In this paper, 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. The penalty term is allowed to be non-smooth, including L ¹ and total variation like penalty functionals, to reconstruct the special features of solutions such as sparsity and discontinuities. 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.

Homotopy perturbation iteration is an effective and fast method for solving nonlinear ill-posed problems. It only needs approximately half the computation time of Landweber iteration to reach the similar recovery precision. In this paper, a Nesterov-type accelerated sequential subspace optimization method based on homotopy perturbation iteration is proposed for solving nonlinear inverse problems. The convergence analysis is provided under the general assumptions for iterative regularization methods. The numerical experiments on inverse potential problem indicate that the proposed method dramatically reduces the total number of iterations and time consumption to obtain satisfying approximations, especially for the problems with costly solution of forward calculation.

In this work, we investigate a stochastic gradient descent (SGD) method for solving inverse problems that can be written as systems of linear or nonlinear ill-posed equations in Banach spaces. The method uses only a randomly selected equation at each iteration and employs the convex function as the penalty term, and thus it is scalable to the problem size and has the ability to detect special features of solutions such as nonnegativity and piecewise constancy. To suppress the oscillation in iterates and reduce the semi-convergence of such methods, by incorporating the spirit of discrepancy principle, an adaptive strategy for choosing the step size is suggested. Under certain conditions, we establish the regularization results of the method under an a priori stopping rule. Further, we study an a posteriori stopping rule for SGD-θ method and show the finite iterations termination property. Several numerical simulations on computed tomography and schlieren imaging are provided to demonstrate the effectiveness of the method.

Inexact Newton regularization methods are a family of prominent regularization methods for solving nonlinear ill-posed problems, which consist of an outer Newton iteration and an inner scheme providing increments by applying the regularization technique to the local linearized equations. In this paper, we propose a heuristic stopping rule for the inexact Newton regularization method, where the inner scheme is defined by Landweber iteration and the strong convex function is incorporated as the penalty term. In contrast to a prior and a posteriori stopping rules, our heuristic rule is purely data driven and does not require the information on noise level, which renders the method feasible when the noise level is unknown or unreliable. Under certain assumptions on the random noise, we establish a new convergence analysis for the inexact Newton-Landweber method under the heuristic rule. The numerical simulations are provided to demonstrate the performance of our heuristic rule.