Discrepancy Sets for Combined Least Squares Projection and Tikhonov Regularization

Abstract

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.