A Discussion on Two Stochastic Elliptic Modeling Strategies

Abstract

Based on the study of two commonly used stochastic elliptic models: I: − ∇· (a(x,w)·∇u(x,w)) = f (x) and II: − ∇·(a(x,w)◊∇u(x,w)) = f (x), we constructed a new stochastic elliptic model III: −∇ ◊ ((a−1)·(−1)◊∇u(x,w)) = f (x), in. The difference between models I and II is twofold: a scaling factor induced by the way of applying the Wick product and the regularization induced by the Wick product itself. In, we showed that model III has the same scaling factor as model I. In this paper we present a detailed discussion about the difference between models I and III with respect to the two characteristic parameters of the random coefficient, i.e., the standard deviation σ and the correlation length lc. Numerical results are presented for both one- and two-dimensional cases.

Numerical Approximation of Elliptic Problems With Log-Normal Random Coefficients

“…We test the parameters σ = 0.2, 0.6, 1 and l c = 20, 2, 0.2. The solution differences of model I and model II is similar to the results in [46] and [47]. So we only sketch the results for two-dimensional case here.…”

“…Based on the properties of Wick product and the assumptions of Theorem 2, we have the following asymptotic results [46] for equation (34) satisfied by u I − u II . With respect to σ, we have the following power series…”

“…Although this is a very nice property for numerical computation, the original equation is changed and the model difference becomes the main concern. In [ 45,46], a new Wick-type model was proposed by modeling the flux as a −1 (−1) ∇u:…”

“…where ρ I,II = σ I,II /(σ I σ II ) is the autocorrelation function of u I,h and u II,h . Due to the fact given by equation (57) and theorem 2, ρ I,II ≈ 1 for small σ or large l c , when u I,h and u II,h are almost linear corresponding to α * ≈ 1 (see more numerical experiments in [46]). This is the reason we choose α = 1 in equation (67).…”

A note on stochastic elliptic models

“…with Eq. (20), we see that the effect of the Wick product for the mean solution is equivalent to omitting the secondorder perturbation term in Eq (19),. which implies that E½u I ðxÞ andE½u II ðxÞ are comparable only when the perturbation of a(x, x) is very small and smooth.We subsequently present some general properties of E½u II and E½u III .Proposition 3.4. kE½u II k e 6 kE½u III k e .…”