On the regularization of the ill-posed logarithmic kernel integral equation of the first kind.

The logarithmic kernel integral equation of the first kind is investigated as improperly posed problem considering its right-hand side as observed quantity in a suitable space with a weaker norm. The improperly posed problem is decomposed into a well-posed one, extensively studied in the literature (cf. e.g. [11], [13], [14]), and an ill-posed imbedding problem. For the ill-posed part a modified truncated singular value decomposition regularization method is proposed that allows an easily performable a-posteriori parameter choice. The whole problem is then solved by combining the regularization method with a numerical procedure from [13] for the well-posed part. Finally, an error estimate is given revealing the influence of the observation error on the approximation error of the numerical procedure. For a specification of the discretization parameter as a known function of the noise level only, the optimal convergence order is achieved