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This manuscript is superseded by Constantine, Dow, and Wang's "ActiveSubspaces in Theory and Practice: Applications to Kriging Surfaces" [SIAM J. ofSci. Comput., 36 (2014), pp. A1500-A1524].  Many multivariate functions encountered in practice vary primarily along afew directions in the space of input parameters. When these directionscorrespond with coordinate directions, one may apply global sensitivitymeasures to determine the parameters with the greatest contribution to thefunction's variability. However, these methods perform poorly when thedirections of variability are not aligned with the natural coordinates of theinput space. We present a method for detecting the directions of variability ofa function using evaluations of its derivative with respect to the inputparameters. We demonstrate how to exploit these directions to construct asurrogate function that depends on fewer variables than the original function,thus reducing the dimension of the original problem. We apply this procedure toan exercise in uncertainty quantification using an elliptic PDE with a modelfor the coefficients that depends on 250 independent parameters. The dimensionreduction procedure identifies a 5-dimensional subspace suitable forconstructing surrogates.

Input Subspace Detection for Dimension Reduction in High Dimensional Approximation