Study On Nesterove-Type Accelerated Methods For Nonlinear Inverse Problems

In recent years,driven by the needs of other disciplines as well as many engineering and technical fields,nonlinear inverse problems have aroused great interest and attention from scholars at home and abroad.Inversion theory,algorithm and applied research in-volving diffuse optical tomography,photoacoustic tomography,inverse doping problems for semiconductors,parameter identification problems based on multi-boundary measure-ment,etc.are of particular concern.From a mathematical point of view,these problems can be abstracted as mathematical models that contain multiple nonlinear ill-posed op-erator equations.However,solving such problems not only faces the dual problem of nonlinearity and ill-posedness,but also puts forward higher requirements for the storage space and computational efficiency of algorithms due to the huge scale of the problem itself.Therefore,it is particularly important to construct iterative regularization methods with simple format and fast convergence speed.In addition,with the increase of practical application needs,people pay more and more attention to the inversion of some com-plex problems,such as the non-smooth nonlinear ill-posed problems.So far,most of the research results are based on the non-differentiable regularization methods in Banach s-pace,while the related research work in Hilbert space is very limited,and some of the developed inversion theories and algorithms are also imperfect.Hence,there is still a lot of space for further exploration.Based on the above research situation,several fast algo-rithms for solving nonlinear inverse problems are constructed by introducing the Nesterov acceleration strategy.The main research work is as follows:To solve a system that contains a finite number of nonlinear ill-posed operator equa-tions,we based on the fast homotopy perturbation iteration and the Kaczmarz accelera-tion strategy,first construct a homotopy perturbation-Kaczmarz iterative method.Then by introducing the generalization form of the Nesterov acceleration scheme,an Nesterov-type accelerated homotopy perturbation-Kaczmarz iterative method is further proposed.Under the Morozov discrepancy principle and some basic assumptions,we provide the corresponding convergence and regularity analysis of the two methods,respectively.At the same time,the effectiveness and acceleration of the proposed methods are verified by numerical simulations.For solving a system with multiple nonlinear ill-posed operator equations,we deal with a hybrid regularization scheme by combining the respective advantages of L~1-norm based penalty and the general L~2-norm stabilizing term,so that the sparsity and smooth-ness of the solution can be taken into account simultaneously.To begin with,we introduce a proximal regularized Gauss-Newton-Kaczmarz(PRGNK)method which is constructed by combining the Kaczmarz strategy with a proximal regularized Gauss-Newton(PRGN)iteration.Its convergence analysis is presented under appropriate assumptions,and the nu-merical experiments on large-scale diffuse optical tomography and parameter identifica-tion problems indicate that,PRGNK is clearly faster than the PRGN iteration.Moreover,we incorporate the generalization form of the Nesterov acceleration scheme into PRGNK in order to further accelerate the convergence,which leads to a so-called Nesterov-type accelerated proximal regularized Gauss-Newton-Kaczmarz(APRGNK)method.Based on the discussion for PRGNK,we also establish the convergence analysis of APRGNK.Meanwhile,the numerical simulations explicitly show that APRGNK makes a remarkable acceleration effect compared with its non-accelerated counterpart.By introducing the Bouligand subdifferential of the forward operator and a gen-eral case of Nesterov acceleration scheme,an Nesterov-type acceleration Bouligand-Landweber iterative method is proposed for solving non-smooth ill-posed inverse prob-lems where the forward operator is merely directionally but not G(?)teaux differentiable.Since the forward mapping is not G(?)teaux differentiable in our case,the standard analysis is not applicable to the convergence analysis.Under certain assumptions on the com-bination parameters,we therefore provide a new convergence analysis of the proposed method also with the help of the concept of asymptotic stability and a generalized tan-gential cone condition.The design of our method involves the choices of the combina-tion parameters which are carefully discussed.Moreover,the Nesterov-type acceleration Bouligand-Landweber iterative method is applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing G(?)teaux derivative,and the corresponding iterative method is shown to be a convergent regularization scheme.Numerical simulations are presented to illustrate the advantages over the Bouligand-Landweber iteration.