A special modified Tikhonov regularization matrix for discrete ill-posed problems

Abstract

In this paper, we investigate the solution of large-scale linear discrete ill-posed problems with error-contaminated data. If the solution exists, it is very sensitive to perturbations in the data. Tikhonov regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one with a fidelity term and a penalty term, can reduce this sensitivity and is a popular approach to determine meaningful approximate solutions of such problems. The penalty term is determined by a regularization matrix. The choice of this matrix may significantly affect the quality of the computed approximate solution. In order to get an appropriate solution with improved accuracy, the paper constructs a special modified Tikhonov regularization (SMTR) matrix so as to include more useful information. The parameters involved in the presented regularization matrix are discussed. Besides, the corresponding preconditioner PSMTR is designed to accelerate the convergence rate of the CGLS method for solving Tikhonov regularization least-squares system. Furthermore, numerical experiments illustrate that the SMTR method and PSMTR preconditioner significantly outperform the related methods in terms of solution accuracy.