A predictor–corrector iterated Tikhonov regularization for linear ill-posed inverse problems

Abstract

Many questions in science and engineering often give rise to linear ill-posed inverse problems. To enable meaningful approximate solutions of the inverse problems, various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations. One of the most popular regularization techniques is iterated Tikhonov regularization, which has attracted considerable attention due to its interesting applications in practical. However, this regularization often suffers from inaccurate results and low computational efficiency in some situations. In this paper, we proposed an accelerated predictor–corrector iterated Tikhonov regularization. This method combined the classical iterated Tikhonov regularization with modified Euler method, which could improve computational efficiency without sacrificing numerical accuracy. The convergence rate of the accelerated regularization was investigated theoretically when the right-hand side was corrupted by noise. In particular, we derived an error estimate of optimal order for the convergence of the accelerated regularization. Both one-and two-dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated Tikhonov regularization.