  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.
