兰州大学机构库 >数学与统计学院
几类反问题的正则化方法研究
Alternative TitleStudies on the regularization methods for some inverse problems
邱春雨
Subtype博士
Thesis Advisor傅初黎
2011-12-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword反问题 不适定问题 正则化 非特征Cauchy问题 积分方程 Fourier截断 压缩映像原理 Meyer小波 小波投影 小波-Galerkin
Abstract反问题往往是不适定问题, 特别是数据的微小改变会导致解的巨大变化. 因此在反问题的研究中, 人们最关心的是恢复解的稳定性. 为了解决这一问题, 数学家们引入了各种正则化方法, 如Tikhonov正则化方法, Landweber迭代正则化方法等等. 近些年来, 一些非经典的正则化方法出现在了反问题的研究中, 如Fourier截断方法, 小波方法等等. 本文利用两类非经典的正则化方法对几类反问题进行较系统的研究, 在理论上给出收敛性分析, 并且对其数值求解给出可行的算法. 全文共分为四章. 第一章对反问题和正则化理论做了简要的介绍. 第二章对于一类非特征Cauchy问题进行了研究. 我们首先将问题转化为第二类积分方程, 对于这一积分方程我们通过三种方法, 即积分方程方法、 Fourier截断方法和改进的积分方程方法分别构造了它的近似问题. 我们证明了所构造的三个近似问题的适定性, 即解的存在性, 唯一性和稳定性. 最后, 对于每一种方法我们都给出了问题的近似解与准确解之间的误差估计. 第三章考虑了小波方法在三类反问题中的应用. 首先考虑了反向时间扩散问题. 对于这一问题, 我们利用了小波投影方法, 将给定数据投影到小波空间Vj<下标!>中, 并利用投影数据计算了近似解, 给出了近似解的稳定性分析. 然后我们又分别考虑了时间分数次逆扩散问题和空间分数次反向扩散问题. 由于小波投影方法只对数据进行了投影, 所以得到的近似解不能保证属于小波空间Vj<下标!>. 为了克服这一缺点, 我们将小波和Galerkin方法相结合, 也即利用小波--Galerkin方法处理了后两类问题, 将解强制在小波空间Vj<下标!>之中, 并给出了此时的近似解与准确解之间的收敛性分析. 第四章考虑前两章所给出方法的数值实现. 对于非特征Cauchy问题, 我们利用线方法, 将问题看做常微分方程初值问题并对其进行正则化和离散化; 对于小波投影方法, 由于能够得到解的解析表达式, 过程相对简单. 我们仅对数据进行投影和滤波, 然后利用解的表达式直接进行了求解. 而对后两类方程, 通过直接求解Galerkin方法得到的无穷维常微分方程组的有限维近似. 其中的矩阵Dj<下标!>由同时在时间分数次导数或空间分数次导数的差分近似矩阵的两边作用小波变换得到. 数值模拟的效果显示, 我们所给出的方法都获得了原问题的很好近似.
Other AbstractInverse problems are usually ill-posed, and small change in the given data may cause dramatically large change in the solution. Thus in the study of inverse problems, we are most concerned about the restoration of the stability of the solution. In order to solve this problem, mathematicians introduced various kinds of regularization methods, such as Tikhonov regularization method, Landweber's iteration method and so on. In recent years, some non-classical regularization methods appeared in the research of inverse problems, such as Fourier truncation method and wavelet method, etc. In this thesis, we use two non-classical regularization methods to systematically study several kinds of inverse problems, and obtain convergence analysis in theory. Furthermore, the algorithm which is applied to compute the numerical solutions is also presented. This thesis includes four chapters. The first chapter briefly introduces inverse problems and the regularization theory for ill-posed problems. The second chapter studies one kind of non-characteristic Cauchy problem. We firstly transform the problem into the second kind of integral equation, and three methods are used to construct its approximate problem: the integral equation method, Fourier truncation method and modified integral equation method. We prove the well-posedness of the three constructed approximate problems, i.e., existence, uniqueness and stability of the solution. Finally, the convergence analysis between exact solution and its approximation is given for the suggested methods. The third chapter considers the application of wavelet method in dealing with three kinds of inverse problems. First of all, for the problem of backward heat equation, we use wavelet projection method to project the given data on the wavelet space Vj<下标!> and compute the approximate solution from the projection data, the stability analysis of approximate solution is obtained. Afterwards we discuss the time fractional inverse diffusion problem and space fractional backward diffusion problem, respectively. As the wavelet projection method only project the data, the approximate solution can not be ensured to belong to the wavelet space Vj<下标!>. In order to overcome this deficiency, we combine wavelet method with Galerkin method, i.e., we solve the latter two problems by using wavelet-Galerkin method so that the solution is restricted on wavelet space Vj<下标!>, and we obtain the convergence analysis bet...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225054
Collection数学与统计学院
Recommended Citation
GB/T 7714
邱春雨. 几类反问题的正则化方法研究[D]. 兰州. 兰州大学,2011.
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