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International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2018025837
pages 447-482

BEYOND BLACK-BOXES IN BAYESIAN INVERSE PROBLEMS AND MODEL VALIDATION: APPLICATIONS IN SOLID MECHANICS OF ELASTOGRAPHY

L. Bruder
Mechanics and High Performance Computing Group, Technical University of Munich, Parkring 35, 85748 Garching, Germany
Phaedon-Stelios Koutsourelakis
Continuum Mechanics Group, Technical University of Munich, Boltzmannstrasse 15, 85748 Garching, Germany

ABSTRACT

The present paper is motivated by one of the most fundamental challenges in inverse problems, that of quantifying model discrepancies and errors. While significant strides have been made in calibrating model parameters, the overwhelming majority of pertinent methods is based on the assumption of a perfect model. Motivated by problems in solid mechanics which, as all problems in continuum thermodynamics, are described by conservation laws and phenomenological constitutive closures, we argue that in order to quantify model uncertainty in a physically meaningful manner, one should break open the black-box forward model. In particular, we propose formulating an undirected probabilistic model that explicitly accounts for the governing equations and their validity. This recasts the solution of both forward and inverse problems as probabilistic inference tasks where the problem's state variables should not only be compatible with the data but also with the governing equations as well. Even though the probability densities involved do not contain any black-box terms, they live in much higher-dimensional spaces. In combination with the intractability of the normalization constant of the undirected model employed, this poses significant challenges which we propose to address with a linearly scaling, double layer of stochastic variational inference. We demonstrate the capabilities and efficacy of the proposed model in synthetic forward and inverse problems (with and without model error) in elastography.


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