论文标题
在全球随机块kaczmarz算法上，用于求解大规模矩阵方程
On global randomized block Kaczmarz algorithm for solving large-scale matrix equations
论文作者
论文摘要
随机Kaczmarz算法是由于其简单性和效率而解决大型线性系统的最流行方法之一。在本文中，我们提出了两类全局随机Kaczmarz方法，用于求解大规模线性矩阵方程$ axb = c $，全局随机块kaczmarz算法和全球随机块kaczmarz算法。全局随机块kaczmarz算法的特征是，在每次迭代处将电流迭代投影到素描的矩阵方程的解决方案空间上，而全局随机平均块kaczmarz方法是无用的，因此可以在平行计算单元上部署显着改善计算时间。我们证明，这两种方法在均值中汇聚到最小规范解决方案$ x _*= a^†cb^†$的给定线性矩阵方程的$。收敛速率取决于数据矩阵及其子膜的几何特性以及块的大小。数值结果表明，我们提出的算法对于求解大规模基质方程是有效的。特别是，当应用于图像过度问题时，它们也可以实现令人满意的性能。
The randomized Kaczmarz algorithm is one of the most popular approaches for solving large-scale linear systems due to its simplicity and efficiency. In this paper, we propose two classes of global randomized Kaczmarz methods for solving large-scale linear matrix equations $AXB=C$, the global randomized block Kaczmarz algorithm and global randomized average block Kaczmarz algorithm. The feature of global randomized block Kaczmarz algorithm is the fact that the current iterate is projected onto the solution space of the sketched matrix equation at each iteration, while the global randomized average block Kaczmarz approach is pseudoinverse-free and therefore can be deployed on parallel computing units to achieve significant improvements in the computational time. We prove that these two methods linearly converge in the mean square to the minimum norm solution $X_*=A^†CB^†$ of a given linear matrix equation. The convergence rates depend on the geometric properties of the data matrices and their submatrices and on the size of the blocks. Numerical results reveal that our proposed algorithms are efficient and effective for solving large-scale matrix equations. In particular, they can also achieve satisfying performance when applied to image deblurring problems.
