Regularization in Hilbert scales under general smoothing conditions

Abstract

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high- and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience.