Science and engineering models typically contain multiple parameters representing input data---e.g., boundary conditions or material properties. The map from model inputs to model outputs can be viewed as a multivariate function. One may naturally be interested in how the function changes as inputs are varied. However, if computing the model output is expensive given a set of inputs, then exploring the high-dimensional input space is infeasible. Such issues arise in the study of uncertainty quantification, where uncertainty in the inputs begets uncertainty in model predictions.

Fortunately, many practical models with high-dimensional inputs vary primarily along only a few directions in the space of inputs. I will describe a method for detecting and exploiting these directions of variability to construct a response surface on a low-dimensional linear subspace of the full input space; detection is accomplished through analysis of the gradient of the model output with respect to the inputs, and the subspace is defined by a projection. I will show error bounds for the low-dimensional approximation that motivate computational heuristics for building a kriging response surface on the subspace. As a demonstration, I will apply the method to a nonlinear heat transfer model on a turbine blade, where a 250-parameter model for the heat flux represents uncertain transition to turbulence of the flow field. I will also discuss the range of existing applications of the method - including the motivating application from Stanford\'s NNSA PSAAP center - and the future research challenges.