An accelerated inexact Newton-type regularizing algorithm for ill-posed operator equations

We propose and analyze a new iterative regularization approach, called IN-SETPG, for efficiently solving nonlinear ill-posed operator equations in the Hilbert-space setting. IN-SETPG consists of an outer iteration and an inner iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method, while the inner iteration is performed by a sequential subspace optimization method based on the two-point gradient iteration. The key idea behind IN-SETPG is that, unlike the standard Landweber method, it uses multiple search directions per iteration in combination with an adaptive step size in order to reduce the total number of iterations. The regularization property of IN-SETPG has been established, i.e., the iterate converges to a solution of the nonlinear problem with exact data when the noise level tends to zero. Various numerical experiments are presented to demonstrate that, compared with the original inexact Newton iteration, IN-SETPG can achieve better reconstruction results and remarkable acceleration.