Analysis of Iterated Tikhonov: Defect Correction for Ill-posed Problems

Given a compact operator G, we consider the ill-posed problem, given y , solve Gφ = y (approximately). Typically, in the presence of noise y / ∈ Range(G). We consider the iterated Tikhonov method for this problem. The method selects a regularization parameter based on stability and corrects several times to increase accuracy. We show that it gives a higher accuracy approximation to the noise-free solution at each step, both in the regular and not regular cases, and that it alleviates the parameter selection difficulty of Tikhonov regularization. We also show that each update also computes the sensitivity of the Tikhonov approximation with respect to the user selected regularization parameter.