The Wick--Malliavin Approximation of Elliptic Problems with Log-Normal Random Coefficients

Abstract

In this work, we discuss the approximation of elliptic problems with log-normal random coefficients using the Wick product and the Mikulevicius--Rozovskii formula. The main idea is that the multiplication between the log-normal coefficient and the gradient of the solution can be regarded as a Taylor-like expansion in terms of the Wick product and the Malliavin derivative. For the classical model, the coefficients of Wiener chaos expansion are fully coupled together in the uncertainty propagator, while the Wick--Malliavin model yields an uncertain propagator with weak coupling in the upper-triangular part for a relatively small truncation order in the Mikulevicius--Rozovskii formula. In this paper we focus on the difference between the classical model and the Wick--Malliavin model with respect to the standard deviation of the underlying Gaussian random process. Both theoretical and numerical discussions are presented.

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