  Response surfaces are common surrogates for expensive computer simulations in
engineering analysis. However, the cost of fitting an accurate response surface
increases exponentially as the number of model inputs increases, which leaves
response surface construction intractable for high-dimensional, nonlinear
models. We describe ridge approximation for fitting response surfaces in
several variables. A ridge function is constant along several directions in its
domain, so fitting occurs on the coordinates of a low-dimensional subspace of
the input space. We review essential theory for ridge approximation---e.g., the
best mean-squared approximation and an optimal low-dimensional subspace---and
we prove that the gradient-based active subspace is near-stationary for the
least-squares problem that defines an optimal subspace. Motivated by the
theory, we propose a computational heuristic that uses an estimated active
subspace as an initial guess for a ridge approximation fitting problem. We show
a simple example where the heuristic fails, which reveals a type of function
for which the proposed approach is inappropriate. We then propose a simple
alternating heuristic for fitting a ridge function, and we demonstrate the
effectiveness of the active subspace initial guess applied to an airfoil model
of drag as a function of its 18 shape parameters.
