Fast realization algorithms for determining regularization parameters in linear inverse problems (Q2720387)

For the stable solution of linear ill-posed problems, Morozov's discrepancy principle is one of the well-known regularization parameter choice strategies for the Tikhonov regularization. NEWLINENEWLINENEWLINEFor the practical determination of the corresponding regularization parameter \( \alpha = \alpha(\delta) \), where \( \delta \) denotes the noise level, in this paper four different methods are considered. The first method is a cubically converging iterative scheme, where at iteration step \( n \), in the equation corresponding to the discrepancy principle the squared residual norm is approximated by an appropriate Taylor expansion at the current iterate \( \alpha_n \). From the resulting equation an approximation \( \alpha_{n+1} \) to the parameter \( \alpha(\delta) \) is then easily determined. NEWLINENEWLINENEWLINEIn the second iteration method under consideration, a similar approach is applied to a damped discrepancy principle. More precisely, in the equation corresponding to the discrepancy principle the squared residual norm is replaced by the Tikhonov functional, and at each iteration step the Tikhonov functional is approximated then by an appropriate Taylor expansion at the actual iterate. NEWLINENEWLINENEWLINEThe authors consider also an iterative scheme where at each iteration step, in the equation corresponding to the discrepancy principle, the squared residual norm is approximated by some functional which is rational with respect to \( \alpha \) and which contains coefficients that depend on the current iterate. Finally a hybrid method is considered. Numerical experiments are included to compare the presented algorithms.