Arnoldi Projection Fractional Tikhonov for Large Scale Ill-Posed Problems

Abstract

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems, so fractional Tikhonov methods have been introduced to remedy this shortcoming. And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining a partial Arnoldi decomposition of the given matrix. In this paper, we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process. The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators. A suitable value of the regularization parameter is determined by the discrepancy principle. Numerical examples with application to image restoration are carried out to examine that the performance of the method.