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

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.

💡 Research Summary

This paper addresses a fundamental question in uncertainty quantification for parametric partial differential equations (PDEs): whether the established upper bounds for the dimension truncation error are sharp, meaning they cannot be improved. When modeling uncertain inputs like lognormal random fields with infinite-dimensional parameters, a common numerical approach is to truncate the parameterization to a finite number s of dimensions. While prior works have derived convergence rates for the error introduced by this truncation, the optimality of these rates remained an open question.

The authors remedy this gap by rigorously proving that the known error rates are indeed optimal. They focus on two concrete model problems that are representative yet amenable to explicit analysis. The first model problem is a one-dimensional elliptic PDE with a Dirichlet-Neumann boundary condition and a lognormal diffusion coefficient α(x,y)=exp(∑ y_j ψ_j(x)). The second is a d-dimensional elliptic PDE with a homogeneous Dirichlet boundary condition and a diffusion coefficient β(y)=exp(∑ b_j y_j) that is constant in space. Both problems assume the parameters y are distributed according to a product Gaussian measure.

The first part of the analysis recaps the derivation of upper bounds for the dimension truncation error, measured in the H^1 norm of the expected value of the PDE solution. Under the assumption that the sequences (‖ψ_j‖_∞) or (b_j) belong to the ℓ^p space for some p in (0,1), the error is shown to be O(s^{-2/p+1}). This aligns with the general result from earlier work