兰州大学机构库 >数学与统计学院
具奇异或退化性质的二阶抛物型方程的系数反演问题
Alternative TitleInverse coefficient problems for second order parabolic equations with singular or degenerate properties
杨柳
Thesis Advisor魏婷
2016-05-29
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
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
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 AbstractThis 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
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224986
Collection数学与统计学院
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.
Files in This Item:
There are no files associated with this item.
Related Services
Recommend this item
Bookmark
Usage statistics
Export to Endnote
Altmetrics Score
Google Scholar
Similar articles in Google Scholar
[杨柳]'s Articles
Baidu academic
Similar articles in Baidu academic
[杨柳]'s Articles
Bing Scholar
Similar articles in Bing Scholar
[杨柳]'s Articles
Terms of Use
No data!
Social Bookmark/Share
No comment.
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.