论文标题
线性系统的概率迭代方法
Probabilistic Iterative Methods for Linear Systems
论文作者
论文摘要
本文介绍了迭代方法的概率观点，用于近似解决方案$ \ mathbf {x} _* \ in \ mathbb {r}^d $的非线性线性系统$ \ mathbf {a} \ mathbf {a} \ mathbf {x}} _} _} _* = \ \ mathbf {在该方法中，$ \ mathbb {r}^d $上的标准迭代方法被提起以在概率分布的空间上作用$ \ mathcal {p}（\ Mathbb {r}^d）$。通常，迭代方法产生序列$ \ mathbf {x} _m $的近似值，将收敛到$ \ mathbf {x} _*$。相反，本文提出的迭代方法的输出是一系列概率分布$μ_m\ in \ Mathcal {p}（\ Mathbb {r}^d）$。分发输出两者都为$ \ mathbf {x} _*$提供了一个“最佳猜测”，例如，当尚未确切确定时，$ \ mathbf {x} _*$值的概率不确定性量化也提供了概率的不确定性量化。理论分析是在固定线性迭代方法的原型情况下提供的。在这种情况下，我们既表征了$μ_m$的收缩率，$ \ mathbf {x} _*$以及提供的不确定性量化的性质。我们以一个经验例证结束，该例证突出了概率迭代方法可以提供的解决方案不确定性的洞察力。
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.
