Global sensitivity analysis using low-rank tensor approximations

Abstract

In the context of global sensitivity analysis, the Sobol' indices constitute a powerful tool for assessing the relative significance of the uncertain input parameters of a model. We herein introduce a novel approach for evaluating these indices at low computational cost, by postprocessing the coefficients of polynomial meta-models belonging to the class of low-rank tensor approximations. Meta-models of this class can be particularly efficient in representing responses of high-dimensional models, because the number of unknowns in their general functional form grows only linearly with the input dimension. The proposed approach is validated in example applications, where the Sobol' indices derived from the meta-model coefficients are compared to reference indices, the latter obtained by exact analytical solutions or Monte-Carlo simulation with extremely large samples. Moreover, low-rank tensor approximations are confronted to the popular polynomial chaos expansion meta-models in case studies that involve analytical rank-one functions and finite-element models pertinent to structural mechanics and heat conduction. In the examined applications, indices based on the novel approach tend to converge faster to the reference solution with increasing size of the experimental design used to build the meta-model.

Tensor Methods for Generating Compact Uncertainty Quantification and Deep Learning Models

“…For instance, in samplingbased techniques, the number of simulation samples may increase exponentially as d increases. Some tensor solvers have been developed to address this fundamental challenge: [38], [39] high-dim stochastic collocation tensor completion to estimate unknown simulation data [37] hierarchical uncertainty quantification tensor-train decomposition for high-dim integration [33] uncertainty analysis with non-Gaussian correlated uncertainty functional tensor train to compute basis functions [40] spatial variation pattern prediction statistical tensor completion to predict variation pattern resulting generalized polynomial chaos expansion is sparse. This technique has been successfully applied to electronic IC, photonics and MEMS with up to 57 random parameters.…”

Extending Classical Surrogate Modeling to High Dimensions Through Supervised Dimensionality Reduction: A Data-Driven Approach

“…, c M } T are non-negative constants. In this application, we chose M = 20 and the constants c given by Konakli and Sudret (2016a); Kersaudy et al (2015): 2, 5, 10, 20, 50, 100, 500, 500, . . .…”

Challenges of order reduction techniques for problems involving polymorphic uncertainty

“…A possible redemption is provided by PLS‐PCE, which couples a nonlinear partial least squares‐based (PLS) order reduction method with a PCE surrogate. For all of the above approaches, the sensitivity indices can be obtained immediately from postprocessing the model coefficients (see Sudret for classical and sparse PCEs, for LRAs and Ehre et al for PLS‐PCEs).…”