A truly predictive science for many problems of scientific and national interest requires advanced mathematical and computational tools that explain how the uncertainties, ubiquitous in all modeling efforts, affect our predictions and understanding of complex phenomena. This remains a fundamental difficulty in most applications that dominate the focus of the DOE mission. Examples include enhancement of reliability of smart energy grids, development of renewable energy technologies, vulnerability analysis of water and power supplies, understanding complex biological networks, climate change estimation and design and licensing of current and future nuclear energy reactors. Often these systems describe physical and biological processes exhibiting highly nonlinear, or even worse, discontinuous/bifurcating phenomena at a diverse set of length and/or time scales. Moreover, predictive simulation of these systems requires significantly more computational effort than high-fidelity deterministic simulations; particularly when the random input data (coefficients, forcing terms, initial and boundary conditions, geometry, etc.) are affected by large amounts of uncertainty. Even for these high-dimensional stochastic problems, simulation code verification, model validation, and uncertainty quantification (UQ) are indispensable tasks demanded to justify a predictive capability in a mathematically and scientifically rigorous manner.