Optimally scaled vector regularization method to solve ill-posed linear problems

Abstract

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization method. The presently proposed vector regularization method is shown to be well conditioning to overcome the ill-posedness of the linear equations system. The OSVRM causes a significant improvement of stability and accuracy in the numerical solution of ill-posed linear problem, and its convergence speed is as fast as by solving the well-posed linear problem. Some tests of the linear inverse problems confirm the efficiency and accuracy of the OSVRM.