AI 中文总结
研究一般赋范空间中克拉索夫斯基-曼恩不动点迭代，借助与\(\mathbb{Z}^2\)上马尔可夫链的联系及组合学计数论证，得出迭代间距离和不动点残差的非渐近误差界，还推导了不精确迭代的误差界。
AI 中文摘要
我们重新审视一般赋范空间中用于压缩映射和非扩张映射的经典克拉索夫斯基-曼恩不动点迭代。该迭代在广泛领域普遍存在，包括凸优化、单调包含、马尔可夫决策过程、非线性偏微分方程的欠松弛方法等。利用与\(\mathbb{Z}^2\)上马尔可夫链的显著联系，借助格路径枚举组合学的计数论证，我们得出迭代间距离的显式估计以及不动点残差的非渐近误差界。当压缩参数趋近于1时，这些界能平稳恢复非扩张映射的已知估计。在此基础上，我们进一步推导了不精确克拉索夫斯基-曼恩迭代的误差界。
英文摘要
We revisit the classical Krasnosel'skii-Mann fixed point iteration for contractions and nonexpansive maps in general normed spaces. This iteration is ubiquitous across a wide range of areas, including convex optimization, monotone inclusions, Markov decision processes, under-relaxed methods for nonlinear PDEs, and more. Drawing on a remarkable connection with a Markov chain on $\mathbb{Z}^2$, and using counting arguments from enumerative combinatorics of lattice paths, we derive explicit estimates for the distance between iterates, as well as non-asymptotic error bounds for the fixed point residuals. As the contraction parameter approaches one, these bounds smoothly recover the known estimates for nonexpansive maps. Building upon these estimates, we further derive error bounds for inexact Krasnosel'skii-Mann iterations.
