发表机构
Eindhoven University of Technology（埃因霍温理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究谱有限记忆预条件器中簇点选择对舍入误差和敏感性的影响，提出加权中位数和加权算术平均最小化误差界，并强调有限精度下需综合考虑收敛、误差和谱信息不准确性。
AI 中文摘要
谱有限记忆预条件器（sLMP）将对称正定矩阵的前导特征值聚簇，以加速共轭梯度（CG）收敛。在实践中，簇点常被选为1。然而，在某些情况下，即使可获得高度精确的谱信息，这种选择也可能无法相对于未预条件的CG加速收敛。基于精确算术收敛分析提出了替代簇点，但此类分析无法解释这种有限精度行为。我们研究了簇点如何影响sLMP预条件CG中两个数值误差来源。首先，我们分析了预条件器应用过程中浮点舍入误差的传播，并推导了可计算的相对误差界。对于主子空间（由未预条件系统前导特征值对应的特征向量张成）及其正交补（由其余特征向量张成），这些界分别由前导特征值的加权中位数和加权算术平均数最小化。我们的分析解释了为何小簇点会强烈放大主子空间中的误差。其次，我们研究了构建预条件器时对主谱信息扰动的敏感性。由此产生的扰动界由扰动后主特征值的加权中位数最小化，权重由特征向量扰动幅度决定。在合成问题上的数值实验展示了预测的舍入误差和敏感性行为。综合这些结果，表明在有限精度下选择簇点应考虑精确算术收敛、舍入误差和可用谱信息的不准确性。
英文摘要
The spectral limited-memory preconditioner (sLMP) clusters leading eigenvalues of symmetric positive definite matrices to accelerate conjugate gradient (CG) convergence. In practice, the cluster point is often chosen to be unity. In some cases, however, this choice can fail to accelerate convergence relative to unpreconditioned CG, even when highly accurate spectral information is available. Alternative cluster points have been proposed based on exact-arithmetic convergence analysis, but such analysis does not explain this finite-precision behaviour. We study how the cluster point influences two sources of numerical error in sLMP-preconditioned CG. First, we analyse the propagation of floating-point rounding errors during application of the preconditioner and derive computable relative-error bounds. For the dominant subspace (spanned by the eigenvectors associated with the leading eigenvalues of the unpreconditioned system) and its orthogonal complement (spanned by the remaining eigenvectors), these bounds are minimized by a weighted median and a weighted arithmetic mean of the leading eigenvalues, respectively. Our analysis explains why small cluster points can strongly amplify errors in the dominant subspace. Second, we investigate sensitivity to perturbations in the dominant spectral information when constructing the preconditioner. The resulting perturbation bound is minimized by a weighted median of the perturbed dominant eigenvalues, with weights determined by the eigenvector perturbation magnitudes. Numerical experiments on synthetic problems illustrate the predicted rounding-error and sensitivity behaviour. Together, these results show that cluster-point selection in finite precision should account for exact-arithmetic convergence, rounding errors, and inaccuracies in the available spectral information.
CommentsSubmitted to SIAM Journal on Matrix Analysis and Applications (SIMAX)
