交替投影的多项式加速：切比雪夫加速、收敛速率与收缩

中文摘要 AI 辅助

当光滑流形在干净交点附近相交时，若弗里德里希角较小，交替投影（AP）可能收敛缓慢。我们提出了不动点流形加速（FPMA）框架，用于逐点固定流形的光滑映射的局部收敛与多项式加速。在适当的正则性和一致法向谱稳定性条件下，FPMA建立了局部R线性收敛，并导出一个光滑极限映射，该映射诱导局部收缩。在不动点处，当变换后的法向块一致谱稳定时，满足p(1)=1的一致多项式加速保持所选极限对法向初始扰动的一阶响应。在充分光滑性下，我们给出了二阶诱导收缩的曲率准则，并证明稳定的一致多项式加速同时保持二次收缩展开和该准则。作为特例，我们为固定阶切比雪夫加速的精确AP和二阶诱导收缩建立了新的局部R线性收敛保证。对于均匀弗里德里希角θ_F∈(0,π/2)，加速严格提高了每次AP评估的局部收敛速率。在固定θ_F下，大阶极限谱因子预测当θ_F→0时，固定相对误差减少所需的评估次数为O(θ_F^{-1})，而普通AP为O(θ_F^{-2})。该框架还涵盖不精确、松弛和广义AP，在各自的正则性和稳定性条件下适用。

英文摘要

Alternating projections (AP) near a clean intersection of smooth manifolds can converge slowly when the Friedrichs angle is small. We develop the \emph{Fixed-Point Manifold Acceleration (FPMA) framework} for {local convergence and polynomial acceleration of} smooth maps that fix a manifold pointwise. {Under suitable regularity and uniform normal spectral stability, FPMA establishes local R-linear convergence and a smooth limiting map inducing a local retraction. At fixed points, consistent polynomial acceleration with $p(1)=1$ preserves the selected limit's first-order response to normal initial perturbations whenever the transformed normal blocks are uniformly spectrally stable. Under sufficient smoothness, we give a curvature criterion for second-order induced retractions and show that stable consistent polynomial acceleration preserves both the quadratic retraction expansion and this criterion. {As a special case, we establish new local R-linear convergence guarantees for fixed-degree Chebyshev-accelerated exact AP and second-order induced retractions. For uniform Friedrichs angle $θ_F\in(0,π/2)$, acceleration strictly improves the local convergence rate per AP evaluation. The large-degree limiting spectral factor at fixed $θ_F$ predicts $O(θ_F^{-1})$ evaluations for fixed relative error reduction as $θ_F\downarrow0$, versus $O(θ_F^{-2})$ for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}

发表机构

• University of Science and Technology of China（中国科学技术大学）

机构由 AI 辅助整理，请以论文原文为准。