On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaces

Abstract

ABSTRACTThis article is devoted to the study of nonstationary Iterated Tikhonov (nIT) type methods (Hanke M, Groetsch CW. Nonstationary iterated Tikhonov regularization. J Optim Theory Appl. 1998;98(1):37–53; Engl HW, Hanke M, Neubauer A. Regularization of inverse problems. Vol. 375, Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group; 1996. MR 1408680) for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an a posteriori strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method's efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem.