A Bayesian numerical homogenization method for elliptic multiscale inverse problems

TL;DR

This paper introduces a Bayesian numerical homogenization approach for efficiently solving multiscale elliptic inverse problems, enabling the recovery of microscopic tensor properties from boundary measurements.

Contribution

It develops a rigorous Bayesian framework for multiscale inverse problems using numerical homogenization and model reduction, linking effective and fine-scale posteriors.

Findings

The method is computationally efficient for multiscale inverse problems.

Theoretical analysis confirms the well-posedness of the Bayesian formulation.

Numerical experiments demonstrate the approach's accuracy and robustness.

Abstract

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. 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.