In view of the advantages of simplicity and effectiveness of the Kaczmarz method, which was originally employed to solve the large-scale system of linear equations $Ax=b$, we study the greedy randomized block Kaczmarz method (ME-GRBK) and its relaxation and deterministic versions to solve the matrix equation $AXB=C$, which is commonly encountered in the applications of engineering sciences. It is demonstrated that our algorithms converge to the unique least-norm solution of the matrix equation when it is consistent and their convergence rate is faster than that of the randomized block Kaczmarz method (ME-RBK). Moreover, the block Kaczmarz method (ME-BK) for solving the matrix equation $AXB=C$ is investigated and it is found that the ME-BK method converges to the solution $A^{+}CB^{+}+X^{0}-A^{+}AX^{0}BB^{+}$ when it is consistent. The numerical tests verify the theoretical results and the methods presented in this paper are applied to the color image restoration problem to obtain satisfactory restored images.

翻译：鉴于Kaczmarz方法在求解大规模线性方程组$Ax=b$时所展现的简洁性与有效性优势，本文研究贪婪随机分块Kaczmarz方法（ME-GRBK）及其松弛与确定性版本，以求解工程科学应用中常见的矩阵方程$AXB=C$。理论证明表明：当矩阵方程相容时，所提算法收敛于其唯一最小范数解，且收敛速率优于随机分块Kaczmarz方法（ME-RBK）。此外，本文探究了求解矩阵方程$AXB=C$的分块Kaczmarz方法（ME-BK），发现当方程相容时，ME-BK方法收敛于解$A^{+}CB^{+}+X^{0}-A^{+}AX^{0}BB^{+}$。数值实验验证了理论结果，并将所提方法应用于彩色图像复原问题，获得了令人满意的复原图像。