On the optimality of dimension truncation error rates for a class of parametric partial differential equations

TL;DR

This paper investigates the sharpness of dimension truncation error rates in uncertainty quantification for parametric PDEs with infinite-dimensional random inputs, demonstrating the optimality of existing bounds in specific models.

Contribution

It provides the first analysis confirming the sharpness of dimension truncation error bounds for certain parametric PDE models, filling a gap in the literature.

Findings

Dimension truncation error bounds are sharp for the studied models.

Existing error estimates accurately reflect the true error behavior.

The results validate the optimality of current truncation strategies.

Abstract

In uncertainty quantification for parametric partial differential equations (PDEs), it is common to model uncertain random field inputs using countably infinite sequences of independent and identically distributed random variables. The lognormal random field is a prime example of such a model. While there have been many studies assessing the error in the PDE response that occurs when an infinite-dimensional random field input is replaced with a finite-dimensional random field, there do not seem to be any analyses in the existing literature discussing the sharpness of these bounds. This work seeks to remedy the situation. Specifically, we investigate two model problems where the existing dimension truncation error rates can be shown to be sharp.