A probabilistic graphical model approach to stochastic multiscale partial differential equations

We develop a probabilistic graphical model based methodology to efficiently perform uncertainty quantification in the presence of both stochastic input and multiple scales. Both the stochastic input and model responses are treated as random variables in this framework. Their relationships are modeled by graphical models which give explicit factorization of a high-dimensional joint probability distribution. The hyperparameters in the probabilistic model are learned using sequential Monte Carlo (SMC) method, which is superior to standard Markov chain Monte Carlo (MCMC) methods for multi-modal distributions. Finally, we make predictions from the probabilistic graphical model using the belief propagation algorithm. Numerical examples are presented to show the accuracy and efficiency of the predictive capability of the developed graphical model.