A Note On The Randomized Kaczmarz Method With A Partially Weighted Selection Step

In this note we reconsider two known algorithms which both usually converge faster than the randomized Kaczmarz method introduced by Strohmer and Vershynin(2009), but require the additional computation of all residuals of an iteration at each step. As already indicated in the literature, e.g. arXiv:2007.02910 and arXiv:2011.14693, it is shown that the non-randomized version of the two algorithms converges at least as fast as the randomized version, while still requiring computation of all residuals. Based on that observation, a new simple random sample selection scheme has been introduced by arXiv:2011.14693 to reduce the required total of residuals. In the same light we propose an alternative random selection scheme which can easily be included as a `partially weighted selection step' into the classical randomized Kaczmarz algorithm without much ado. Numerical examples show that the randomly determined number of required residuals can be quite moderate.