  Active subspaces can effectively reduce the dimension of high-dimensional
parameter studies enabling otherwise infeasible experiments with expensive
simulations. The key components of active subspace methods are the eigenvectors
of a symmetric, positive semidefinite matrix whose elements are the average
products of partial derivatives of the simulation's input/output map. We study
a Monte Carlo method for approximating the eigenpairs of this matrix. We offer
both theoretical results based on recent non-asymptotic random matrix theory
and a practical approach based on the bootstrap. We extend the analysis to the
case when the gradients are approximated, for example, with finite differences.
Our goal is to provide guidance for two questions that arise in active
subspaces: (i) How many gradient samples does one need to accurately
approximate the eigenvalues and subspaces? (ii) What can be said about the
accuracy of the estimated subspace, both theoretically and practically? We test
the approach on both simple quadratic functions where the active subspace is
known and a parameterized PDE with 100 variables characterizing the
coefficients of the differential operator.
