A nonparametric belief propagation method for uncertainty quantification with applications to flow in random porous media

A probabilistic graphical model approach to uncertainty quantification for flows in random porous media is introduced. Model reduction techniques are used locally in the graph to represent the random permeability. Then the conditional distribution of the multi-output responses on the low dimensional representation of the permeability field is factorized into a product of local potential functions. An expectation–maximization algorithm is used to learn the nonparametric representation of these potentials using the given input/output data. We develop a nonparametric belief propagation method for uncertainty quantification by employing the loopy belief propagation algorithm. The nonparametric nature of our model is able to capture non-Gaussian features of the response. The proposed framework can be used as a surrogate model to predict the responses for new input realizations as well as our confidence on these predictions. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed framework for solving uncertainty quantification problems in flows through porous media using stationary and non-stationary permeability fields.