Many physical systems of fundamental and industrial importance are significantly affected by the underlying fluctuations/variations in boundary, initial conditions as well as variabilities in operating and surrounding conditions. There has been increasing interest in analyzing and quantifying the effects of uncertain inputs in the solution of partial differential equations that describe these physical phenomena. Such analysis naturally leads to a rigorous methodology to design/control physical processes in the presence of multiple sources of uncertainty. Spectral stochastic methods and Monte-Carlo based sampling methods are two approaches that have been used to analyze these problems. As the complexity of the problem or the number of random variables involved in describing the input uncertainties increases, these approaches become highly impractical from implementation and convergence points-of-view, creating a bottle-neck to the utility of stochastic analysis. In this thesis, the sparse grid collocation method based on the Smolyak algorithm is developed as a viable alternate method for solving high-dimensional stochastic partial differential equations. The second bottleneck to the utility of stochastic modelling is the construction of realistic, viable models of the input variability. In the second part of the thesis, a framework to construct realistic input models that encode the variabilities in initial and boundary conditions as well as other parameters using data-driven strategies are developed.

In the third part of the thesis, the data-driven input model generation strategies coupled with the sparse grid collocation strategies are utilized to analyze systems characterized by multi-length scale uncertainties. A stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features. The framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given.