Approximation of the Tikhonov regularization parameter through Aitken's extrapolation

Abstract

In the present work, we study the determination of the regularization parameter and the computation of the regularized solution in Tikhonov regularization, by the Aitken's extrapolation method. In particular, this convergence acceleration method is adjusted for the approximation of quadratic forms that appear in regularization methods, such as the generalized cross-validation method, the quasi-optimality criterion, the Gfrerer/Raus method and the Morozov's discrepancy principle. We present several numerical examples to illustrate the effectiveness of the derived estimates for approximating the regularization parameter for several linear discrete ill-posed problems and we compare the described method with further existing methods, for the determination of the regularized solution.