残差加权随机雅可比：通过残差集中性改进界及异步扩展
Residual-Weighted Randomized Jacobi: Sharpened Bounds via Residual Concentration and Asynchronous Extension
AI总结：
针对对称正定线性系统，提出残差幂加权随机雅可比方法，利用逆参与比（IPR）收紧收敛界，并扩展至异步并行，实验验证了界和IPR轨迹的并发不敏感性。
摘要：
我们研究对称正定线性系统的随机平稳方法，其中分量$j$以正比于$|r_j|^\ell$的概率被选中。这个幂加权族在均匀随机雅可比（$\ell \to 0$）和高斯-索斯韦尔贪婪松弛（$\ell \to \infty$）之间连续插值。对于中心情况$\ell = 2$，我们使用逆参与比（IPR）$\nu^2(r) = n\|r\|_4^4/\|r\|_2^4$来收紧标准的一步收敛分析，当残差均匀时IPR等于1，当残差集中时增长至$n$。得到的界将期望的每步进展精确地放大$\nu^2$倍于均匀采样基线。IPR可以在$O(n)$成本下在线计算，并作为每次迭代的诊断指标。 我们通过Avron-Druinsky-Gupta框架将分析扩展到异步幂加权雅可比，得到基于轮次的收敛定理，其中IPR同时控制进展系数和允许延迟窗口。在共享内存硬件上的数值实验支持收紧的界，并显示IPR轨迹本质上对并发不敏感。出乎意料的是，一致读取执行（ADG分析中较容易的情况）在高并发下使幂加权采样不稳定，而不一致读取保持稳定；相同的IPR放大了进展，也放大了线程碰撞率，而不一致读取似乎吸收了这种碰撞率。我们提出了一种反馈阻尼机制，并验证了关于其对问题规模依赖性的两个预测。
英文摘要：
We study randomized stationary methods for symmetric positive definite linear systems in which component $j$ is selected with probability proportional to $|r_j|^\ell$. This power-weighted family interpolates continuously between uniform randomized Jacobi as $\ell \to 0$ and Gauss--Southwell greedy relaxation as $\ell \to \infty$. For the central case $\ell = 2$, we sharpen the standard one-step convergence analysis using the inverse participation ratio (IPR) $ν^2(r) = n\|r\|_4^4/\|r\|_2^4$, which equals $1$ when the residual is uniform and grows toward $n$ as it concentrates. The resulting bound amplifies the expected per-step progress by exactly $ν^2$ over the uniform-sampling baseline. The IPR can be computed online at $O(n)$ cost and doubles as a per-iteration diagnostic. We extend the analysis to asynchronous power-weighted Jacobi via the Avron--Druinsky--Gupta framework, obtaining an epoch-based convergence theorem in which the IPR controls both the progress coefficient and the allowed-delay window. Numerical experiments on shared-memory hardware support the sharpened bound and show the IPR trajectory is essentially concurrency-insensitive. Unexpectedly, consistent-reads execution, the easier case for the ADG analysis, destabilizes power-weighted sampling at high concurrency while inconsistent reads remain stable; the same IPR that amplifies progress amplifies a thread-collision rate that inconsistent reads appear to absorb. We propose a feedback-damping mechanism and verify two predictions about its dependence on problem size.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
