Controlling semi-convergence phenomenon in non-stationary simultaneous iterative methods

When applying the non-stationary simultaneous iterative methods for solving an ill-posed set of linear equations, the error usually initially decreases but after some iterations, depending on the amount of noise in the data, and the degree of ill-posedness, it starts to increase. This phenomenon is called semi-convergence. We study the semi-convergence behavior of the non-stationary simultaneous iterative methods and obtain an upper bound for data error (noise error). Based on this bound, we propose new ways to specify the relaxation parameters to control the semi-convergence. The performance of our strategies is shown by examples taken from tomographic imaging.