Semi-convergence and relaxation parameters for a class of SIRT algorithms

Abstract

Abstract. This paper is concerned with the Simultaneous Iterative Reconstruction Technique (SIRT) class of iterative methods for solving inverse problems. Based on a careful analysis of the semi-convergence behavior of these methods, we propose two new techniques to specify the relaxation parameters adaptively during the iterations, so as to control the propagated noise component of the error. The advantage of using this strategy for the choice of relaxation parameters on noisy and ill-conditioned problems is demonstrated with an example from tomography (image reconstruction from projections). Key words. SIRT methods, Cimmino and DROP iteration, semi-convergence, relaxation parameters, tomographic imaging AMS subject classifications. 65F10, 65R32 1. Introduction. Large-scale discretizations of ill-posed problems (such as imaging problems in tomography) call for the use of iterative methods, because direct factorization methods are infeasible. In particular, there is an interest in regularizing iterations, where the iteration vector can be considered as a regularized solution with the iteration index playing the role of the regularizing parameter. Initially the iteration vector approaches a regularized solution,