MATHICSE Technical Report : A Bayesian numerical homogenization method for elliptic multiscale inverse problems

A new strategy based on numerical homogenization and Bayesian techniques for solvingmultiscale inverse problems is introduced. We consider a class of elliptic problems which vary ata microscopic scale, and we aim at recovering the highly oscillatory tensor from measurements ofthe fine scale solution at the boundary, using a coarse model based on numerical homogenizationand model order reduction. We provide a rigorous Bayesian formulation of the problem, takinginto account different possibilities for the choice of the prior measure. We prove well-posednessof the effective posterior measure and, by means of G-convergence, we establish a link betweenthe effective posterior and the fine scale model. Several numerical experiments illustrate theefficiency of the proposed scheme and confirm the theoretical findings.