is applied to conquer the difficulty. The forward operator is decomposed into four
independent operators, and then every adjoint operator is resolved
respectively. In the end, the adjoint operator for
the forward problem is obtained by combining the four adjoint
operators together. The numerical experiments are done, and
some typical numerical examples are also presented. Numerical
results show that our algorithm is stable and effective, and the two unknown functions are reconstructed quite well. |
| 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/262010/224986
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
杨柳. 具奇异或退化性质的二阶抛物型方程的系数反演问题[D]. 兰州. 兰州大学,2016.
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