| Author | Jayanth Jagalur-Mohan, Onkar Sahni, Alireza Doostan, Assad Oberai |
|---|---|
| Title | Variational multiscale analysis: the fine-scale Green's function for stochastic partial differential equations |
| Year | 2013 |
| Journal | Submitted to SIAM/ASA Journal on Uncertainty Quantification |
| Abstract | 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) 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 about the physical support of the stochastic fine-scale function, and demonstrate the benefit of using the VMS method when the fine-scale Green's function is approximated by an easier to implement, element Green's function. |
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