Study On Numerical Methods For Nonlinear Ill-posed Problems

The study of nonlinear ill-posed problems has been widely used in the fields of natural science and engineering, especially in geophysics, inverse scattering and inverse problems of differential equations. Due to the nonlinearity and ill-posedness of such problems, we can not find the exact solution. So the study of numerical methods plays a very important role. This paper, based on the Urysohn-type nonlinear operator equation, we focus on the nonlinear ill-posed problems in the gravimetry and inverse scattering. The main research work of this paper carried out as follows:(1) In this paper, two kinds of iterative regularization methods for solving nonlinear ill-posed problems are introduced, which are iterative regularized Newton method and iterative regularized Gauss-Newton method. The specific discretization process is given by using complex trapezoid formula and Simpson formula. In addition, based on the Sigmoid-function,the method of determining the regularization parameter is put forward.(2) In the study of Inverse Gravimetry, two kinds of iterative regularization methods are used for numerical simulation, and the numerical results are compared and analyzed. The results show that the proposed algorithms are feasible and effective for solving such problems.(3) As for study on the Inverse Scattering Problem, which is solved numerically using the iterative regularization Gauss-Newton method, we select different regularization operators for numerical simulation, and the numerical results are compared and analyzed.