An asymptotical regularization with convex constraints for inverse problems. (1st April 2022) - Zhong, Min; Wang, Wei; Tong, Shanshan
Abstract: We investigate the method of asymptotical regularization for the stable approximate solution of nonlinear ill-posed problems F ( x ) = y in Hilbert spaces. The method consists of two components, an outer Newton iteration and an inner scheme providing increments by solving a local coupling linearized evolution equations. In addition, a non-smooth uniformly convex functional has been embedded in the evolution equations which is allowed to be non-smooth, including L 1 -liked and total variation-like penalty terms. We establish convergence properties of the method, derive stability estimates, and perform the convergence rate under the Hölder continuity of the inverse mapping. Furthermore, based on Runge–Kutta (RK) discretization, different kinds of iteration schemes can be developed for numerical realization. In our numerical experiments, four types iterative scheme, including Landweber type, one-stage explicit, implicit Euler and two-stage RK are presented to illustrate the performance of the proposed method.