Stable Algorithms Design And Application On Ill-Posed Problems

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.