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In this paper we consider the computation of approximate solutions forinverse problems in Hilbert spaces. In order to capture the special feature ofsolutions, non-smooth convex functions are introduced as penalty terms. Byexploiting the Hilbert space structure of the underlying problems, we propose afast iterative regularization method which reduces to the classicalnonstationary iterated Tikhonov regularization when the penalty term is chosento 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 secondstep involves only the penalty term. This splitting character has the advantageof making the computation efficient. In case the data is corrupted by noise, astopping rule is proposed to terminate the method and the correspondingregularization property is established. Finally, we test the performance of themethod by reporting various numerical simulations, including the imagedeblurring, the determination of source term in Poisson equation, and thede-autoconvolution problem.

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