Abstract

Co-authors: Paul Constantine (University of Colorado), Clémentine Prieur (University Joseph Fourier), Youssef Marzouk (MIT)

Approximation of multivariate functions is a difficult task when the number of input parameters is large. Identifying the directions where the function does not significantly vary is a key preprocessing step to reduce the complexity of the approximation algorithms.

Among other dimensionality reduction tools, the active subspace is defined by means of the gradient of a scalar-valued function. It can be interpreted as the subspace in the parameter space where the gradient varies the most. In this talk, we propose a natural extension of the active subspace for vector-valued functions, e.g. functions with multiple scalar-valued outputs or functions taking values in function spaces. Our methodology consists in minimizing an upper-bound of the approximation error obtained using Poincaré-type inequalities.

We also compare the proposed gradient-based approach with the popular and widely used truncated Karhunen-Loève decomposition (KL). We show that, from a theoretical perspective, the truncated KL can be interpreted as a method which minimizes a looser upper bound of the error compared to the one we derived. Also, numerical comparisons show that better dimension reduction can be obtained provided gradients of the function are available.