Fast CG-Based Methods for Tikhonov--Phillips Regularization

Tikhonov--Phillips regularization is one of the best-known regularization methods for inverse problems. A posteriori criteria for determining the regularization parameter α require solving (*) (A^*A+α I) x =A^* yδ for different values of α. We investigate two methods for accelerating the standard cg-algorithm for solving the family of systems (*). The first one utilizes a stopping criterion for the cg-iterations which depends on α and δ. The second method exploits the shifted structure of the linear systems (*), which allows us to solve (*) simultaneously for different values of α. We present numerical experiments for three test problems which illustrate the practical efficiency of the new methods. The experiments as well as theoretical considerations show that run times are accelerated by a factor of at least 3.