We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution based on the current residue and effective orthogonality between blocks. This improved method provides significant gains in solving high-condition number linear systems that are either sparse, or dense least-squares problems that are significantly over/under determined. Considering the poor generalizability of preconditioners for such problems, it can also serve as a pre-solver for other iterative numerical methods when required, and as an inner iteration in certain types of GMRES solvers for linear systems.

翻译：本文提出了一种改进的行随机化块Kaczmarz方法，用于求解线性方程组。该改进方法在解的块更新过程中引入了正则化技术，并基于当前残差及块间有效正交性构建了动态建议分布。这种增强方法在求解高条件数线性系统时表现出显著优势，适用于稀疏系统或严重超定/欠定的稠密最小二乘问题。考虑到此类问题预条件子普遍泛化能力不足的特点，本方法在需要时可作为其他迭代数值方法的预求解器，也可作为特定类型GMRES线性求解器的内层迭代。