On the asymptotical regularization with convex constraints for inverse problems

TL;DR

This paper introduces a new asymptotical regularization method with convex constraints for nonlinear ill-posed inverse problems, capable of handling non-smooth penalties like L1 and total variation, with proven convergence and novel Runge-Kutta based discretizations.

Contribution

It develops a regularization framework accommodating non-smooth penalties and introduces Runge-Kutta type discretizations for iterative solution construction.

Findings

Convergence properties established under certain conditions.

Method effectively handles sparsity and piecewise constancy.

New iterative regularization methods proposed.

Abstract

In this paper, we consider the asymptotical regularization with convex constraints for nonlinear ill-posed problems. The method allows to use non-smooth penalty terms, including the L1-like and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy. Under certain conditions we give convergence properties of the methods. Moreover, we propose Runge-Kutta type methods to discrete the initial value problems to construct new type iterative regularization methods.