Abstract. Many multivariate functions in engineering models vary primarily along a few directions in the space of input parameters. When these directions correspond to coordinate directions, one may apply global sensitivity measures to determine the most influential parameters. However, these methods perform poorly when the directions of variability are not aligned with the natural coordinates of the input space. We present a method to first detect the directions of the strongest variability using evaluations of the gradient and subsequently exploit these directions to construct a response surface on a low-dimensional subspace-i.e., the active subspace-of the inputs. We develop a theoretical framework with error bounds, and we link the theoretical quantities to the parameters of a kriging response surface on the active subspace. We apply the method to an elliptic PDE model with coefficients parameterized by 100 Gaussian random variables and compare it with a local sensitivity analysis method for dimension reduction. Key words. active subspace methods, kriging, Gaussian process, uncertainty quantification, response surfaces 1. Introduction & motivation. As computational models of physical systems become more complex, the need increases for uncertainty quantification (UQ) to enable defensible predictions. Monte Carlo methods are the workhorse of UQ, where model inputs are sampled according to a characterization of their uncertainty and the corresponding model outputs are treated as a data set for statistical analysis. However, the slow convergence of Monte Carlo methods coupled with the high computational cost of the models has led many to employ response surfaces trained on a few carefully selected runs in place of the full model. This strategy has had great success in forward [19, 26] and inverse uncertainty propagation problems [29, 4, 12] as well as optimization [24, 39] . However, most response surfaces suffer from the curse of dimensionality, where the cost of constructing an accurate surface increases exponentially as the dimension (i.e., the number of input parameters) increases. To make construction tractable, one may first perform sensitivity analysis [36] to determine which variables have the most influence on the model predictions. With a ranking of the inputs, one may construct response surfaces that concentrate the approximation on the most influential variables, e.g., through a suitably anisotropic design; the same concept applies to mesh refinement strategies for solving PDEs. Methods for sensitivity analysis are typically classified as local perturbation or global methods. Local methods perturb the inputs-often along coordinate directionsaround a nominal value and measure the effects on the outputs. Though relatively inexpensive, local methods are fraught with difficulties like sensitivity to noise and the choice of the perturbation step. Also, the local sensitivity measured at the nominal condition may be very different elsewhere in the parameter space. Global methods address these issues by providing integrated measures of the output's variability over the full range of parameters; consequently, they are computationally more expensive. Methods based on variance decompositions [30, 36] require approximating
The Active Subspace method of REF uses the information in the gradient to discover a subspace of maximum variation of f .
Active subspace methods in theory and practice: applications to kriging surfaces
Mathematics Computer Science