Variational Multiscale Analysis: The Fine-Scale Green's Function for Stochastic Partial Differential Equations

We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect of the unresolved fine scales through a model term that contains a fine-scale stochastic Green's function. For a natural choice of an “optimal” coarse-scale solution and $L^2$-orthogonal stochastic basis functions, we demonstrate that the fine-scale stochastic Green's function is intimately linked to its deterministic counterpart. In particular, (i) we demonstrate that whenever the deterministic fine-scale function vanishes, the stochastic fine-scale function satisfies a weaker and discrete notion of vanishing stochastic coefficients, and (ii) we derive an explicit formula for the fine-scale stochastic Green's function that only involves quantities needed to evaluate the fine-scale deterministic Green's function. We present numerical results that support our claims abo...