Fine scale uncertainty in parameter estimation for elliptic equations

We study the problem of estimating the coefficients in an elliptic partial differential equation using noisy measurements of a solution to the equation. Although the unknown coefficients may vary on many scales we aim only at estimating their slowly varying parts, thus reducing the complexity of the inverse problem. However, ignoring the fine-scale fluctuations altogether introduces uncertainty in the estimates, even in the absence of measurement noise. We propose a strategy for quantifying the uncertainty due to the fine scale fluctuations in the coefficients by modeling their effect on the solution of the forward problem using the central limit theorem. When this is possible, the Bayesian estimation of the coefficients reduces to a weighted least squares problem with a covariance matrix whose rank is low regardless of the number of measurements and does not depend on the details of the coefficient fluctuations.