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Certified dimension reduction of the input parameter space of 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
UGA [2016-2019] - Université Grenoble Alpes [2016-2019], Grenoble INP [2007-2019] - Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019], LJK - Laboratoire Jean Kuntzmann , Inria Grenoble - Rhône-Alpes
Abstract : 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 Poincaré-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-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.
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Contributor : Olivier Zahm <>
Submitted on : Friday, December 14, 2018 - 4:01:14 PM
Last modification on : Wednesday, October 14, 2020 - 4:19:50 AM


  • HAL Id : hal-01955806, version 1



Olivier Zahm, Paul Constantine, Clémentine Prieur, Youssef Marzouk. Certified dimension reduction of the input parameter space of vector-valued functions. FrontUQ 18 - Frontiers of Uncertainty Quantification, Sep 2018, Pavie, Italy. ⟨hal-01955806⟩



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