The Study Of Proximal Modified Landweber Iteration Method For Nonlinear And Nonsmooth Problems

This paper mainly studies the proximal modified Landweber iteration method for the nonlinear and nonsmooth problems.Many practical applications involve inverse problems,the greatest characteristic of the inverse problems is the ill-posedness.Due to the ill-posedness of the inverse problems,it needs to use regularization methods to get stable approximate solutions of the real solutions.Iterative regularization methods are usually utilized to solve the large-scale nonlinear inverse problems.We adopt the modified Landweber iteration method.However,L~2-norm penalty function can cause over-smoothing of the solution,which can lead to appear large deviation with actual situation.In order to overcome this difficulty,we introduce the sparse regularization method,adding a L~1-norm penalty term,but it can lead to the nonsmooth.In order to solve the problem,we introduce the proximal operator.Combining the proximal operator with the modified Landweber iteration method,it is used to solve the nonlinear and nonsmooth inverse problems.This paper firstly reviews the definitions of the proximal operator and the modified Landweber iteration method.Then it states the properties of the proximal operator and the modified Landweber iteration method.On this basis,combining the proximal operator with the modified Landweber iteration method gets the final proximal modified Landweber iteration method.By the analysis,we prove the convergence of the algorithm and the corresponding convergence rate.Finally,the algorithm is used to solve the parameter identification problems,which involve the one dimensional and the two dimensional numerical experiments.The effectiveness of the proposed algorithm is proved.The results of theoretical analysis and numerical experiments show that the proximal modified Landweber iteration method can effectively solve the problems of nonlinear and nonsmooth parameter identification.