A Bayesian Numerical Homogenization Method for Elliptic Multiscale Inverse Problems
Assyr Abdulle, Andrea Di Blasio · SIAM/ASA Journal on Uncertainty Quantification · 2020
A new strategy based on numerical homogenization and Bayesian techniques for solving multiscale inverse problems is introduced. We consider a class of elliptic problems which vary at a microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements of the fine scale solution at the boundary, using a coarse model based on numerical homogenization and model order reduction. Assuming a known micro structure, our aim is to recover a macroscopic scalar parameterization of the microscale tensor. We provide a rigorous Bayesian formulation of the problem, taking into account different possibilities for the choice of the prior measure. We prove well-posedness of the effective posterior measure and, by means of G-convergence, we establish a link between the effective posterior and the fine scale model. Several numerical experiments illustrate the efficiency of the proposed scheme and confirm the theoretical findings.