【作者】 朱华平；

【导师】 吴传生；

【作者基本信息】 武汉理工大学 ， 应用数学， 2007， 硕士

【摘要】 反问题与不适定问题是现在数学中的一个研究热点问题。问题是适定的指的是问题的解存在，唯一并且稳定，如果有一个不满足，则称为不适定的。不适定问题的求解面临的最大困难是解的不稳定性，即：原始资料小的观测误差(这在实际中是不可避免的)会导致近似解与真解的严重偏离，这就是不适定性问题的本质困难。求解不适定问题的普遍方法是正则化方法。如何建立有效的正则化方法及算法是反问题和不适定问题研究领域中的重要内容。本文从一些实例出发，介绍了反问题和不适定问题的基本概念，并讨论了线性紧算子方程的Moore-Penrose广义解，得出了Moore-Penrose广义解的不稳定性的结论。介绍了不适定问题正则化的一般理论，主要讨论了各种常用的正则化方法包括Tikhonov正则化，Landweber迭代法，并研究了正则解的误差估计及正则参数的选取问题。Landweber迭代法对于求解大规模问题是十分有利的，而且比较稳定。目前，Landweber迭代法已进一步发展于求解非线性的不适定问题。但是，Landweber迭代序列收敛速度(特别是当真实解的“光滑性”很差时)是相当慢的。本文给出了一种新的迭代格式，这种迭代格式能够大大加快收敛速度。本文还将Landweber迭代法应用于数值微分问题，将数值微分问题转化为一个特殊的第一类Fredholm积分方程的求解问题，应用文中设计的算法给出了具体的数值实验。线性微分方程中的边界函数识别问题是数学物理中常见的并且是非常重要的问题，该问题的难点在于端点的确定。本文通过Tikhonov正则化方法，将边界函数识别问题归结为泛函的极小值问题，利用Morzove原理确定适当的正则参数后，应用演化算法，求得相应的极小值，从而确定边界函数，并给出了相应的数值实验。更多还原

【Abstract】 Inverse problems and ill-posed problems are very hot in mathematics nowadays. The well-posed problems is that the solution of problems is existent, unique and stable, if there is one or more to be dissatisfied, it is ill-posed. However, the main difficulty about the solution of ill-posed problems lies in the instability of solution, which will produce a great error between the approximate solution and the correct solution where the observation data in sourcebook are small error (which in practice is inevitable), this is the nature of difficulties of ill-posed problems. A general way that we solute ill-posed problems is regularization method. How to build up effective regularization method and algorithm are very important parts of ill-pose problems researching in inverse problems field.Beginning with some cases, the article gives basic definitions of inverse problems and ill-posed problems. Then it discusses the Moore-Penrose generalized solution, and makes a conclusion that the Moore-Penrose generalized solution of linear compact operator equations is unstable. The article introduces general regularization theories about ill-posed problems and a variety of regularization method besides Tikhonov Regularization and Landweber Iterative Method, it also discusses the calculation of regularization resolution’s error and the choice of regularization parameter.Landweber iterative methods for solving large-scale problems is very beneficial, and relatively stable. Currently, Landweber iterative method has been further development in the ill-posed nonlinear problems. However, Landweber iterative sequence convergence speed (especially when the real solution "smoothness" very poor) is very slow. This paper presents a new iteration format, this format can greatly accelerate the iterative convergence rate.This paper is returned back applying Landweber iterative method to numerical differential, numerical differential will be converted into a special category I Fredholm integral equation and given specific numerical experiments.Boundary function identification in linear differential equation is common and very important problem in mathematical physics, the difficulty of the problem is determining endpoint. Using Tikhonov regularization method, the identification of boundary function is written as the minimum problem of functional, the regularization parameters is chosen by Morzove principle, and the corresponding minimum is obtained based on evolutionary algorithm and numerical experiment is given.更多还原