  This manuscript is superseded by Constantine, Dow, and Wang's "Active
Subspaces in Theory and Practice: Applications to Kriging Surfaces" [SIAM J. of
Sci. Comput., 36 (2014), pp. A1500-A1524].
  Many multivariate functions encountered in practice vary primarily along a
few directions in the space of input parameters. When these directions
correspond with coordinate directions, one may apply global sensitivity
measures to determine the parameters with the greatest contribution to the
function's variability. 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 for detecting the directions of variability of
a function using evaluations of its derivative with respect to the input
parameters. We demonstrate how to exploit these directions to construct a
surrogate function that depends on fewer variables than the original function,
thus reducing the dimension of the original problem. We apply this procedure to
an exercise in uncertainty quantification using an elliptic PDE with a model
for the coefficients that depends on 250 independent parameters. The dimension
reduction procedure identifies a 5-dimensional subspace suitable for
constructing surrogates.
