【作者】 范小平；

【导师】 李功胜；

【作者基本信息】 山东理工大学 ， 计算机应用技术， 2006， 硕士

【摘要】 反问题广泛存在于自然科学与工程技术诸多领域之中。反问题一个突出的特性就是“不适定”性，所以反问题也称作不适定问题。反问题的这一性质使得它的求解比正问题困难的多。因而，不适定问题的求解算法，成为广大数学工作者，自然科学工作者及工程技术人员努力开拓的一个崭新的研究方向。 随着计算方法与计算机技术的进步，不适定问题的算法研究进展迅速。正则化方法、脉冲谱方法、最佳摄动量方法以及非线性优化等算法获得广泛应用。近年来，有许多学者把遗传算法应用到数理方程反问题与不适定问题的求解中，并收到了很好的效果。 本文基于最优化求解思路，对于不适定问题的稳定化算法设计及其实现进行了研究。主要做了以下工作： (1) 应用Tikhonov正则化算法，在算子的奇异系统已知情形下，将正则解近似表为有限项之和，简化了编程设计，进而对于数值微分问题进行了数值实验，实现了较高的计算精度； (2) 通过引进适当的正则化子，建立了一种改进的Tikhonov正则化算法，并对于一次数值微分及地质勘察中提出的一个第一类Fredholm积分方程进行了数值试验，结果表明改进的Tikhonov算法有可能使正则解取得更高的精度与收敛阶； (3) 采用最佳摄动量算法对数据有扰动的抛物型偏微分方程的扩散系数识别反问题进行了求解，并对该算法应用中正则参数与迭代次数及初始迭代点的选取进行了初步研究；结果表明正则参数选取不当会增加迭代次数，降低解的收敛速度；初始迭代点选取不当则可能造成解的不收敛或迭代次数的增加。 (4) 根据基本的遗传算法思想，提出了一种新的遗传演化算法。对于抛物型方程的参数反演问题进行了数值试验，并应用该算法于一个区域地下水硫酸污染源强度的反演问题中，所得计算结果与实际部门的估算值基本吻合。更多还原

【Abstract】 There are many inverse problems in the fields of natural science and engineering technology. Ill-posedness is a primary characteristic of inverse problems such that inverse problem is also called ill-posed problem. Due to the ill-posedness of inverse problems, it is more difficult to solve an inverse problem than a direct problem. So, numerical methods of solving such problems have been widely researched by many mathematicans, natural science researchers and engineers, and become a new field in natural science and engineering technology with the development of computational methods and computer techniques.There are many researches on numerical algorithms for inverse and ill-posed problems so far. Especially, regularization method, PST method and nonlinear optimization method have been extensively studied and utilized in solving ill-posed problems. Recently, genetic algorithm is applied to solve some actual inverse problems showing its effectivity and maneuverability in solving such problems. In this paper, several stable algorithms based on optimal ideas in solving ill-posed problems are investigated. The main works are listed in the following:(1) Tlkhonov regularization method is studied and applied to compute stable solutions of the first kind of operator equations in the case of knowing singular system of the operator. With the help of Matlab software, programming of the algorithm here is simplied, and numerical tests for numerical differentiation problems are carried out. (2 ) A modified Tikhonov regularization is constructed by using a suitable regularizing filter. With this method, the regularized solution could have higher precision. Numerical simulations for numerical differentiation problem and a Fredholm integral equation of the first kind arising in geological prospecting are carried out showing that the modified Tikhonov regularization algorithm can get better precision and convergence rate. (3) The best perturbation method is applied to solve inverse problems of determining coefficients in parabolic PDE in the presentation of noise data. Some relations on the optimal regularization parameter, and the initial value of iteration with the precision and convergence rate are discussed. (4) A new genetic algorithm based on the basic genetic idea is established with which an inverse coefficient problem in parabolic PDE is solved. Furthermore, this modified genetic algorithm is also applied to solve an actual inverse problem of determining source magnitude in regional groundwater pollution.更多还原