A discretizing Tikhonov regularization method via modified parameter choice rules

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke-Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper.