  In this work we develop a Hessian-based sampling method for the construction
of goal-oriented reduced order models with high-dimensional parameter inputs.
Model reduction is known very challenging for high-dimensional parametric
problems whose solutions also live in high-dimensional manifolds. However, the
manifold of some quantity of interest (QoI) depending on the parametric
solutions may be low-dimensional. We use the Hessian of the QoI with respect to
the parameter to detect this low-dimensionality, and draw training samples by
projecting the high-dimensional parameter to a low-dimensional subspace spanned
by the eigenvectors of the Hessian corresponding to its dominating eigenvalues.
Instead of forming the full Hessian, which is computationally intractable for a
high-dimensional parameter, we employ a randomized algorithm to efficiently
compute the dominating eigenpairs of the Hessian whose cost does not depend on
the nominal dimension of the parameter but only on the intrinsic dimension of
the QoI. We demonstrate that the Hessian-based sampling leads to much smaller
errors of the reduced basis approximation for the QoI compared to a random
sampling for a diffusion equation with random input obeying either uniform or
Gaussian distributions.
