On Morozov's method for Tikhonov regularization as an optimal order yielding algorithm
Many problems in science and engineering have their mathematical formulations as an operator equation \[ Kx= y,\tag{1} \] where \(K:X\to Y\) is a bounded linear operator between Hilbert spaces \(X\) and \(Y\) with its range \(R(K)\) not closed in \(Y\). It is well known that if \(R(K)\) is not closed, the equation (1) or the problem of solving (1) is ill-posed. Regularization procedures are employed for obtaining stable approximate solutions of ill-posed equations of the type (1). These procedures are especially useful when the data available is inexact. That is, we may have an approximation \(\widetilde y\) of \(y\) with a known error level \(\delta> 0\), \(\|y-\widetilde y\|\leq\delta\). In this paper the author shows 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 ordinary Tikhonov regularization and also estimates available under the Hilbert scales approach.
scientific article; zbMATH DE number 1278885