Deanna Needell Iterative projective approach for highly corrupted linear systems ABSTRACT. We consider solving large-scale systems of linear equations Ax = b that are inconsis- tent. We propose a hybrid approach utilizing ideas from the randomized Kaczmarz method and those put forth by Agmon, Motzkin et al. that exhibits a faster initial convergence rate without sacrificing solution accuracy. We show that significant improvements in the convergence rate are possible when the residual vector has a high dynamic range, as is the case when the system has been corrupted by a small number of large errors in b. With this as our motivating example, we further develop an approach for this setting that allows detection of the corrupted entries and thus convergence to the “true” solution of the original uncorrupted system. We provide analytical justification for our approaches as well as experimental evidence on real and synthetic systems