Truncated low‐rank methods for solving general linear matrix equations
Truncated low‐rank methods for solving general linear matrix equations
复制标题
求解一般线性矩阵方程的截断低秩方法
DOI：
10.1002/nla.1973
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发表时间：
2015
影响因子：
4.3
通讯作者：
Petar Sirkovic
中科院分区：
文献类型：
--
作者：
D. Kressner;Petar Sirkovic
本文研究大规模线性矩阵方程A1XB1T+⋯+AKXBKT=C的数值解。最直接的方法是从一个mN×mN线性系统的解计算X∈Rm×n，通常将m，n的可行值限制在最多几百个。我们的新方法利用了这样一个事实，即X通常可以很好地被低阶矩阵近似。它将贪婪的低阶数技术与Galerkin投影和预条件梯度相结合。反过来，只需要解m×m和n×n的线性系统。此外，这些线性系统继承了系数矩阵的稀疏性，这允许处理大到m=n=O(105)的线性矩阵方程。数值实验表明，所提出的方法对广义Lyapunov方程具有较好的求解效果。即使对于标准的Lyapunov方程，我们的方法也是有利的，因为我们不需要假设C具有低的秩数。版权所有©2015 John Wiley&Sons，Ltd.
This work is concerned with the numerical solution of large‐scale linear matrix equations A1XB1T+⋯+AKXBKT=C . The most straightforward approach computes X∈Rm×n from the solution of an mn × mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low‐rank matrix. It combines greedy low‐rank techniques with Galerkin projection and preconditioned gradients. In turn, only linear systems of size m × m and n × n need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as m = n = O(105). Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as we do not need to assume that C has low rank. Copyright © 2015 John Wiley & Sons, Ltd.
影响因子：
3.1
作者：
Dolgov, Sergey V.;Savostyanov, Dmitry V.
通讯作者：
Savostyanov, Dmitry V.