论文标题
非线性随机PDE的在线多尺度模型具有乘法噪声
Online multiscale model reduction for nonlinear stochastic PDEs with multiplicative noise
论文作者
论文摘要
在本文中，为具有乘法噪声的随机部分微分方程（SPDE）提供了一种在线多尺度模型减少方法，其中扩散系数在空间上是多尺度的，并且噪声扰动非线性取决于扩散动力学。有必要有效计算随机动力学的所有可能轨迹，以量化模型的不确定性和统计矩。多尺度扩散和非线性可能会导致计算棘手。为了克服多尺度的难度，使用约束能量最小化通用的多尺度有限元方法（CEM-GMSFEM）来定位计算并获得有效的粗制模型。但是，在将CEM-GMSFEM的Galerkin投影应用于非线性SPDES之后，非线性项仍在细节空间上定义。这显着影响了CEM-GMSFEM的仿真效率。为此，提出了一种随机在线离散的经验插值法（DEIM）来治疗随机的非线性。随机在线Deim合并了离线快照和在线快照。离线快照由随机动力学的近似平均值组成，并用于构建离线还原模型。在线快照包含一些当前新轨迹的信息，用于以增量方式纠正离线减少模型。随机在线Deim大大降低了非线性动力学的维度，并提高了还原模型的预测准确性。因此，通过使用CEM-GMSFEM和随机在线Deim构建在线多尺度模型。对非线性SPDE进行了先验误差分析。我们提出了一些数值示例，这些示例在异质多孔介质中扩散，并显示了提出的模型还原的有效性。
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.
