Active Subspace Methods in Theory and Practice: Applications to Kriging Surfaces

By Paul G. Constantine, Eric Dow, Qiqi Wang. SIAM Journal on Scientific Computing, 2013

621 citations recorded by the source corpus.

Many multivariate functions in engineering models vary primarily along a few directions in the space of input parameters. When these directions correspond to coordinate directions, one may apply global sensitivity measures to determine the most influential parameters. However, these methods perform poorly when the directions of variability are not aligned with the natural coordinates of the input space. We present a method to first detect the directions of the strongest variability using evaluations of the gradient and subsequently exploit these directions to construct a response surface on a low-dimensional subspace---i.e., the active subspace---of the inputs. We develop a theoretical framework with error bounds, and we link the theoretical quantities to the parameters of a kriging response surface on the active subspace. We apply the method to an elliptic PDE model with coefficients parameterized by 100 Gaussian random variables and compare it with a local sensitivity analysis method for dimension reduc...

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8f83f11dfec3f4f21ac4983e83f146f71ddb8516

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Citations in the corpus

Active Subspaces - Emerging Ideas for Dimension Reduction in Parameter Studies (2015, background): “Much of the technical development in Chapters 3 and 4 comes from our paper [28].”
Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification (2018, background): “A recent advancement in dimensionality reduction is active subspaces (AS) [31, 32, 33, 34, 35, 36].”
Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier-Stokes simulations: A data-driven, physics-informed Bayesian approach (2015, background): “, active subspace methods [51], likelihood informed dimension-reduction [52]) and sampling techniques (e.”
Ready Policy One: World Building Through Active Learning (2020, background): “Compute Active Subspace Uact as the result of stacking together some orthonormal basis of Lactive def = span{u1, ...,ur} where the vectors ui correspond to the top k eigenvalues of the covariance matrix Covt.”
Machine Learning in Aerodynamic Shape Optimization (2022, methodology): “PCA and ASM modes are linear, so the shape deformation shows a linear superposition effect when the mode coefficient is doubled.”
Active learning for structural reliability: Survey, general framework and benchmark (2021, methodology): “For the former, dimensionality reduction techniques are often combined with surrogate modelling as in manifold learning (Zheng and Xue, Zheng and Xue), active subspace method (Constantine et al., 2014; Zhou and Peng, 2020) or in a more general setting as in Lataniotis et al.”
Convergence rates for a class of estimators based on Stein’s method (2016, background): “the observation that many test functions of interest are of low \eective dimension" suggests that stronger assumptions could reasonably be placed on the function space. Active subspace methods (Constantine et al., 2014) and weighted function spaces (Niederreiter, 2010) may have an important rol”
Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks (2019, methodology): “The recently popularized method of active subspaces (AS) [34, 35, 36, 37, 38, 39, 40], on the other hand, performs linear dimensionality reduction by performing eigendecomposition on an empirical covariance matrix constructed from samples of the model output gradients - thereby alleviating the secon”