Active subspace methods in theory and practice: applications to kriging surfaces

Abstract

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 reduction.

Cited by in corpus (67)

• Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification
• Machine Learning in Aerodynamic Shape Optimization
• Quantifying and Reducing Model-Form Uncertainties in Reynolds-Averaged Navier-Stokes Simulations: A Data-Driven, Physics-Based Bayesian Approach
• Active learning for structural reliability: survey, general framework and benchmark
• Dimension-independent likelihood-informed MCMC
• Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks
• Exploiting Active Subspaces to Quantify Uncertainty in the Numerical Simulation of the HyShot II Scramjet
• Simulator-free Solution of High-Dimensional Stochastic Elliptic Partial Differential Equations using Deep Neural Networks
• Calibration of a SEIR-SEI epidemic model to describe the Zika virus outbreak in Brazil
• Compressive sensing adaptation for polynomial chaos expansions
• Accelerating MCMC with active subspaces
• Airfoil GAN: Encoding and Synthesizing Airfoils for Aerodynamic Shape Optimization
• A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces
• Data-driven polynomial ridge approximation using variable projection
• Surrogate Modeling of Aerodynamic Simulations for Multiple Operating Conditions Using Machine Learning
• Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations
• A near-stationary subspace for ridge approximation
• Manifold learning for parameter reduction
• Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification
• Latent Map Gaussian Processes for Mixed Variable Metamodeling
• Surrogate assisted active subspace and active subspace assisted surrogate -- A new paradigm for high dimensional structural reliability analysis
• A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime
• Bayesian calibration and sensitivity analysis for a karst aquifer model using active subspaces
• Adaptive active subspace-based metamodeling for high-dimensional reliability analysis
• Generalization Bounds for Sparse Random Feature Expansions
• Projection pursuit adaptation on polynomial chaos expansions
• On the unsteady Darcy-Forchheimer-Brinkman equation in local and nonlocal tumor growth models
• Discovering an active subspace in a single-diode solar cell model
• A supervised learning approach involving active subspaces for an efficient genetic algorithm in high-dimensional optimization problems
• Quantifying the influence of conformational uncertainty in biomolecular solvation
• Enhancing CFD predictions in shape design problems by model and parameter space reduction
• Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes
• Divide and Conquer: An Incremental Sparsity Promoting Compressive Sampling Approach for Polynomial Chaos Expansions
• Bayesian, frequentist, and information geometric approaches to parametric uncertainty quantification of classical empirical interatomic potentials
• Reliable emulation of complex functionals by active learning with error control
• Multi-fidelity Bayesian Optimization: A Review
• Modeling and Optimization with Gaussian Processes in Reduced Eigenbases -- Extended Version
• A Dimensionality Reduction Approach for Convolutional Neural Networks
• Kernel-based active subspaces with application to computational fluid dynamics parametric problems using the discontinuous Galerkin method
• Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems
• Adaptive Dimension Reduction to Accelerate Infinite-Dimensional Geometric Markov Chain Monte Carlo
• Interpretable Approximation of High-Dimensional Data
• Gaussian processes for Bayesian inverse problems associated with linear partial differential equations
• Efficient parameter estimation for a methane hydrate model with active subspaces
• Goal-oriented adaptive surrogate construction for stochastic inversion
• Federated Optimization of Smooth Loss Functions
• A Training Set Subsampling Strategy for the Reduced Basis Method
• Learning Active Subspaces and Discovering Important Features with Gaussian Radial Basis Functions Neural Networks
• Tensor rank reduction via coordinate flows
• A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness
• Nonlinear dimension reduction for surrogate modeling using gradient information
• Optimal Design of Validation Experiments for the Prediction of Quantities of Interest
• Microstructure-Aware Bayesian Materials Design
• Discovering Active Subspaces for High-Dimensional Computer Models
• Variational Bayesian surrogate modelling with application to robust design optimisation
• Bayesian optimisation of poloidal field coil positions in tokamaks
• An efficient estimation of time-varying parameters of dynamic models by combining offline batch optimization and online data assimilation
• Generalized bounds for active subspaces
• A gradient-enhanced univariate dimension reduction method for uncertainty propagation
• An Efficient Global Optimization Algorithm with Adaptive Estimates of the Local Lipschitz Constants
• Multi-level informed optimization via decomposed Kriging for large design problems under uncertainty
• Accelerating Dimensionality Reduction in Wave-Resistance Problems through Geometric Operators
• PISP: Projected-Space Inference of Stellar Parameters
• Surrogate to Poincaré inequalities on manifolds for dimension reduction in nonlinear feature spaces
• Multi-Scenario and Stochastic Thermo-Electro-Mechanical Modeling of Failure in Power Transmission Lines
• Conservative Surrogate Models for Optimization with the Active Subspace Method
• Identifying sensitivity-dominant parameters via active subspaces in reduced-order modeling of fluid dynamics