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Center for Nonlinear Studies |
Mathematical models with uncertainties can be represented as stochastic partial differential equations (SPDEs). The model's output can be accurately predicted by efficiently solving the associated SPDEs. It is quite challenging to solve the SPDEs when the random inputs (e.g., coefficient) vary over multiple scales in space and contain inherent uncertainties. In this talk we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the new MsFEM with sparse grid collocation methods, the longstanding computation difficulty, curse of dimensionality, can be substantially alleviated. We rigorously analyze the convergence of the proposed method for both deterministic and stochastic elliptic equations. A computational complexity discussion is also offered. A number of numerical results are presented to confirm the theoretical findings.
Host: Shiv K. Sambasivan, T-5 665-8075