RT Journal Article ID 6b673c5535731fb2 A1 Efendiev, Yalchin A1 Leung, Wing Tat A1 Cheung, S. W. A1 Guha, N. A1 Hoang, V. H. A1 Mallick, B. T1 BAYESIAN MULTISCALE FINITE ELEMENT METHODS. MODELING MISSING SUBGRID INFORMATION PROBABILISTICALLY JF International Journal for Multiscale Computational Engineering JO JMC YR 2017 FD 2017-04-27 VO 15 IS 2 SP 175 OP 197 K1 Bayesian K1 multiscale K1 MCMC K1 Gibbs K1 multiscale finite element method AB 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 cannot resolve all scales. Various subgrid models are proposed to represent unresolved scales. Here, we consider a probabilistic approach for modeling unresolved scales using the multiscale finite element method [cf., Chkrebtii et al., Bayesian Anal., vol. 11, no. 4, pp. 1239-1267, 2016; Mallick et al., Bayesian Anal., vol. 11, no. 4, p. 1279, 2016]. 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. PB Begell House LK http://dl.begellhouse.com/journals/61fd1b191cf7e96f,381b14b515667722,6b673c5535731fb2.html