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.

翻译：近期，文献[17]提出了用于求解大规模线性系统的多步惯性随机Kaczmarz（MIRK）方法。本文在MIRK方法中引入贪心概率准则，并针对该准则提出了更紧凑的阈值参数。我们证明，所提出的贪心MIRK（GMIRK）方法在确定性线性收敛速度上优于MIRK方法和贪心随机Kaczmarz方法。此外，我们从几何角度揭示多步惯性外推方法可视为一种正交投影方法，并阐明其与草图投影方法[15]及斜投影技术[22]的内在联系。数值实验验证了理论结果。