【作者】 武苗；

【导师】 闵涛；

【作者基本信息】 西安理工大学 ， 计算数学， 2010， 硕士

【摘要】 信赖域算法是一种新发展起来的求解不适定问题的方法,而第一类Fredholm积分方程的求解问题,是一类特殊的反问题。反问题求解的特征即不适定性为了得到稳定的数值解,必须采用正则化方法解决该类问题。本文主要研究的是信赖域算法在求解第一类Fredholm积分方程中的应用。具有弱奇异核第一类Fredholm积分方程以及一般二维、三维第一类Fredholm积分方程的求解一直是反问题研究领域的一个重要课题,用信赖域算法求解此类问题是本文的研究重点。首先,给出Fredholm积分方程的基本模型和应用,阐述了该问题求解的困难所在；其次,系统的描述信赖域算法,并给出收敛性分析；然后,对具有弱奇异核的一维第一类Fredholm积分方程和非奇异核的二维、三维第一类Fredholm积分方程进行积分离散分析和信赖域方法求解,并与迭代Tikhonov正则化方法、TSVD正则化方法的计算结果进行比对。数值模拟和对实验结果的分析,验证了文中给出的信赖域算法求解具有弱奇异核的一维第一类Fredholm积分方程是可行的。求解非奇异核的二维、三维第一类Fredholm积分方程依赖于核的变化及其真解的光滑度,此外,数据的扰动和网格的剖分也有关系。更多还原

【Abstract】 A trust region algorithm is a new developed algorithm for solving ill-posed problems, and solving the first kind Fredholm integral equation is a special kind of inverse problem. Ill-posedness is the characteristics of this problem. In order to obtain a stable numerical solution, regularization method must be used.The main idears of this study are using trust region algorithms for solving first kind Fredholm integral equations. For solving with weakly singular Fredholm integral equation kernel and general two-dimensional, three-dimensional first kind Fredholm integral equation has always been an important topic in research fields, and used trust region algorithm for solving such problems is the focus of this research.First, we gave the basic model of Fredholm integral equations and applications and described the difficulties of the problem-solving; Secondly, the system description of trust region algorithm, and gives convergence analysis;With trust region algorithms solving and discreting for weakly singular kernel of Fredholm integral equation and general two-dimensional, three-dimensional first kind Fredholm integral equation and with the calculation results of Tikhonov regularization method and TSVD Regularization Method compared.The results of numerical simulation and experimental analysis verify that paper gives the trust region algorithm with a one-dimensional weakly singular kernel Fredholm integral equation of the first category is feasible; solving the kernel of non-singular two-dimensional and three-dimensional first kind Fredholm integral equation depends on the kernel changes and smoothness of true solution. In addition, there are also a relationship with disturbance of data grid and subdivision.更多还原