A relaxed iterated Tikhonov regularization for linear ill-posed inverse problems

Abstract

Iterated Tikhonov regularization methods are prevalent regularization methods that can make the solution of ill-posed problems less sensitive to noise. As the iteration number grows, their solution gradually converges to the optimum approximation of the desired solution but then changes to the worse, causing a semiconvergence. This difficulty can be reduced by a reliable early termination rule or a strategy for choosing the appropriate regularization parameter at each iteration. Different from the existing methods, this paper describes a novel iterative Tikhonov regularization based on introducing a non-decreasing sequence of relaxation parameters into the stationary iterated Tikhonov regularization, providing a fast and roughly stable convergence. After investigating theoretically the convergence rate of the novel iterative Tikhonov regularization, we propose a series of numerical experiments for the evaluation of its accuracy and finally describe a real-world application in the field of 3-dimensional image restoration problems to verify its efficiency.