关于Lyapunov方程共轭梯度法中迭代项秩增长的界
On bounds for rank growth of iterands in conjugate gradients for Lyapunov equation
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中文总结 AI 辅助
针对Lyapunov方程，研究共轭梯度法迭代中解近似、残差和方向向量秩的界，为低秩求解提供理论保证。
中文摘要 AI 辅助
我们关注求解Lyapunov方程$AX + XA^T = F$，其中$A$、$X$和$F$为方阵，$A$对称正定（SPD）且稀疏，$F$对称且低秩。解$X$同样对称且通常稠密，但可用低秩矩阵近似。若$n$很大，$X$无法直接计算，但可通过所谓的低秩算术（LRA）获得。该方程通过共轭梯度法（MCG）的矩阵重构求解。我们关注迭代过程中解近似$X_\ell$、残差$R_\ell$和方向向量$P_\ell$的秩的行为。
英文摘要
We focus on solving the Lyapunov equation $AX + XA^T = F$, where $A$, $X$ and $F$ are square matrices, $A$ is symmetric positive definite (SPD) and sparse, and $F$ is symmetric and of a low-rank. The solution $X$ is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If $n$ is large, $X$ cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations $X_\ell$, residuals $R_\ell$, and direction vectors $P_\ell$ during iterations.
发表机构
- Department of Mathematics, Technical University of Liberec（捷克理工大学数学系）
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