A near-stationary subspace for ridge approximation

Paul G. Constantine, Armin Eftekhari, Jeffrey Hokanson, Rachel A. Ward

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

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.
Original languageEnglish
Pages (from-to)402-421
JournalComputer Methods in Applied Mechanics and Engineering
Volume326
Early online date24 Aug 2017
DOIs
Publication statusPublished - 1 Nov 2017

Keywords / Materials (for Non-textual outputs)

  • active subspace
  • ridge function
  • projection pursuit

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