A data-driven Kaczmarz iterative regularization method with non-smooth constraints for ill-posed problems

Abstract

A data-driven type Kaczmarz method with uniformly convex constraints is derived and analyzed for solving ill-posed problems. By using a black box strategy, the training data are utilized to learn an operator, which is used to construct a data-driven term in iterative scheme. The introduced convex penalty is allowed to include L1 and total variation functionals, which contributes to recovering special features of solutions such as sparsity and piecewise constancy. In addition, the acceleration version by two-point strategy of the proposed method is also considered. The convergence and regularity of the method is established under suitable assumptions. The proposed method is validated through numerical experiments on both linear problems and nonlinear problems. The numerical results show that the introduced data-driven term helps to reduce the numerical consumptions and improve the reconstruction accuracy, especially for the case that the desired solution is well approximated by the training data. Meanwhile, the convex penalty of the proposed algorithm efficiently captures the special features of the solutions and removes the artifacts brought by the data-driven term when data information is not close to the true solution.