三类不适定问题的正则化方法研究 Alternative Title Regularization methods for three kinds of ill-posed problems 杨帆 Thesis Advisor 傅初黎 ; 魏婷 2014-05-29 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 不适定问题 正则化方法 正则化参数先验和后验选取规则 未知源识别 Laplace 方程 Cauchy 问题 非线性反向热传导 Abstract 本论文考虑三类不适定问题, 即反应扩散方程未知源识别问题、带型区域上的 Laplace 方程 Cauchy 问题和非线性反向热传导问题的正则化方法. 这些问题虽然有过讨论, 但绝大部分结果是先验正则化范畴, 其数值结果受到未知先验信息影响很大. 本文在使用新方法讨论这些问题的同时, 特别突出了对有关问题若干非经典后验正则化方法的探索和研究, 并建立了相对严密完善的理论, 这些结果都是新的. 论文第二章集中讨论扩散方程未知源识别问题. 我们首先考虑了抛物方程只含有空间变量的热源识别. 接着考虑了抛物方程只含有时间变量的热源识别问题. 其次考虑了时间分数阶和空间分数阶扩散方程只含有一个独立变量的未知源识别问题. 最后讨论了空间分数阶扩散方程只含有空间变量的未知源识别问题. 对本章讨论的所有问题我们都进行了数值试验. 本文第三章讨论带型区域上的 Laplace 方程 Cauchy 问题. 这是一个经典的不适定问题, 但对同时具有非齐次 Dirichlet 边界条件和 Neumann 边界条件的研究结果非常少. 本文利用修正的Tikhonov给出正则解并证明了收敛性误差估计并进行了数值试验. 本文第四章考虑一类非线性反向热传导反问题, 可用的工具和方法非常有限, 我们尝试用 Fourier 正则化方法给出正则解, 并且得到 H\"{o}lder 型误差估计. Other Abstract This thesis investigates the regularization methods for three kinds of ill-posed problems, i.e., the unknown source identification problem for the diffusion equation, the Cauchy problem for the Laplace equation in strip domain, a class of the nonlinear backward heat equation. Although these problems has been discussed before, but most of the results belongs to the category of a-priori regularization, in which the numerical results are much affected by the unknown a-priori information. While using some new methods to study the aforementioned problems, we especially discuss and explore several non-classical a-posteriori regularization methods for related ill-posed problems, and construct relative rigorous and complete theory analysis, these results are completely new. In Chapter 2, we mainly discuss the source identification problem for several kinds of diffusion equations. First, we consider identifying the unknown heat source which depends only on spatial variable. Second, we consider identifying the unknown heat source which depends only on time variable. Third, we propose to identify the unknown source which depends only on one variable for fractional diffusion equation. For these methods, we obtain the error estimates between the regularization solutions and the exact solution. Fourth, we identify the unknown source which depends only on spatial variable for the spatial-fractional diffusion equation. In Chapter 3, we study the Cauchy problem of Laplace equation in strip domain. This is a classical ill-posed problem, however, to the best of the authors' knowledge, there is very limited literature on the case that both given the nonhomogeneous Dirichlet boundary condition and the nonhomogeneous Neumann boundary condition. We propose to use the modified Tikhonov regularization method for obtaining the regularization solution. We also obtain the error estimates between the regularization solutions and the exact solution, respectively. In Chapter 4, we discuss a class of non-linear heat equation backward in time. Up to the present moment, the available methods and tools for dealing with this ill-posed problem are very limited, we use the Fourier regularization method to obtain the regularization solution and obtain the H\"{o}lder type error estimate. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224751 Collection 数学与统计学院 Recommended CitationGB/T 7714 杨帆. 三类不适定问题的正则化方法研究[D]. 兰州. 兰州大学,2014.
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