Much of the uncertainty quantification (UQ) research over the last fifteen year has given little attention to critical problems necessary for predictive modelling of stochastic multiscale problems. They include modelling of correlations in space and time necessary to alleviate issues related to high stochastic dimensionality, ability to perform UQ tasks with limited data, accounting for the phenomenology of coarse graining and selection of effective variables, and many more. In this presentation, we will advocate the exploration of synergies between the machine learning and uncertainty quantification research communities towards addressing the aforementioned problems. In particular, we will present a data-‐driven probabilistic graphical model based methodology to efficiently perform uncertainty quantification in multiscale systems. 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 the high-‐dimensional joint probability distribution. The hyperparameters in the probabilistic model can be learned locally in the graph using various techniques including sequential Monte Carlo (SMC) method, EM or variational methods. The effective coarse grained variables arise naturally in the graphical model and their marginal distributions can be computed non-‐parametrically in a data-‐driven manner. We make predictions from the probabilistic graphical model using loopy belief propagation algorithms. Numerical examples will be presented to show the accuracy and efficiency of the predictive capability of the developed graphical model in multiscale fluid flow and materials simulations. We will conclude with a discussion of the many exciting open problems and unexplored research directions
Graph Theoretic Models for the Solution of Stochastic Multiscale Problems
Monday, September 19, 2016 4:15 PM – 5:15 PM
127 Hayes-‐Healy Center Colloquium Tea 3:45 PM to 4:15 PM 154 Hurley Hall
Nicholas Zabaras Department of Aerospace & Mechanical
Engineering University of Notre Dame
Department of Applied and Computational Mathematics and Statistics Colloquium