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 $\mu_m \in \mathcal{P}(\mathbb{R}^d)$. The distributional output both provides a "best guess" for $\mathbf{x}_*$, for example as the mean of $\mu_m$, 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 $\mu_m$ 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.
翻译:本文展示了一种有关迭代方法的比方观点, 以套用迭代方法来表示 $\\ mathbf{A}\\ mathbf{A}\ \\ mathbf{x\\ =\ mathbf{x{b{b}美元。 在方法中, $\ mathbb{R\\\\\\\ f{b} 美元的标准迭代方法被解除, 以在概率分布空间上操作 $\\ mathb{P}(\ mathb{R\b{x}) (\ mathb{x{x} 美元。 通常地, 一种迭代用方法产生一个序列, 用于计算 美元( max) 美元( 美元) 的值 。 以 美元( ) 美元( ) 美元( 美元) 的內值提供“ 最有可能的數據 ” 。 以 美元( 美元) 的內歐( ) 的數( 美元) 的數( 美元) 的內值) 的數數( ) 。