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A Bayesian approach for multiscale inverse problems Abdulle, Assyr


In this talk we discuss a Bayesian approach for inverse problems involving elliptic differential equations with multiple scales. Computing repeated forward problems in a multiscale context is computationnally too expensive and we propose a new strategy based on the use of "effective" forward models originating from homogenization theory. Convergence of the true posterior distribution for the parameters of interest towards the homogenized posterior is established via G-convergence for the Hellinger metric. A computational approach based on numerical homogenization and reduced basis methods is proposed for an efficient evaluation of the forward model in a Markov Chain Monte Carlo procedure. We also discuss a methodology to account for the modeling error introduced by the effective forward model and the combination of the Bayesian multiscale method with a probabilisitic approach to quantify the uncertainty in building the effective forward model for a multiscale elastic problem in random media. References: A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, to appear in SIAM MMS. A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems, Preprint submitted for publication.

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