On the Adaptive Selection of the Parameter in Regularization of Ill-Posed Problems

We study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko [Z. Vychisl. Mat. Mat. Fiz., 4 (1964), pp. 564-571] and to an adaptation scheme proposed in a statistical context by Lepskii [Theory Probab. Appl., 36 (1990), pp. 454-466] is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for nonlinear ill-posed problems, and an inverse problem of the two-dimensional profile reconstruction.