Nonstationary asymptotical regularization method with convex constraints for linear ill-posed problems

Abstract

We investigate the method of nonstationary asymptotical regularization for solving linear ill-posed problems in Hilbert spaces. This method introduces the convex constraints that are proper lower semicontinuous and allowed to be non-smooth, therefore can be used for sparsity and discontinuity reconstruction. Under some suitable conditions , the convergence and regularity of the proposed method are established. Under the discretion of Runge–Kutta method, different iteration modes can be deduced for numerical implementation. The numerical results of iteration modes under one-stage explicit Euler, one-stage implicit Euler and two-stage explicit Runge–Kutta are presented to illustrate the efficiency of the proposed method.