This proposal concerns the analysis of transport phenomena in heterogeneous random media with emphasis on the multi-length scale variations in properties that these phenomena exhibit, and the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochastic processes. The non-intrusive stochastic multiscale framework being developed has three key components: (a) A computational framework that encodes the limited information available about the variability of the (multiscale) material properties (permeability) into a reduced-order stochastic input model, (b) An adaptive sparse grid collocation framework for solving the stochastic PDEs involved and (c) A mathematically consistent strategy to exchange information across length scales for the solution of stochastic multiscale problems. The key concept explored in the data-driven reduced-order stochastic input model construction is the low-dimensional parametrization of manifolds embedded in high-dimensional spaces. The sparse grid collocation approach constructs the stochastic solution solely based on function calls to the corresponding deterministic physical simulator. The framework is based on hierarchical basis functions in multiple dimensions. Adaptivity and convergence are ensured by utilizing a local support while scalability is guaranteed by the careful choice of appropriate data structures. The information transfer strategies are based on the decoupled structure of the stochastic and multiscale algorithms.
The results of this research will impact the understanding of flow processes in random media. Thermal and hydrodynamic transport in random heterogeneous media are ubiquitous processes occurring in various scales ranging from the large scale (e.g. geothermal energy systems, oil recovery, geological heating of the earth?s crust) to smaller scales (e.g. heat transfer through composites, polycrystals, flow through pores, inter-dendritic flow in solidification, heat transfer through fluidized beds). There has been increasing scientific, technological and economic interests in predictive modeling of the thermal and hydrodynamic behavior of such media. In addition, this work can be valuable in understanding other systems that are poorly understood and/or controlled due to the gappy and inaccurate data available for their description. The problems addressed provide a unique and valuable training opportunity for students to learn, develop and apply cutting edge computational mathematics techniques to a variety of complex systems.
