A new heuristic parameter choice rule in Tikhonov regularization applied for Ritz approximation of an ill-posed problem
Volume 48 / 2021
Abstract
The paper concerns a certain parameter selection criterion for a combination of Ritz approximation of an ill-posed problem and Tikhonov regularization in the case when the level of data noise is unknown. A new minimization-based noise-level-free choice rule is considered. The new functional depends on the discretization level, the Tikhonov regularization parameter, and noisy data. The functional is minimized over an interval depending on a number expressing how well the discrete operator approximates the initial one. In order to obtain the convergence of the discrete regularized solutions to the exact solution when the data noise tends to 0, a certain specific qualitative assumption on the noise is introduced. The convergence is proved under source conditions on the exact solution, the noise condition and certain additional assumptions on the operator approximation.
