The Research Of Iterative Regularization Methods For â…¢-posed

Now the research of iterative regularization methods for ill-posed problems is a hot issue for the study in the inverse problem of mathematical physics. There are a lot of ill-posed problems in physics, biology, medicine, geology and engineering technology. For example, parameter identification problems, nonlinear ill-posed Hammerstein integral equation, the solution for the first kind of Fredholm integral equation and so on. While the solution of the ill-posed problem is facing a serious problems that the instability of approximate solutions. when the right data of equations for solving ill-posed problems have an error, between the equation’s solutions and exact solutions can exist a larger error. Usually the typical method of solving the ill-posed problem is the regularization method. Then how to construct an effective regularization method for solving the ill-posed problem has an important theoretical and practical significance.First, aiming at how to establish an effective regularization method of solving the ill-posed problem. The ill-posed problem can be divided into linear ill-posed problems and nonlinear ill-posed problems. According to some practical problems, we have introduced the general theory of inverse problems, ill-posed problems, linear and nonlinear ill-posed problems. Then we have combined with ill-posed problems that are given in this paper and come up with the corresponding iterative stopping criterion of regularization methods.Second, we mainly discuss several commonly used regularization methods in solving ill-posed problems such as the Tikhonov regularization method, the Landweber iterative method and the modified Landweber iterative method. At the same time, according to different properties(linear or nonlinear) of ill-posed problems we give different iterative formats of the Landweber iterative method. Among these regularization methods, we also raise several accelerated Landweber iterative methods for the linear Landweber iterative method. What is specially, An accelerated iterative format is applied to solving the first kind of Fredholm integral equation and made numerical tests. Then those experimental results show that this accelerated Landweber iterative algorithm can more effectively solve the ill-posed problem. What is more, it can quicken the convergence of the iterative sequence.Finally, the nonlinear Landweber iterative method and the modified Landweber iterative method that the nonlinear ill-posed problem has given are applied to solve the nonlinear ill-posed Hammerstein integral equation and made numerical tests. Through those numerical experiments are compared, these results show that the modified Landweber iterative method is more superior.