Stochastic Galerkin method for elliptic spdes: A white noise approach

An equation that arises in mathematical studies of the transport of pollutants in groundwater and of oil recovery processes is of the form: irx ¢ (•(x;¢)rxu(x;!)) = f(x), for x 2 D, where •(x;¢), the permeability tensor, is random and models the properties of the rocks, which are not know with certainty. Further, geostatistical models assume •(x;¢) to be a log-normal random fleld. The use of Monte Carlo methods to approximate the expected value of u(x;¢), higher moments, or other functionals of u(x;¢), require solving similar system of equations many times as trajectories are considered, thus it becomes expensive and impractical. In this paper, we present and explain sev- eral advantages of using the White Noise probability space as a natural frame- work for this problem. Applying properly and timely the Wiener-It^o Chaos decomposition and an eigenspace decomposition, we obtain a symmetric pos- itive deflnite linear system of equations whose solutions are the coe-cients of a Galerkin-type approximation to the solution of the original equation. More- over, this approach reduces the simulation of the approximation to u(x;!) for a flxed !, to the simulation of a flnite number of independent normally distributed random variables.