With the growth of data, it is more important than ever to develop an efficient and robust method for solving the consistent matrix equation AXB=C. The randomized Kaczmarz (RK) method has received a lot of attention because of its computational efficiency and low memory footprint. A recently proposed approach is the matrix equation relaxed greedy RK (ME-RGRK) method, which greedily uses the loss of the index pair as a threshold to detect and avoid projecting the working rows onto that are too far from the current iterate. In this work, we utilize the Polyak's and Nesterov's momentums to further speed up the convergence rate of the ME-RGRK method. The resulting methods are shown to converge linearly to a least-squares solution with minimum Frobenius norm. Finally, some numerical experiments are provided to illustrate the feasibility and effectiveness of our proposed methods. In addition, a real-world application, i.e., tensor product surface fitting in computer-aided geometry design, has also been presented for explanatory purpose.

翻译：随着数据量增长，开发高效稳健的方法求解相容矩阵方程AXB=C比以往更为重要。随机Kaczmarz（RK）方法因其计算效率高和内存占用少而备受关注。最新提出的矩阵方程松弛贪心RK（ME-RGRK）方法，通过贪心地利用索引对的损失作为阈值来检测并避免将工作行投影到距离当前迭代过远的行上。本文利用Polyak动量和Nesterov动量进一步加速ME-RGRK方法的收敛速度。理论证明，所提方法线性收敛至具有最小Frobenius范数的最小二乘解。最后通过数值实验验证了所提方法的可行性和有效性，并辅以计算机辅助几何设计中的张量积曲面拟合这一实际应用进行说明。