Projected randomized Kaczmarz methods

Abstract

For a constrained linear ill-posed problems, the projected algebra reconstruction technique (PART) usually has a faster convergence rate than the projected simultaneous iterative reconstruction technique (PSIRT), but can not be applied to the inconsistent case. In this work we propose the projected randomized Kaczmarz method (PRK) to further accelerate the convergence of PART and analyze its convergence. With the noise-free right-hand side, we show that the PRK method can converge exponentially in expectation to the solution for the consistent system. With the noisy right-hand side, we present some insights into the semiconvergence property of the PRK method and propose an extended version of PRK (PREK) method to seek the least squares solution. We prove that the PREK method can converge exponentially in expectation to the least squares solution for the inconsistent systems.