兰州大学机构库 >数学与统计学院
数学物理反问题的正则化
Alternative TitleRegularization of Inverse Problems in Mathematical Physics
钱志
Subtype博士
Thesis Advisor傅初黎
2008-05-17
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 Fourier变换 有限差分 修改核
Abstract在这篇论文中我们提出研究反问题的“修改核”思想,并依据此思想对四类经典数学物理反问题: 高阶数值微分、 逆热传导问题、 Laplace方程Cauchy问题和反向热传导问题进行了系统的研究。 我们分析这些反问题的不适定性, 即解不连续依赖于定解数据,并且讨论它们不适定的程度。 为稳定地计算这些问题,首先通过在频域空间分析一维逆热传导问题的多种正则化方法,我们找到了这些方法的一个有趣联系并指出它们形成正则化的本质原因,从而归结出一个所有一维逆热传导问题的正则化方法均满足的重要性质。借助该性质的思想,运用扰动核方法和Fourier截断方法我们研究了高阶数值微分和一个二维逆热传导问题。基于前面的分析和讨论,我们提出了研究解在频域空间具有某种共同形式的不适定问题的``修改核"思想。遵循该思想, 我们用扰动方法研究了非标准逆热传导问题,Laplace方程Cauchy问题和反向热传导问题。 我们讨论了所有这些正则化方法的稳定性,给出并且证明了正则化解与精确解之间的收敛性估计。 此外, 我们还讨论了所有这些方法的数值实现,详细阐述了Fourier变换和有限差分的应用技巧。而且用大量的数值例子测试了所提出的正则化方法各方面的性质。这些测试表明我们提出的方法是有效的和数值可行的。
Other AbstractIn this thesis, we propose the idea of modifying ``kernel" for inverse problems. According to the idea, we systematically investigate four kinds of classical inverse problems in mathematical physics: high order numerical differentiation; inverse heat conduction problem; Cauchy problem for Laplace equation; backward heat conduction problem. We analyze the ill-posedness of these inverse problems, i.e., the solution does not depend continuously on the data, and discuss their degree of ill-posedness. For computing these problems stably,we firstly analyze many regularization methods for a one-dimensional inverse heat conduction problem(1D IHCP) in the frequency space. We find an interesting relation among these methods and point out the natural cause of regularization. Consequently, we conclude an important property: all regularization methods for 1D IHCP should satisfy the property. Following the idea of the property, we employ a method of perturbing kernel and a Fourier truncation method for high order numerical differentiation and a two-dimensional inverse heat conduction problem. Concluding previous analysis and discussion, we propose the idea of modifying kernel for ill-posed problems whose solutions have a common form in the frequency space. Based on the idea, we employ the turbation method to study a non-standard inverse heat conduction problem, Cauchy problem for Laplace equation and backward heat conduction problem. For all these regularization methods, we discuss the stability, and give and prove the convergence estimate between the exact solution and its regularized approximation. In addition, we discuss the numerical implementation of all these methods: we expatiate on the skill of applying Fourier transform and finite difference. Moreover, we give a large number of numerical examples to test various properties of the proposed regularization methods. These tests show that our methods are effective and numerically feasible.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224693
Collection数学与统计学院
Recommended Citation
GB/T 7714
钱志. 数学物理反问题的正则化[D]. 兰州. 兰州大学,2008.
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