A Bayesian extension to FEMU for identification of spatially varying stochastic elastic properties from digital image and volume correlation measurements

We present a Bayesian framework for the identification of stochastic and spatially varying elastic parameters using noisy displacement observations obtained with DIC or DVC trials. Our method is a generalization of identification procedures such as FEMU or I-DIC to materials with spatially varying properties and stochastic mesostructures, where the elasticity tensor is modelled as a parametric non-Gaussian random field. Both the elastic parameters and the parameters of the random field model are identified jointly from the displacement measurement. We formulate the approach as a hierarchical Bayesian PDE-constrained inverse problem and MAP estimates are obtained through gradient based optimization. We resort to an adjoint based formulation and leverage automatic differentiation to derive the parameter sensitivities. We show how modelling unknown parameters with Gaussian Random Fields leads to a natural Bayesian regularization and leverage the use of Whittle-Matérn priors. Covariance parameter estimation is discussed, and we propose an empirical Bayes approach to avoid numerical shortcomings related to a standard hierarchical model. A set of numerical examples is presented to assess the performance of the proposed method, based on synthetic data generated through Matérn Random fields. In particular, we show how data noise is naturally modelled by the Bayesian formulation and impacts spatial covariance of identified parameters.