  Many multivariate functions in engineering models vary primarily along a few
directions in the space of input parameters. When these directions correspond
to coordinate directions, one may apply global sensitivity measures to
determine the most influential parameters. However, these methods perform
poorly when the directions of variability are not aligned with the natural
coordinates of the input space. We present a method to first detect the
directions of the strongest variability using evaluations of the gradient and
subsequently exploit these directions to construct a response surface on a
low-dimensional subspace---i.e., the active subspace---of the inputs. We
develop a theoretical framework with error bounds, and we link the theoretical
quantities to the parameters of a kriging response surface on the active
subspace. We apply the method to an elliptic PDE model with coefficients
parameterized by 100 Gaussian random variables and compare it with a local
sensitivity analysis method for dimension reduction.
