  Recently, the multi-step inertial randomized Kaczmarz (MIRK) method for
solving large-scale linear systems was proposed in [17]. In this paper, we
incorporate the greedy probability criterion into the MIRK method, along with
the introduction of a tighter threshold parameter for this criterion. We prove
that the proposed greedy MIRK (GMIRK) method enjoys an improved deterministic
linear convergence compared to both the MIRK method and the greedy randomized
Kaczmarz method. Furthermore, we exhibit that the multi-step inertial
extrapolation approach can be seen geometrically as an orthogonal projection
method, and establish its relationship with the sketch-and-project method [15]
and the oblique projection technique [22]. Numerical experiments are provided
to confirm our results.
