The Wick-Malliavin approximation of elliptic problems with log-normal random coefficients (Q2870649)

The authors consider elliptic problems with log-normal random coefficients of the form NEWLINE\[NEWLINE\begin{aligned} -\nabla\cdot(a(\mathbf{x},\omega)\nabla u(\mathbf{x},\omega)) & =f(\mathbf{x}),\;\mathbf{x}\in D,\\ u(\mathbf{x},\omega) & =0,\;\mathbf{x}\in\partial D, \end{aligned}NEWLINE\]NEWLINE where \(\ln a(\mathbf{x},\omega)=G(\mathbf{x},\omega)\) and \(G(\mathbf{x} ,\omega)\) is a homogeneous Gaussian random process.NEWLINENEWLINEThey use the Wick product and the Mikulevicius-Rozovskii formula, see [\textit{R. Mikulevicius} and \textit{B. L. Rozovskii}, Probab. Theory Relat. Fields 154, No. 3--4, 787--834 (2012; Zbl 1277.60109)] for the approximation of the elliptic problems. In the paper, both theoretical and numerical discussions are presented.