Skip to Main content Skip to Navigation
Journal articles

Gradient-based dimension reduction of multivariate vector-valued functions

Olivier Zahm 1 Paul Constantine 2 Clementine Prieur 1 Youssef Marzouk 3
1 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Inria Grenoble - Rhône-Alpes, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these directions and using them to construct ridge approximations of such functions, in a setting where the functions are vector-valued (e.g., taking values in Rn). The methodology consists of minimizing an upper bound on the approximation error, obtained by subspace Poincaré inequalities. We provide a thorough mathematical analysis in the case where the parameter space is equipped with a Gaussian probability measure. The resulting method generalizes the notion of active subspaces associated with scalar-valued functions. A numerical illustration shows that using gradients of the function yields effective dimension reduction. We also show how the choice of norm on the codomain of the function has an impact on the function's low-dimensional approximation.
Document type :
Journal articles
Complete list of metadata

Cited literature [56 references]  Display  Hide  Download

https://hal.inria.fr/hal-01701425
Contributor : Olivier Zahm Connect in order to contact the contributor
Submitted on : Friday, November 8, 2019 - 10:43:09 AM
Last modification on : Tuesday, October 19, 2021 - 11:25:56 AM
Long-term archiving on: : Sunday, February 9, 2020 - 9:41:35 PM

File

main.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Olivier Zahm, Paul Constantine, Clementine Prieur, Youssef Marzouk. Gradient-based dimension reduction of multivariate vector-valued functions. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2020, 42 (1), pp.A929-A956. ⟨10.1137/18M1221837⟩. ⟨hal-01701425v3⟩

Share

Metrics

Record views

174

Files downloads

679