In a stochastic noise setting the Lepskij balancing principle for choosing the regularization parameter in the regularization of inverse problems is depending on a parameter $tau$ which in the currently known proofs is depending on the unknown noise level of the input data. However, in practice this parameter seems to be obsolete. We will present an explanation for this behavior by using a stochastic model for noise and initial data. Furthermore, we will prove that a small modification of the algorithm also improves the performance of the method, in both speed and accuracy.
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