Active subspace methods for high-dimensional function approximation | Department of Mathematics

Active subspace methods for high-dimensional function approximation

Event Information
Event Location: 
GAB 461, 4-5 PM; Refreshments: GAB 472, 3:30 PM
Event Date: 
Monday, December 2, 2013 - 4:00pm
Science and engineering models typically contain multiple parameters representing input data---e.g., boundary conditions or material properties. The map from model inputs to model outputs can be viewed as a multivariate function. One may naturally be interested in how the function changes as inputs are varied. However, if computing the model output is expensive given a set of inputs, then exploring the high-dimensional space of possible inputs is infeasible. Such issues arise in the study of uncertainty quantification, where uncertainty in the inputs begets uncertainty in model predictions. Fortunately, many practical models with high-dimensional inputs vary primarily along only a few directions in the space of inputs. I will describe a method for detecting and exploiting these primary directions of variability to construct a response surface on a low dimensional linear subspace of the full input space; detection is accomplished through analysis of the gradient of the model output with respect to the inputs, and the subspace is defined by a projection. I will show error bounds for the low-dimensional approximation that motivate computational heuristics for building a Kriging response surface on the subspace. As a demonstration, I will apply the method to a nonlinear heat transfer model on a turbine blade, where a 250-parameter model for the heat flux represents uncertain transition to turbulence of the flow field. I will also discuss the range of existing applications of the method and the future research challenges.