Near-optimal approximation methods for elliptic PDEs with lognormal coefficients

TL;DR

This paper develops and analyzes near-optimal polynomial approximation methods for elliptic PDEs with lognormal coefficients, achieving convergence rates comparable to best n-term Hermite polynomial approximations, even with rough Gaussian random fields.

Contribution

It introduces a convergence analysis for weighted least-squares approximants that are optimal for smooth and rough lognormal coefficients, overcoming limitations of stochastic Galerkin methods.

Findings

Weighted least-squares approximants achieve near-optimal convergence rates.

Numerical tests with Brownian bridge validate theoretical results.

Method is effective for both smooth and rough Gaussian random fields.

Abstract

This paper studies numerical methods for the approximation of elliptic PDEs with lognormal coefficients of the form $-{\rm div}(a\nabla u)=f$ where $a=\exp(b)$ and $b$ is a Gaussian random field. The approximant of the solution $u$ is an $n$-term polynomial expansion in the scalar Gaussian random variables that parametrize $b$. We present a general convergence analysis of weighted least-squares approximants for smooth and arbitrarily rough random field, using a suitable random design, for which we prove optimality in the following sense: their convergence rate matches exactly or closely the rate that has been established in \cite{BCDM} for best $n$-term approximation by Hermite polynomials, under the same minimial assumptions on the Gaussian random field. This is in contrast with the current state of the art results for the stochastic Galerkin method that suffers the lack of coercivity due to the lognormal nature of the diffusion field. Numerical tests with $b$ as the Brownian bridge confirm our theoretical findings.