Numerical approximation of elliptic problems with log-normal random coefficients

TL;DR

This paper introduces a Wick-type elliptic model as an efficient preconditioner for classical elliptic equations with log-normal random coefficients, improving numerical approximation methods like Monte Carlo and stochastic Galerkin.

Contribution

The work proposes a novel Wick-type model that approximates the classical elliptic problem and accelerates stochastic numerical methods, with theoretical and numerical validation.

Findings

Wick-type model provides a second-order approximation of the classical model.

The model yields the same solution as the classical one when the Gaussian process correlation length tends to infinity.

Numerical results demonstrate the efficiency of the Wick-type preconditioning strategy.

Abstract

In this work, we consider a non-standard preconditioning strategy for the numerical approximation of the classical elliptic equations with log-normal random coefficients. In \cite{Wan_model}, a Wick-type elliptic model was proposed by modeling the random flux through the Wick product. Due to the lower-triangular structure of the uncertainty propagator, this model can be approximated efficiently using the Wiener chaos expansion in the probability space. Such a Wick-type model provides, in general, a second-order approximation of the classical one in terms of the standard deviation of the underlying Gaussian process. Furthermore, when the correlation length of the underlying Gaussian process goes to infinity, the Wick-type model yields the same solution as the classical one. These observations imply that the Wick-type elliptic equation can provide an effective preconditioner for the classical random elliptic equation under appropriate conditions. We use the Wick-type elliptic model to accelerate the Monte Carlo method and the stochastic Galerkin finite element method. Numerical results are presented and discussed.