A near-stationary subspace for ridge approximation
- University of Colorado, Boulder, CO (United States)
- Alan Turing Institute, London (United Kingdom)
- University of Texas, Austin, TX (United States)
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. Here 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.
- Research Organization:
- Colorado School of Mines, Golden, CO (United States); Univ. of Colorado, Boulder, CO (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Defense Advanced Research Projects Agency (DARPA); Alan Turing Institute; National Science Foundation (NSF)
- Grant/Contract Number:
- SC0011077; EP/N510129/1; 1255631; SC-0011077
- OSTI ID:
- 1538114
- Alternate ID(s):
- OSTI ID: 1495603
- Journal Information:
- Computer Methods in Applied Mechanics and Engineering, Vol. 326, Issue C; ISSN 0045-7825
- Publisher:
- ElsevierCopyright Statement
- Country of Publication:
- United States
- Language:
- English
Web of Science
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