Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification

TL;DR

This paper demonstrates that high-dimensional functions with effective low-dimensional structure can be approximated efficiently without the curse of dimensionality, with applications to uncertainty quantification.

Contribution

It introduces a general framework for approximating effectively low-dimensional functions in high dimensions, including specific results for Sobolev spaces and practical error estimation methods.

Findings

Approximation does not suffer from curse of dimensionality for effectively low-dimensional functions.

Derived efficient error bounds for identifying low-dimensional structure.

Applied methods to parametric PDEs in uncertainty quantification.

Abstract

In this paper, we show that the approximation of high-dimensional functions, which are effectively low-dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space set-up and then specifically for Sobolev and mixed-regularity Sobolev spaces. Finally, efficient estimates are derived for deciding whether a high-dimensional function is effectively low-dimensional by studying error bounds in weighted reproducing kernel Hilbert spaces. The results are applied to parametric partial differential equations, a typical problem from uncertainty quantification.