Research On Some Regularization Methods For Nonlinear Inverse Problems

As an emerging discipline,the inverse problem has been highly focused on since its extensiveness in practice and scientific research.However,it is hard to obtain a stable solution of an inverse problem due to its inherent ill-posedness.Therefore,regulariza-tion methods are desired in order to solve nonlinear inverse problems.There are many regularization methods,such as variational regularization methods and iterative methods.Among these iterative methods,Landweber-type and Newton-type methods are signifi-cant to solve nonlinear inverse problems.In the thesis,we study acceleration Landweber iteration and various Newton-type methods and use them to find solutions of nonlinear inverse problems.The detailed work is as follows.To solve nonlinear inverse problems with non-smooth forward mappings in Hilbert spaces,we study the projected Bouligand-Landweber iteration which is an acceleration version of constant step Bouligand-Landweber iteration.Regularization of the method is given via using its property of asymptotically stable.Numerical simulations on semi-linear operator equations are given to test its acceleration effect.Numerical results show that the projected Bouligand-Landweber iteration has acceleration effect when solving non-smooth nonlinear inverse problems.In order to find non-smooth solutions of nonlinear inverse problems in Hilbert s-paces,we formulate a Levenberg-Marquardt iteration with general convex penalty terms.Furthermore,we propose a new strategy to determine regularization parameters.Using tools from convex analysis,we give the convergence analysis of this method.We use some numerical simulations to show its efficiency to reconstruct non-smooth solutions of nonlinear inverse problems.In addition,reconstruction results of Levenberg-Marquardt scheme using geometric regularization parameters are presented,which shows the neces-sity to use regularization parameters determined by our strategy.To deal with nonlinear inverse problems in Banach spaces,we propose a modified in-exact Newton method which is based on the non-stationary Tikhonov iteration.Since the proposed method use general convex function to be penalty terms,it can reconstruct non-smooth solutions of nonlinear inverse problems.Convergence analysis of this method is given:when the data is noise-free,the iterated sequence converges to a solution of the problem;when the data is contaminated by noise,the method is proven to be a reg-ularization method.To verify the efficiency of this method,numerical experiments on parameter identification problems are given.In addition,reconstruction results show that the proposed REGINN-IT method can efficiently reconstruct solutions which are sparse or piece-wise constant.Discrepancy principle is commonly used to terminate an iteration method.Howev-er,discrepancy principle depends heavily on the accurate knowledge of the noise level.Overestimation or underestimation on noise level may lead to significant loss of recon-struction accuracy when using this stopping rule.Therefore,we consider a heuristic rule to terminate the Gauss-Newton iteration.Posterior error estimate is given when the op-erator satisfies variational source condition.The Gauss-Newton iteration terminated by heuristic rule is proven to be a regularization method when no source condition fulfills.Numerical simulations show that the Gauss-Newton iteration terminated by heuristic rule can efficiently reconstruct non-smooth solutions of nonlinear inverse problems.Further-more,numerical results imply that the Gauss-Newton iteration terminated by discrepancy principle could not reconstruct non-smooth solutions of inverse problems efficiently when noise levels are overestimated or underestimated.