A Regularization Parameter in Discrete Ill-Posed Problems

Abstract

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.