标题： 贪心随机块Kaczmarz方法用于矩阵方程AXB=C及其在彩色图像恢复中的应用 显示英文标题

标题： Greedy randomized block Kaczmarz method for matrix equation AXB=C and its applications in color image restoration

摘要： 鉴于Kaczmarz方法在求解大规模线性方程组$Ax=b$时的简单性和有效性，我们研究了贪心随机块Kaczmarz方法 (ME-GRBK) 及其松弛和确定性版本，以求解在工程科学应用中常见的矩阵方程$AXB=C$。结果表明，当矩阵方程相容时，我们的算法收敛于矩阵方程的唯一最小范数解，且其收敛速度比随机块Kacz马尔兹方法 (ME-RBK) 更快。此外，研究了求解矩阵方程$AXB=C$的块Kaczmarz方法 (ME-BK)，发现当矩阵方程相容时，ME-BK 方法收敛于解$A^{+}CB^{+}+X^{0}-A^{+}AX^{0}BB^{+}$。数值测试验证了理论结果，本文提出的的方法被应用于彩色图像恢复问题，以获得令人满意的恢复图像。 显示英文摘要

摘要： 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.