In this paper, we develop a Bayesian multiscale approach based on a multiscale finite element method. Because of scale disparity in many multiscale applications, computational models can not resolve all scales. Various subgrid models are proposed to represent un-resolved scales. Here, we consider a probabilistic approach for modeling un-resolved scales using the Multiscale Finite Element Method (cf., [1, 2]). By representing dominant modes using the Generalized Multiscale Finite Element, we propose a Bayesian framework, which provides multiple inexpensive (computable) solutions for a deterministic problem. These approximate probabilistic solutions may not be very close to the exact solutions and, thus, many realizations are needed. In this way, we obtain a rigorous probabilistic description of approximate solutions. In the paper, we consider parabolic and wave equations in heterogeneous media. In each time interval, the domain is divided into subregions. Using residual information, we design appropriate prior and posterior distributions. The likelihood consists of the residual minimization. To sample from the resulting posterior distribution, we consider several sampling strategies. The sampling involves identifying important regions and important degrees of freedom beyond permanent basis functions, which are used in residual computation. Numerical results are presented. We consider two sampling algorithms. The first algorithm uses sequential sampling and is inexpensive. In the second algorithm, we perform full sampling using the Gibbs sampling algorithm, which is more accurate compared to the sequential sampling. The main novel ingredients of our approach consist of: defining appropriate permanent basis functions and the corresponding residual; setting up a proper posterior distribution; and sampling the posteriors.

中文翻译：

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贝叶斯多尺度有限元方法。对缺失的子电网信息进行概率建模

在本文中，我们开发了一种基于多尺度有限元方法的贝叶斯多尺度方法。由于许多多尺度应用中的尺度差异，计算模型无法解决所有尺度。提出了各种子网格模型来表示未解析的尺度。在这里，我们考虑使用多尺度有限元方法对未解析尺度进行建模的概率方法（参见 [1, 2]）。通过使用广义多尺度有限元表示主导模式，我们提出了一个贝叶斯框架，它为确定性问题提供了多种廉价（可计算）的解决方案。这些近似概率解可能不太接近精确解，因此需要许多实现。通过这种方式，我们获得了近似解的严格概率描述。在论文中，我们考虑异质介质中的抛物线方程和波动方程。在每个时间间隔内，域被划分为子区域。使用残差信息，我们设计适当的先验和后验分布。似然由残差最小化组成。为了从产生的后验分布中采样，我们考虑了几种采样策略。抽样涉及识别重要区域和重要的自由度，超出永久基函数，用于残差计算。给出了数值结果。我们考虑两种采样算法。第一种算法使用顺序采样并且成本低廉。在第二种算法中，我们使用 Gibbs 采样算法进行全采样，它比顺序采样更准确。我们方法的主要新颖成分包括：定义适当的永久基函数和相应的残差；建立适当的后验分布；并采样后验。