Application of dimension truncation error analysis to high-dimensional function approximation in uncertainty quantification

TL;DR

This paper analyzes the $L^2$ dimension truncation error in high-dimensional parametric models used in uncertainty quantification, providing theoretical estimates and numerical validation, with invariance properties under variable transformations.

Contribution

It introduces a new analysis of the $L^2$ dimension truncation error for high-dimensional models, extending prior work mainly focused on numerical integration.

Findings

The dimension truncation error rate is invariant under certain transformations.

Theoretical error estimates are validated by numerical experiments.

Results demonstrate the sharpness of the derived error bounds.

Abstract

Parametric mathematical models such as parameterizations of partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.