Regularized Gradient Method For Ill-posed Operator Equation

The dissertation investigates the regularized iterative gradient-type methods for ill-posed nonlinear operator equation. We obtain the convergence and stability of the methods. Some numerical results are reported.In chapter two, for the following type nonlinear ill-posed operator equationswhere instead of F an approximation F is accessible and the solution is not continuous with respect to the operator perturbationsδ(i.e. the equation is ill-posed), a modified iterative gradient-type method is proposed. Under some common nonlinear assumptions or the modified source condition, the method is convergent and is stable with respect to the operator perturbations whether a-posteriori or a-priori stopping rule is employed. If a-priori rule is employed, the method is of optimal order under the modified source condition. The linear operator P may have some physical meaning. The numerical results show that the operator P may effect the performance of the method.In chapter three, the continuous version of above method (i,e. an asymptotical regularization gradient method)is presented for nonlinear ill-posed operator equation. Employing a-priori rule or a-posteriori rule respectively, we obtain some convergence results under a modified source condition. The method is stable with respect to perturbations of the source condition. If the modified source condition holds, the method is of optimal order under a priori rule. The method can be looked as the generalization of the existing asymptotical gradient method. From the method, many other convergent iterative methods can be constructed by solving above Cauchy problem.In chapter four, for the following type of nonlinear ill-posed problemwhere only the right-hand term is noisy data, a preconditioning iterative gradient-type method is presentd. In order to obtain the preconditioning method, we consider the operator F is defined on a Hibert Scale. The theoretical analysis and the numerical results show that the preconditioning method improves the performance greatly.