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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.

We consider determining $\R$-minimizing solutions of linear ill-posed problems $A x = y$, where $A: \X \to \Y$ is a bounded linear operator from a Banach space $\X$ to a Hilbert space $\Y$ and $\R: \X \to [0, \infty]$ is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of $y$ are available, we consider a dual gradient method to reconstruct the $\R$-minimizing solution using the average of these data. By terminating the method by either an {\it a priori} stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method. 

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