Near-optimal parameters for Tikhonov and other regularization methods (Q2780572)

For overdetermined discrete ill-posed linear problems having coefficient matrix with full rank for which the discrete Picard condition is satisfied a heuristic for choosing the regularization parameter in Tikhonov regularization and Lavrentiev's simplified Tikhonov regularization is proposed. While some previously proposed methods involve minimizing a bound for the error in the approximate solution or exploit asymptotics of the approximate solution, the idea of this method is to minimize the deviation of the regularized approximate solution from the noise-free true solution of the discrete problem. NEWLINENEWLINENEWLINEA singular value decomposition analysis suggests that a near optimal regularization parameter is the smallest positive root of a function that depends on the singular values, the standard deviation of the noise in the discrete data and an index related to the discrete Picard condition. This choice of the parameter is illustrated on two large scale discrete ill-posed problems and compared with good results to a method of \textit{M. Hanke} and \textit{T. Raus} [SIAM J. Sci. Comput., 17, No. 4, 956-972 (1996; Zbl 0859.65051)], the discrepancy methods and the generalized cross validation method.