Seminar in Numerical Analysis: Assyr Abdulle (EPFL)

Probabilistic numerical methods, Bayesian inference and multiscale inverse problems

In this talk we discuss several challenges that arise in Bayesian inference for ordinary and partial differential equations. The numerical solvers used to compute the forward model of such problems induce a propagation of the discretization error into the posterior measure for the parameters of interest. This uncertainty originating from the numerical approximation error can be accounted for using probabilistic numerical methods. New probabilistic numerical methods for ordinary differential equations that share geometric properties of the true solution will be presented in the first part of this talk. 
In the second part of the talk, we will discuss a Bayesian approach for inverse problems involving elliptic partial 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. 

References:
A. Abdulle, G. Garegnani, Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration, Preprint (2018), submitted for publication.
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 (2018), submitted for publication.

For further information about the seminar, please visit this <link de forschung mathematik seminar-in-numerical-analysis internal link in current>webpage.

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