Lavrentiev regularization for linear ill-posed problems under general source conditions
The Lavrentiev regularization \( A x_\alpha^\delta + \alpha x_\alpha^\delta = y^\delta \) for the approximate solution of ill-posed problems (1) \( Ax = y \) is considered. Here, \( \alpha > 0 \) denotes a regularization parameter and \( A: X \to X \) is a linear bounded injective selfadjoint nonnegative operator in a Hilbert space \( X \). In addition, \( y^\delta \in X \) is an approximation to the right-hand side \( y \in R(A) \) of equation (1) satisfying \( \| y^\delta - y \| \leq \delta \), where \( \| \cdot \| \) denotes the norm of the Hilbert space \( X \). For general source representations of the form \( x^\dagger = \varphi(A)z, \, \| z \| \leq E \), and under some additional conditions, the authors prove estimates of the form \( \| x_\alpha^\delta - x^\dagger \| \leq cE \rho^{-1}(\delta/E) \) for a priori- as well as a posteriori parameter choices of \( \alpha \), respectively. Here, \( x^\dagger \) denotes the solution of equation (1), and \( \rho(\lambda)= \lambda \varphi^{-1}(\lambda) \) holds. In addition, \( c \) is a nonnegative constant. The presented results then are extended to an iterated version of the method of Lavrentiev as well as to infinitely smoothing operators, respectively. Finally the results are applied to an inverse heat conduction problem.
scientific article; zbMATH DE number 2111233