Operator shifting for noisy elliptic systems

• Published in: Research in the mathematical sciences Vol. 10; no. 4
• Main Authors: Etter, Philip A., Ying, Lexing
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2023; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computational Mathematics and Numerical Analysis; Error reduction; Mathematical models; Mathematics; Mathematics and Statistics; Model accuracy; Parameters; Polynomials; System effectiveness; Elliptic systems; Random matrices; Monte Carlo; Polynomial expansion; Operator shifting
• ISSN: 2522-0144; 2197-9847
• DOI: 10.1007/s40687-023-00414-x

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 an elliptic linear system with the operator corrupted by noise. We assume the noise preserves positive definiteness, but otherwise, we make no additional assumptions about the structure of the noise. Under these assumptions, we propose the operator shifting framework, a collection of easy-to-implement algorithms that augment a noisy inverse operator by subtracting an additional shift term. In a similar fashion to the James–Stein estimator, this has the effect of drawing the noisy inverse operator closer to the ground truth by reducing both bias and variance. We develop bootstrap Monte Carlo algorithms to estimate the required shift magnitude for optimal error reduction in the noisy system. To improve the tractability of these algorithms, we propose several approximate polynomial expansions for the operator inverse and prove desirable convergence and monotonicity properties for these expansions. We also prove theorems that quantify the error reduction obtained by operator shifting. In addition to theoretical results, we provide a set of numerical experiments on four different graph and grid Laplacian systems that all demonstrate the effectiveness of our method.