Hierarchical Bayesian level set inversion

The level set approach has proven widely successful in the study of inverse problems for inter- faces, since its systematic development in the 1990s. Re- cently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be cir- cumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful con- sideration of the development of algorithms which en- code probability measure equivalences as the hierar- chical parameter is varied, this leads to well-defined Gibbs based MCMC methods found by alternating Metropolis-Hastings updates of the level set function and the hierarchical parameter. These methods demon- strably outperform non-hierarchical Bayesian level set methods

Robustness analysis of an uncertain computational model to predict well integrity for geologic CO2 sequestration

International audienceGeologic storage of CO2 must respond to demonstrations of safety, control and acceptability with authorities and public. The wells are essential elements of the storage system and constitute the only man-made intrusive element in the geologic systems. The role of containment of components of wells must then be ensured for hundreds of years, despite degradation mechanisms that affect their properties. Probabilistic approaches are used to take into account the uncertainties on the quantities of CO2 which migrate from the reservoir of CO2 towards the surface and towards the aquifer. Uncertainties are taken into account by using the generalized probabilistic approach which allows both the system-parameter uncertainties and the model uncertainties induced by modeling errors to be performed in the stochastic computational model. These probabilistic tools, applied to industrial projects, allow owners and operators to set up decisions and provide a strong support to long term safety demonstration with a high level of confidence, even in presence of uncertainties in the computational models

Random matrix models and nonparametric method for uncertainty quantification

Springer ReferenceInternational audienceThis paper deals with the fundamental mathematical tools and the associated computational aspects for constructing the stochastic models of random matrices that appear in the nonparametric method of uncertainties and in the random consti-tutive equations for multiscale stochastic modeling of heterogeneous materials. The explicit construction of ensembles of random matrices, but also the presentation of numerical tools for constructing general ensembles of random matrices are presented and can be used for high stochastic dimension. The developments presented are illustrated for the nonparametric method for multiscale stochastic mod-eling of heterogeneous linear elastic materials and for the nonparametric stochas-tic models of uncertainties in computational structural dynamics