A fast nonstationary iterative method with convex penalty for inverse problems in Hilbert spaces

Abstract: In this paper we consider the computation of approximate solutions for inverse problems in Hilbert spaces. In order to capture the special feature of solutions, non-smooth convex functions are introduced as penalty terms. By exploiting the Hilbert space structure of the underlying problems, we propose a fast iterative regularization method which reduces to the classical nonstationary iterated Tikhonov regularization when the penalty term is chosen to be the square of norm. Each iteration of the method consists of two steps: the first step involves only the operator from the problem while the second step involves only the penalty term. This splitting character has the advantage of making the computation efficient. In case the data is corrupted by noise, a stopping rule is proposed to terminate the method and the corresponding regularization property is established. Finally, we test the performance of the method by reporting various numerical simulations, including the image deblurring, the determination of source term in Poisson equation, and the de-autoconvolution problem.