Fixed-point iterations in determining the Tikhonov regularization parameter

We review a Tikhonov parameter criterion based on the search for local minima of the function Psi(mu) (lambda) = x(lambda) y(mu) (lambda), mu > 0 where x(lambda) and y(lambda) are the squared residual norm and the squared solution norm, respectively, proposed earlier by Reginska (1996, SIAM J. Sci. Comput. 3 740). As a consequence, we demonstrate that extreme points of Psi(mu)(lambda) are fixed points of a related function, and then propose a fixed-point algorithm for choosing the Tikhonov parameter. The algorithm constructs a regularization parameter associated with the corner of the L-curve in log-log scale, thus yielding solutions with accuracy comparable to that of the L-curve method but at a lower computational cost. The performance of the algorithm on representative discrete ill-posed problems is evaluated and compared with results obtained by the L-curve method, generalized cross- validation and another fixed-point algorithm from the literature.