We present the variational multiscale (VMS) method for partial differentialequations (PDEs) with stochastic coefficients and source terms. We use it as amethod for generating accurate coarse-scale solutions while accounting for theeffect of the unresolved fine scales through a model term that contains afine-scale stochastic Green's function. For a natural choice of an "optimal"coarse-scale solution and L^2-orthogonal stochastic basis functions, wedemonstrate that the fine-scale stochastic Green's function is intimatelylinked to its deterministic counterpart. In particular, (i) we demonstrate thatwhenever the deterministic fine-scale function vanishes, the stochasticfine-scale function satisfies a weaker, and discrete notion of vanishingstochastic coefficients, and (ii) derive an explicit formula for the fine-scalestochastic Green's function that only involves quantities needed to evaluatethe fine-scale deterministic Green's function. We present numerical resultsthat support our claims about the physical support of the stochastic fine-scalefunction, and demonstrate the benefit of using the VMS method when thefine-scale Green's function is approximated by an easier to implement, elementGreen's function.
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