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

M. T. Nair

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.

Keywords: tikhonov regularization, ill-posed equations, order-optimal algorithms, interpolation inequalities, Hilbert scales

Classification (MSC2000): 65R10, 65J10, 65J20, 65R20, 45B05, 45L10, 47A50

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