The PI and his colleagues study the solution of Stochastic Partial Differential Equations (SPDEs) with emphasis on modeling processes in random heterogeneous media. A holistic mathematical view of the problem is adopted in which the stochastic input is explicitly modeled from experimental data and uncertainty is propagated via a deterministic solver. The key aspects of the proposed work are as follows: (A) Data-driven stochastic input modeling. This includes the development of reduced-order representations of experimentally observed input fields, construction of the mapping from the reduced input space to the high dimensional input space and estimating the probability density of the reduced input space as induced by the observations. (B) Uncertainty propagation. This includes the construction of a hierarchical, local, Bayesian surrogate of the deterministic code made efficient by the informative selection of deterministic runs using active learning, the sequential refining of the discretization of the solver, the explicit treatment of output correlations and/or output dimensionality reduction and analytical/ numerical integration of the surrogate for the calculation of the response statistics as well as error bars.
The work performed in this project allows better appreciation for the need to account for variabilities in the analysis and design of engineered and natural systems. It is such thinking that will contribute towards re-usable multifunctional design of components and systems in an environment prone to natural and manufacturing uncertainties. In particular, this research systematically addresses key problems that arise in modeling random heterogeneous media (existing, very expensive deterministic multiscale solvers, few experimental input realizations, non-isotropy of the response, high-dimensionality of input/output) and poses concrete mathematical questions for their resolution. It serves as a paradigm of a successful combination of seemingly diverse ideas of applied mathematics and computational statistics in order to answer realistic engineering questions. Informative (data-driven) stochastic simulations would improve safety and greatly benefit the design and optimization process for a wide array of industries while at the same time reducing costs. The student that participates in this research is trained in the interface of computational mathematics and Bayesian statistics. The research activity and courses organized benefit many diverse communities that have interests in predictive science and design in the presence of uncertainties. The algorithms, data and software developed are disseminated to the community through the web.
