An asymptotical regularization with convex constraints for inverse problems

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 H\"{o}lder 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, $1$-stage explicit, implicit Euler and $2$-stage RK are presented to illustrate the performance of the proposed method.