A regularization parameter in discrete ill-posed problems

The Tikhonov regularization method for discrete ill-posed problems is considered. For the practical choice of the regularization parameter alpha, some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered. This paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the ''L-corner'' of the plot considered in the logarithmic scale. Moreover, a new criterion for choosing alpha is introduced (independent of the shape of the plot) which gives a new interpretation of the ''corner criterion'' mentioned above. The existence of ''L-corner'' is discussed.