A near-stationary subspace for ridge approximation

Abstract

Response surfaces are common surrogates for expensive computer simulations in engineering analysis. However, the cost of fitting an accurate response surface increases exponentially as the number of model inputs increases, which leaves response surface construction intractable for high-dimensional, nonlinear models. We describe ridge approximation for fitting response surfaces in several variables. A ridge function is constant along several directions in its domain, so fitting occurs on the coordinates of a low-dimensional subspace of the input space. We review essential theory for ridge approximation---e.g., the best mean-squared approximation and an optimal low-dimensional subspace---and we prove that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. Motivated by the theory, we propose a computational heuristic that uses an estimated active subspace as an initial guess for a ridge approximation fitting problem. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. We then propose a simple alternating heuristic for fitting a ridge function, and we demonstrate the effectiveness of the active subspace initial guess applied to an airfoil model of drag as a function of its 18 shape parameters.

References in corpus (3)

• Gradient Descent Converges to Minimizers
• Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation
• Orthogonal polynomials of several variables

Cited by in corpus (16)

• Data-driven polynomial ridge approximation using variable projection
• Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods
• Manifold learning for parameter reduction
• Generalization Bounds for Sparse Random Feature Expansions
• Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems
• An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics
• Interpretable Approximation of High-Dimensional Data
• Computer Model Emulation with High-Dimensional Functional Output in Large-Scale Observing System Uncertainty Experiments
• Data-driven dimensional analysis: algorithms for unique and relevant dimensionless groups
• Polynomial Ridge Flowfield Estimation
• A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration
• A Lipschitz Matrix for Parameter Reduction in Computational Science
• Generalized bounds for active subspaces
• Sparse Mixture Models inspired by ANOVA Decompositions
• AeroVR: Immersive Visualization System for Aerospace Design
• Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations