Online multiscale model reduction for nonlinear stochastic PDEs with multiplicative noise

In this paper, an online multiscale model reduction method is presented for stochastic partial differential equations (SPDEs) with multiplicative noise, where the diffusion coefficient is spatially multiscale and the noise perturbation nonlinearly depends on the diffusion dynamics. It is necessary to efficiently compute all possible trajectories of the stochastic dynamics for quantifying model's uncertainty and statistic moments. The multiscale diffusion and nonlinearity may cause the computation intractable. To overcome the multiscale difficulty, a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) is used to localize the computation and obtain an effective coarse model. However, the nonlinear terms are still defined on a fine scale space after the Galerkin projection of CEM-GMsFEM is applied to the nonlinear SPDEs. This significantly impacts on the simulation efficiency by CEM-GMsFEM. To this end, a stochastic online discrete empirical interpolation method (DEIM) is proposed to treat the stochastic nonlinearity. The stochastic online DEIM incorporates offline snapshots and online snapshots. The offline snapshots consist of the nonlinear terms at the approximate mean of the stochastic dynamics and are used to construct an offline reduced model. The online snapshots contain some information of the current new trajectory and are used to correct the offline reduced model in an increment manner. The stochastic online DEIM substantially reduces the dimension of the nonlinear dynamics and enhances the prediction accuracy for the reduced model. Thus, the online multiscale model reduction is constructed by using CEM-GMsFEM and the stochastic online DEIM. A priori error analysis is carried out for the nonlinear SPDEs. We present a few numerical examples with diffusion in heterogeneous porous media and show the effectiveness of the proposed model reduction.