Two Kinds Of Iterative Methods For Solving Nonlinear Ill-posed Problems

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.