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In this paper, we develop a Bayesian multiscale approach based on amultiscale finite element method. Because of scale disparity in many multiscaleapplications, computational models can not resolve all scales. Various subgridmodels are proposed to represent un-resolved scales. Here, we consider aprobabilistic approach for modeling un-resolved scales using the MultiscaleFinite Element Method (cf., [1, 2]). By representing dominant modes using theGeneralized Multiscale Finite Element, we propose a Bayesian framework, whichprovides multiple inexpensive (computable) solutions for a deterministicproblem. These approximate probabilistic solutions may not be very close to theexact solutions and, thus, many realizations are needed. In this way, we obtaina rigorous probabilistic description of approximate solutions. In the paper, weconsider parabolic and wave equations in heterogeneous media. In each timeinterval, the domain is divided into subregions. Using residual information, wedesign appropriate prior and posterior distributions. The likelihood consistsof the residual minimization. To sample from the resulting posteriordistribution, we consider several sampling strategies. The sampling involvesidentifying important regions and important degrees of freedom beyond permanentbasis functions, which are used in residual computation. Numerical results arepresented. We consider two sampling algorithms. The first algorithm usessequential sampling and is inexpensive. In the second algorithm, we performfull sampling using the Gibbs sampling algorithm, which is more accuratecompared to the sequential sampling. The main novel ingredients of our approachconsist of: defining appropriate permanent basis functions and thecorresponding residual; setting up a proper posterior distribution; andsampling the posteriors.

Bayesian Multiscale Finite Element Methods. Modeling missing subgrid information probabilistically