Operator Augmentation for General Noisy Matrix Systems

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.