Several Type Of Iterative Methods For Solving Nonlinear Ill-Posed Problems

In this thesis, we investigate the method for nonlinear ill-posed problem andinverse problem. In Physics, Mechanics and Engineering, there exist lots of nonlin-ear ill-posed problems, such as parameter identification problem, inverse scatteringproblem, inverse potential problem and the first kind integral equation problem, etc.presently, the theory of linear ill-posed problem has been relatively perfect and obtainsatisfied effect in some realistic fields. However, the theory and practice of nonlinearill-posed problem, for its intrinsic speciality and complexity, need further be perfected.So the research for nonlinear ill-posed problems is significative not only to theory butalso to application.To nonlinear ill-posed problem, there is only few reference from the theory oflinear ill-posed problems, undoubtedly, which add the difficulty to the research ofnonlinear ill-posed problem. For the speciality of nonlinear ill-posed problems, themethod to solve nonlinear ill-posed problems has it owner applied scope, which makepeople have to design appropriate algorithm from the property of concrete problemand analyze the theory of the algorithm based on added condition.At present, people usually solve ill-posed problems by variational method anditerative method. In research, they mainly focus on the two problems that the designof algorithm and choice of regularization parameter. In this thesis, we investigate iter-ative method to solve nonlinear ill-posed problems and develop our research aroundthe design of algorithm and choice of regularization parameter.The computational cost to solve nonlinear ill-posed problem is perhaps great,which to some extent counteract the application of some methods. In the first part ofthis thesis, we propose the mixed Newton-Tikhonov method. In this thesis, we firstinvestigate two kind strategies of the choice of parametersαn,k for the mixed Newton-Tikhonov method with fixed simplified step p(n)≡p, i.e., Bakushinskii method andHanke criteria, for the latter, we obtain the result of convergence and stability. Numer-ical results show that the new methods can obviously decrease computational cost. Atthe following part, we improve the mixed Newton-Tikhonov method with fixed sim-plified step and propose the adaptive mixed Newton-Tikhonov method and analyzethe convergence and stability of this method successfully, numerical results show itsefficiency.In the second part, we study how to choose regularization parameterαk for non- linear implicit iterative method and give its theoretical analysis based on Hanke cri-terion, improve the restriction of the present method that parameterαk must be suffi-cient big. From optimization aspect, we design several algorithms, numerical exper-iments show that these algorithms are very efficient. One improvement for nonlinearTikhonov regularization method is initially proposed and analyzed by Liu Jianjun inhis thesis for Doctorate.At the end of this thesis, we research a replacement functional method ofTikhonov method and a variance of Landweber method to solve nonlinear ill-posedproblems, give two strategies of the choice of parameter which improve the restrictionof the choice of parameter of the initiate method, numerical results its efficiency.