Certified dimension reduction of the input parameter space of vector-valued functions

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. In this talk, we propose a methodology for dimension reduction which consists in minimizing an upper bound of the approximation error obtained using Poincare-type inequalities. This approach is fundamentally gradient-based, and generalizes the so-called active subspace method for vector-valued functions, e.g. functions with multiple scalar-valued outputs or functions taking values in function spaces. We also compare the proposed gradient-based approach with the popular and widely used truncated Karhunen-Loeve 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.