The reconstruction of solutions in statistical inverse problems in Hilbert spaces requires regularization, which is often based on a parametrized family of proposal estimators. The choice of an appropriate parameter in this family is crucial. We propose a modification of the classical discrepancy principle as an adaptive parameter selection. This varying discrepancy principle evaluates the misfit in some weighted norm, and it also has an incorporated emergency stop. These ingredients allow the order optimal reconstruction when the solution owns nice spectral resolution. Theoretical analysis is accompanied with numerical simulations, which highlight the features of the proposed varying discrepancy principle.
Lu, S., & Mathé, P. (2014). Discrepancy based model selection in statistical inverse problems. Journal of Complexity, 30(3), 290–308. https://doi.org/10.1016/j.jco.2014.02.002