Study On Some Iterative Regularization Methods For Nonlinear Inverse Problems In Banach Spaces

Inverse problem,as a new interdisciplinary subject,has been increasingly applied in geophysics,biomedicine,materials science and engineering control.Due to its wide application background,scholars attach great importance to the study of inverse problems.In general,the solutions of some practical inverse problems,such as medical imaging and signal analysis,are piecewise constant or sparse,so we are particularly interested in the theoretical and algorithmic study of nonlinear inverse problems in Banach spaces.In this thesis we study some iterative regularization methods for solving nonlinear inverse problems in Banach spaces that the solutions are discontinuous or sparse.The penalty term of these proposed methods is allowed to be non-smooth to reconstruct special features of solutions such as sparsity and piecewise constancy.Together with the discrepancy principle as stopping rule,we prove,under certain assumptions,convergence and regularization properties of these proposed methods.Finally,the feasibility and effectiveness of these proposed methods are validated by numerical simulations.In what follows,we elaborate on the main research work of this thesis.Landweber iteration is easy to implement,which can be viewed as the gradient descent method for the corresponding functional,however,the slow convergence of this method makes it inefficient in practical applications.To accelerate the convergence of the Landweber iteration,we consider the subspace optimization method for solving nonlinear inverse problems in Banach spaces,which is based on the sequential Bregman projections with uniformly convex penalty term.Instead of just utilizing the current gradient like the Landweber iteration,the method uses multiple search directions in each iteration to accelerate the convergence.Moreover,their step lengths are calculated by the projection onto the subspace that contains the solution set of the unperturbed problem.Under certain assumptions,we present the detailed convergence analysis when the data is given exactly.For the data containing noise,we use the discrepancy principle as stopping rule and then obtain the regularization result of the method.Finally,some numerical simulations for parameter identification problems are provided to illustrate the effectiveness of capturing the property of exact solutions and the acceleration effect of the method.Inexact Newton-Landweber regularization method is an attractive iterative method.The main idea of this method is to linearize the equation around the current iterate and subsequently determine the new iterate as a stable approximation of the resulting linearized system by applying Landweber iterative method.This method consists of two components: an outer Newton iteration and an inner scheme which provides increments by regularizing local linearized equations.In order to speed up the inner scheme,we employ a so-called two-point gradient method as inner regularization scheme,which is based on the Landweber iteration and an extrapolation strategy.We present,under certain assumptions and suitable choices of combination parameter,the detailed analysis of convergence and regularization properties of the proposed method.Finally,some numerical experiments on elliptic parameter identification and Robin coefficient reconstruction problems are provided to demonstrate that the proposed method is more effective and useful compared with the inexact Newton-Landweber iteration.Landweber iteration of Kaczmarz type is an efficient method for solving inverse problems that can be written as systems of nonlinear operator equations in Banach spaces,which cyclically adapt Landweber iteration to solve each equation of systems.To speed up this method,we propose and analyze a novel Kaczmarz type method.The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms.To accelerate the iteration,we introduce a sophisticated rule to determine the step sizes per iteration.Under certain conditions,we present the convergence result of the proposed method in the exact data case.When the data is given approximately,together with a suitable stopping rule,we establish the stability and regularization properties of the method.Finally,we present some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements to investigate the effectiveness and acceleration effect of the proposed method.