This project seeks to develop a new theory, algorithms, and computational tools to enable a linearly scalable characterization and solution of partial differential equations (PDEs) with high-dimensional random inputs. In realistic situations where the dynamics of a system is intrinsically variable or understanding of the underlying physical laws is incomplete, there is a need to represent and quantify the impact of such uncertainties on quantities of interest. A fundamental difficulty arises when the number of independent sources of uncertainty is large. In these situations, standard approaches to the solution of PDEs with uncertain inputs encounter an exponential growth of computational complexity, i.e., the so-called curse-of-dimensionality. The proposed effort tackles this issue through a new approach to stochastic model reduction based on low-rank separated representation of multi-variate solutions. Both reduced-order characterization, via numerical homogenization, and forward propagation of uncertainties will be considered. While this approach is applicable to a wide variety of stochastic PDE-based models, this project is primarily focused on the solution of nonlinear, advection-reaction-diffusion equations with high-dimensional stochastic velocity and diffusion fields.
One of the major problems of interest in science and engineering is the prediction of multi-scale and multi-physics systems with high-dimensional uncertainties. Examples are common in combustion, energy storage systems, fusion energy, among others. The new numerical techniques of this award, based upon recent ideas from nonlinear, multi-linear, and sparse approximation of functions with many variables, will significantly advance the state-of-the-art in computer simulation of such problems. To further broaden the impact of this project, the PIs will develop graduate level service courses to introduce the new uncertainty quantification methods to researchers working on complex systems.
