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Publication# A Bayesian numerical homogenization method for elliptic multiscale inverse problems

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.

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Γ-convergence

In the field of mathematical analysis for the calculus of variations, Γ-convergence (Gamma-convergence) is a notion of convergence for functionals. It was introduced by Ennio de Giorgi. Let be a topological space and denote the set of all neighbourhoods of the point . Let further be a sequence of functionals on . The and the are defined as follows: are said to -converge to , if there exist a functional such that . In first-countable spaces, the above definition can be characterized in terms of sequential -convergence in the following way.

Haar measure

In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory.

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