Active subspace methods in theory and practice: applications to kriging surfaces (Q6486742)

The authors are interested in finding directions of strongest variability of a given function allowing to construct an approximation in a lower-dimensional subspace of the function's inputs. The directions are detected by evaluating the function's gradient at a set of input points and determining a rotation of the input space that separates the directions of relative variability from directions of relative flatness.NEWLINENEWLINEThe contribution of this paper is twofold. First, the theoretical foundation for gradient based dimension reduction and subspace reduction is provided. The authors construct and factorize a covariance-like matrix of the gradient to determine the directions of variability. Further, they provide error bounds for these approximations, and examine the effects of using directions that are slightly perturbed. Second, theoretical quantities are linked to the parameters of a kriging response surface on the active subspace.NEWLINENEWLINEThe procedure is applied to an elliptic partial differential equation model with 100 parameters for the coefficients and scalar quantity of interest. An active subspace approach is compared with the newly developed dimension reduction approach based on local sensitivity analysis.