A Bayesian numerical homogenization method for elliptic multiscale inverse problems
This work provides a rigorous Bayesian framework for multiscale inverse problems, benefiting researchers in computational science and engineering who need to infer fine-scale properties from coarse observations.
The authors introduce a Bayesian numerical homogenization method for elliptic multiscale inverse problems, enabling recovery of highly oscillatory tensors from boundary measurements. Numerical experiments demonstrate efficiency and confirm theoretical well-posedness and convergence.
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.