A modified iterative regularization method for ill-posed problems

In this paper, we study a modified Landweber iteration method via a gradient flow equation induced by a weighted least squares functional. We investigate the proposed scheme for solving ill-posed problems under the setting of compact operator and pseudo-differential operator. The a-priori and the a-posteriori choice rules for regularization parameters are given and both rules yield the order optimal error estimates. Relative to the classical Landweber method, the new method significantly reduces the number of iterations needed to match an appropriate stopping criterion. As applications, we focus on some important ill-posed problems arising from mathematical physics. Numerical experiments are conducted for showing the validity of the scheme.