【作者】 孙婷婷；

【导师】 潘状元；

【作者基本信息】 哈尔滨理工大学 ， 应用数学， 2011， 硕士

【摘要】 由于数学物理反问题在医学成像、无损探伤、气象预报等领域有着越来越广泛的应用,因此反问题受到更多学者的关注。反问题大都具有不适定的特点,该特点也是反问题研究的难点所在。一个问题如果其解存在、唯一并且连续依赖于输入数据,就称该问题是适定的,否则称为是不适定的。本文考虑计算非线性算子方程F ( x )= y精确解的稳定逼近。由于非线性算子F通常是不可逆的,且实际测量得到的数据和精确数据之间存在一定的误差,从而加大了求解的难度。基于线性不适定问题的求解,对于非线性不适定问题多数是把非线性算子转为线性算子求解。本文在现有成果的基础上,展开了如下两方面的工作。研究求解非线性反问题的一种King-Werner迭代法。首先,简单构造了此方法的迭代格式。其次,利用偏差原理终止迭代。当非线性算子和参数满足一定的条件时,证明此方法是收敛的;当精确解满足某种条件时得到最优收敛速率。最后,通过数值算例验证了方法的有效性和可行性。研究基于牛顿型方法的非线性不适定问题。给出了带有初始值和两个参数的改进的迭代格式。由于合适的迭代终止准则会影响初始数据的误差,因而必须选择合适的终止准则。在参数{α_ k},{ g_α}和非线性算子F满足某些条件下,证明此方法的正则解是收敛到精确解的。更多还原

【Abstract】 Inverse problems of mathematical physics is focused on by more scholars for its increasingly wide application in the fields of medical imaging, nondestructive testing, weather and forecasting. Most of inverse problems has the characteristics of the ill-posedness, in which also the difficulty of inverse problems lies. If its solution exists, is unique and depends continuously on the input data, the problem is well-posed, otherwise known as ill-posed. In this paper, the computation of stable approximations to the exact solution of nonlinear operator equation F ( x )= y is considered. The nonlinear operator in general is not invertible, and there is the error between data obtained by measurement and accurate data, thereby increasing the difficulty of solving the equation. Based on linear ill-posed problems, majority of nonlinear operator is converted to linear operator for solving nonlinear ill-posed problems. On the basis of the previous theory, two aspects of the work are studied as following.A King-Werner iteration method for solving nonlinear inverse problem is researched. Firstly, the iterative scheme of the method is introduced. Secondly, the iteration is terminated by the discrepancy principle. When the nonlinear operator and the parameters satisfy certain conditions, it is proved that this method is convergent, and the order optimal convergence rates are obtained when the exact solution satisfies suitable source-wise representations. Finally, the numerical experiments verify the feasibility and effectiveness of the method.A Newton type method for solving nonlinear ill-posed problems is analyzed. The iterative scheme with an initial value and two parameters is given. A suitable stopping rules affecting the errors of an initial data must be choosed. Under certain conditions on parameters{α_ k},{ g_α}and nonlinear operator F , it is proved that the regularized solution obtained by this method converges to the exact solution.更多还原