Using the L--curve for determining optimal regularization parameters

Summary.

The ``L--curve'' is a plot (in ordinary or doubly--logarithmic scale) of the norm of (Tikhonov--) regularized solutions of an ill--posed problem versus the norm of the residuals. We show that the popular criterion of choosing the parameter corresponding to the point with maximal curvature of the L--curve does not yield a convergent regularization strategy to solve the ill--posed problem. Nevertheless, the L--curve can be used to compute the regularization parameters produced by Morozov's discrepancy principle and by an order--optimal variant of the discrepancy principle proposed by Engl and Gfrerer in an alternate way.