Input Subspace Detection for Dimension Reduction in High Dimensional Approximation

TL;DR

This paper introduces a method to identify key input directions in high-dimensional functions using derivatives, enabling the construction of lower-dimensional surrogate models for efficient approximation, demonstrated on a PDE with 250 parameters.

Contribution

The paper presents a novel derivative-based approach for detecting input subspaces that facilitate dimension reduction in high-dimensional approximation problems.

Findings

Identified a 5-dimensional subspace for a 250-parameter PDE model.

Constructed effective surrogate models based on the identified subspace.

Demonstrated significant dimension reduction in uncertainty quantification.

Abstract

This manuscript is superseded by Constantine, Dow, and Wang's "Active Subspaces in Theory and Practice: Applications to Kriging Surfaces" [SIAM J. of Sci. Comput., 36 (2014), pp. A1500-A1524]. Many multivariate functions encountered in practice vary primarily along a few directions in the space of input parameters. When these directions correspond with coordinate directions, one may apply global sensitivity measures to determine the parameters with the greatest contribution to the function's variability. 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 for detecting the directions of variability of a function using evaluations of its derivative with respect to the input parameters. We demonstrate how to exploit these directions to construct a surrogate function that depends on fewer variables than the original function, thus reducing the dimension of the original problem. We apply this procedure to an exercise in uncertainty quantification using an elliptic PDE with a model for the coefficients that depends on 250 independent parameters. The dimension reduction procedure identifies a 5-dimensional subspace suitable for constructing surrogates.