In the computational sciences, one must often estimate model parameters from data subject to noise and uncertainty, leading to inaccurate results. In order to improve the accuracy of models with noisy parameters, we consider the problem of reducing error in a linear system with the operator corrupted by noise. To address this problem, we extend the elliptic operator augmentation framework (Etter, Ying 2020) to the general nonsymmetric matrix case. We show that under the conditions of right-hand-side isotropy and noise symmetry that the optimal operator augmentation factor for the residual error is always positive, thereby making the framework amenable to a necessary bootstrapping step. While the above conditions are unnecessary for positive optimal augmentation factor in the elliptic case, we provide counter-examples that illustrate their necessity when applied to general matrices. When the noise in the operator is small, however, we show that the condition of noise symmetry is unnecessary. Finally, we demonstrate through numerical experiments on Markov chain problems that operator augmentation can significantly reduce error in noisy matrix systems -- even when the aforementioned conditions are not met.

翻译：在计算科学中,人们必须经常从受噪音和不确定性影响的数据中估计模型参数,从而得出不准确的结果。为了提高具有噪音参数的模型的准确性,我们考虑减少线性系统中由噪音腐蚀的操作者造成的错误的问题。为了解决这个问题,我们将椭圆操作者增强框架(Ettter, Ying 2020)扩大到一般的非对称矩阵情况。我们表明,在右侧的同位素和噪音对称条件下,剩余错误的最佳操作者增强系数总是正的,从而使框架适合必要的靴式步骤。虽然上述条件对于在椭圆情况下的正面最佳增强系数是不必要的,但我们提供了反示例,说明在对一般矩阵应用时,它们的必要性。但是,当操作者的噪音小时,我们表明,噪音的对称性条件是不必要的。最后,我们通过对Markov链问题进行的数字实验,表明,操作者增强作用者可以大大减少噪音矩阵系统中的错误 -- -- 即使没有达到上述条件。