A Multiscale Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

In this paper, we propose a multiscale data-driven stochastic method (MsDSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. This method combines the advantages of the recently developed multiscale model reduction method [M. L. Ci, T. Y. Hou, and Z. Shi, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 449–474] and the datadriven stochastic method (DSM) [M. L. Cheng et al., SIAM/ASA J. Uncertain. Quantif., 1 (2013), pp. 452–493]. Our method consists of offline and online stages. In the offline stage, we decompose the harmonic coordinate into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Based on the Karhunen–Lo`eve (KL) expansion of the smooth parts and oscillatory parts of the harmonic coordinates, we can derive an effective stochastic equation that can be well-resolved on a coarse grid. We then apply the DSM to the effective stochastic equation to construct a da...

• Type: Journal
• Year: 2015
• Where: MMAS
• Authors: Zhiwen Zhang, Maolin Ci, Thomas Y. Hou
• Keywords: Intelligent Agents, MMAS 2015