On Morozov’s Method for Tikhonov Regularization as an Optimal Order Yielding Algorithm

Abstract

It is shown that Tikhonov regularization for an ill-posed operator equation $Kx=y$ using a possibly unbounded regularizing operator $L$ yields an order-optimal algorithm with respect to certain stability set when the regularization parameter is chosen according to Morozov’s discrepancy principle. A more realistic error estimate is derived when the operators $K$ and $L$ are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also estimates available under the Hilbert scales approach.