Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

翻译：经典Krylov子空间投影方法求解线性问题$Ax = b$时输出近似解$\widetilde{x}\simeq x$。近年来，人们认识到投影方法可从统计视角进行理解。这类概率投影方法以分布$p(\widetilde{x})$替代点估计$\widetilde{x}$。理论上，由此产生的以分布形式编码的不确定性能与其他不确定性进行有意义的结合，能通过计算管道传播，并应用于概率决策理论框架。我们解决的核心问题是：现有概率投影方法会导致后验分布校准不良。通过改进先前工作的协方差矩阵，我们使其不再包含$A^{-1}$或$A^{-1}A^{-T}$等不良对象，既能产生非平凡不确定性，又能以任意投影方法的均值作为后验分布均值。我们还提出一种数值成本较低的变体方案，适用于不确定性先验已知的场景。由于通常缺乏先验信息，我们进而提出一种实用的不确定性校准方法，该方法在将底层投影方法数值成本大致翻倍的代价下，实现了较为理想的校准效果。