The Study Of Runge-Kutta Type Landweber Iteration For Nonlinear Ill-Posed Problems

In this paper, we investigate the behaviour of Runge-Kutta method for nonlinearill-posed operator equations."Ill-posed"is the important attribute of the inverse problems. Up to now, thereare Tikhonov regularization, iteration and continuous methods to solve the nonlin-ear ill-posed problems. Comparatively speaking, we note that the computation ofLandweber method is more stable with respect to perturbations of the right-hand sideand gives very good results even for large errors. However, to Landweber method, ifwe want to compute more precisely, the times of iteration would much more than ever.In order to keep its stability, we should control the times of iteration to small enough.In order to improve the stability and convergence of the Landweber iteration,2-stage Runge-Kutta method is applied to the continuous analogy of the Landwebermethod to give a new modified Landweber method, which is called R-K type modifiedLandweber method. Under some appropriate conditions, we prove the convergenceand convergence rate of the proposed method. We testify the theory results by solvingnonlinear ill-posed convolution problem, and point out it's superiority in relative tomodified Landweber iteration.Moreover, using m-stage Runge-Kutta discrete the continuous analogy of theLandweber method, we propose m-stage R-K type Landweber method. We provethat it is a stable method for solving nonlinear ill-posed problems. The presentednumerical experiments on nonlinear ill-posed convolution problem and some param-eter identification problem illustrate the theoretical findings and demonstrate that theimplicit schemes outperform the classical Landweber iteration.