Computing active subspaces with Monte Carlo

TL;DR

This paper develops a Monte Carlo approach to efficiently compute active subspaces for high-dimensional problems, providing theoretical analysis and practical guidance on sampling requirements and accuracy, demonstrated on functions and PDE models.

Contribution

It introduces a Monte Carlo method for approximating active subspaces, with theoretical guarantees and practical tools for gradient sampling and subspace accuracy assessment.

Findings

The Monte Carlo method accurately approximates eigenpairs of the active subspace matrix.

Theoretical bounds guide the number of gradient samples needed.

Practical bootstrap approach assesses subspace estimation accuracy.

Abstract

Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace methods are the eigenvectors of a symmetric, positive semidefinite matrix whose elements are the average products of partial derivatives of the simulation's input/output map. We study a Monte Carlo method for approximating the eigenpairs of this matrix. We offer both theoretical results based on recent non-asymptotic random matrix theory and a practical approach based on the bootstrap. We extend the analysis to the case when the gradients are approximated, for example, with finite differences. Our goal is to provide guidance for two questions that arise in active subspaces: (i) How many gradient samples does one need to accurately approximate the eigenvalues and subspaces? (ii) What can be said about the accuracy of the estimated subspace, both theoretically and practically? We test the approach on both simple quadratic functions where the active subspace is known and a parameterized PDE with 100 variables characterizing the coefficients of the differential operator.