Gradient-based dimension reduction of multivariate vector-valued functions

Olivier Zahm (1) , Paul Constantine (2) , Clémentine Prieur (1) , Youssef Marzouk (3)

1 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
2 University of Colorado [Boulder]
3 MIT - Massachusetts Institute of Technology

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.

Mots clés: Sobol' indices; Poincaré inequality; dimension reduction; High-dimensional function approximation; active subspace; ridge approximation; Karhunen-Loève decomposition

Domaines: Equations aux dérivées partielles [math.AP]

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