Multiscale Modeling &Simulation, Volume 18, Issue 4, Page 1595-1620, January 2020.
This work presents a stochastic multiscale model reduction approach to solve PDEs with random coefficients. An ensemble-based low-rank approximation method is proposed to approximate multiscale basis functions used to build a coarse model. To this end, we build local problems with multiple boundary conditions based on the generalized multiscale finite element method (GMsFEM), and construct a variable-separation representation for the corresponding multiscale basis functions, which admits a low-rank approximated form in terms of stochastic basis functions and deterministic physical basis functions. The construction and interrogation of the low-rank approximation for each multiscale basis function may demand fair computational cost. To significantly improve the efficiency of computation in the offline and interrogating the low-rank representation in the online stage, the ensemble-based method is proposed. In the offline stage, we obtain the stochastic basis functions and interpolate points using a variable separation method, with respect to the ensemble equation, then derive the deterministic physical parts for each member of the ensemble through residual decomposition. The resulted low-rank representations for each member share the common stochastic basis functions but have different physical basis functions characterizing their individual properties, offering considerable savings in online computation. This approach provides much flexibility inherited from GMsFEM and derives an efficient surrogate model. We present various numerical examples to demonstrate the accuracy and efficiency of the proposed method.

中文翻译：

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具有随机系数的Pdes的低秩近似多尺度方法

多尺度建模与仿真，第18卷，第4期，第1595-1620页，2020年1月。
这项工作提出了一种随机多尺度模型还原方法来求解具有随机系数的PDE。提出了一种基于整体的低秩逼近方法来逼近用于建立粗糙模型的多尺度基函数。为此，我们基于广义多尺度有限元方法（GMsFEM）建立具有多个边界条件的局部问题，并为相应的多尺度基函数构造变量分离表示形式，该形式表示为随机的低秩近似形式基本功能和确定性的物理基本功能。每个多尺度基函数的低秩逼近的构造和查询可能需要公平的计算成本。为了显着提高离线状态下的计算效率，并在在线状态下查询低秩表示，提出了一种基于集成的方法。在离线阶段，针对集合方程，我们使用变量分离方法获得随机基函数和插值点，然后通过残差分解得出集合中每个成员的确定性物理部分。每个成员所得到的低秩表示具有共同的随机基础函数，但是具有表征其各个属性的不同物理基础函数，从而大大节省了在线计算量。这种方法提供了从GMsFEM继承的很大的灵活性，并得出了有效的替代模型。