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 many models. We propose 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 develop essential theory for ridge approximation---e.g., the best mean-squared approximation and an optimal low-dimensional subspace---and we show that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. We propose practical computational heuristics motivated by the theory including an alternating minimization heuristic that estimates an optimal ridge approximation. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. And we demonstrate a successful example with an airfoil model of drag as a function of its 18 shape parameters.
|Journal||Computer Methods in Applied Mechanics and Engineering|
|Publication status||Submitted - Jun 2016|