Multifidelity Dimension Reduction via Active Subspaces

TL;DR

This paper introduces a multifidelity active subspace method for efficient dimension reduction in high-dimensional models, leveraging models of different fidelities to reduce computational costs while identifying low-dimensional structures.

Contribution

It extends active subspace methodology to a multifidelity setting, providing theoretical analysis and demonstrating effectiveness in high-dimensional problems.

Findings

Sample complexity depends on intrinsic dimension, often much smaller than input dimension.

Numerical experiments show successful dimension reduction in spaces up to 3000 dimensions.

The method reduces computational cost while accurately identifying low-dimensional structures.

Abstract

We propose a multifidelity dimension reduction method to identify a low-dimensional structure present in many engineering models. The structure of interest arises when functions vary primarily on a low-dimensional subspace of the high-dimensional input space, while varying little along the complementary directions. Our approach builds on the gradient-based methodology of active subspaces, and exploits models of different fidelities to reduce the cost of performing dimension reduction through the computation of the active subspace matrix. We provide a non-asymptotic analysis of the number of gradient evaluations sufficient to achieve a prescribed error in the active subspace matrix, both in expectation and with high probability. We show that the sample complexity depends on a notion of intrinsic dimension of the problem, which can be much smaller than the dimension of the input space. We illustrate the benefits of such a multifidelity dimension reduction approach using numerical experiments with input spaces of up to three thousand dimensions.