基于几何平滑动量的大规模线性方程组自适应确定性块Kaczmarz方法

An Adaptive Deterministic Block Kaczmarz Algorithm Based on Geometrically Smoothed Momentum for Solving Large-Scale Linear Systems

摘要: 为高效求解大规模相容线性方程组，本文在已有自适应确定性块Kaczmarz (ADBK)方法的基础上，引入几何平滑动量加速机制，提出了基于几何平滑动量的自适应确定性块Kaczmarz方法。该方法继承了ADBK通过残差范数自适应选取块索引、避免显式计算子矩阵伪逆的计算优势，并在不增加额外计算复杂度的前提下进一步提升了收敛速度。数值实验结果表明，与原始ADBK方法及多种典型算法相比，所提出的方法在迭代次数、计算时间和收敛速度方面均具有明显优势。

Abstract: To efficiently solve large-scale consistent systems of linear equations, this paper proposes an adaptive deterministic block Kaczmarz method based on geometric smooth momentum, building upon the existing Adaptive Deterministic Block Kaczmarz (ADBK) method. The proposed method inherits the computational advantages of ADBK in adaptively selecting block indices through residual norms and avoiding explicit computation of submatrix pseudoinverses. Furthermore, by introducing a geometric smooth momentum acceleration mechanism, the convergence speed is enhanced without increasing additional computational complexity. Numerical experimental results demonstrate that compared with the original ADBK method and various typical algorithms, the proposed method exhibits significant advantages in terms of iteration count, computational time, and convergence rate.