线性方程组的概率迭代法
Probabilistic Iterative Methods for Linear Systems
摘要：
本文提出了一个概率的角度来看，迭代方法逼近的解决方案$\mathbf{x}_* \in \mathbb{R}^d$的非奇异线性系统$\mathbf{A} \mathbf{x}_* = \mathbf{b}$。在该方法中，一个标准的迭代方法$\mathbb{R}^d$提升到作用于概率分布空间$\mathcal{P}(\mathbb{R}^d)$。传统上，迭代方法产生收敛于$\mathbf{x}_*$的近似序列$\mathbf{x}_m$。本文提出的迭代方法的输出是概率分布序列$μ_m \in \mathcal{P}(\mathbb{R}^d)$。分布输出既提供了对$\mathbf{x}_*$的“最佳猜测”，例如作为$μ_m$的平均值，又提供了在没有精确确定时对$\mathbf{x}_*$的值的概率不确定性量化。理论分析提供了一个固定的线性迭代方法的典型情况下。在这种情况下，我们将$μ_m$收缩到$\mathbf{x}_*$上的原子测量的速率和提供的不确定性量化的性质进行比较。我们的结论与实证说明，突出了洞察解决方案的不确定性，可以提供概率迭代方法。
英文摘要：
This paper presents a probabilistic perspective on iterative methods for approximating the solution $\mathbf{x}_* \in \mathbb{R}^d$ of a nonsingular linear system $\mathbf{A} \mathbf{x}_* = \mathbf{b}$. In the approach a standard iterative method on $\mathbb{R}^d$ is lifted to act on the space of probability distributions $\mathcal{P}(\mathbb{R}^d)$. Classically, an iterative method produces a sequence $\mathbf{x}_m$ of approximations that converge to $\mathbf{x}_*$. The output of the iterative methods proposed in this paper is, instead, a sequence of probability distributions $μ_m \in \mathcal{P}(\mathbb{R}^d)$. The distributional output both provides a "best guess" for $\mathbf{x}_*$, for example as the mean of $μ_m$, and also probabilistic uncertainty quantification for the value of $\mathbf{x}_*$ when it has not been exactly determined. Theoretical analysis is provided in the prototypical case of a stationary linear iterative method. In this setting we characterise both the rate of contraction of $μ_m$ to an atomic measure on $\mathbf{x}_*$ and the nature of the uncertainty quantification being provided. We conclude with an empirical illustration that highlights the insight into solution uncertainty that can be provided by probabilistic iterative methods.
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。
