ISSN 打印: 1543-1649
ISSN 在线: 1940-4352

# 国际多尺度计算工程期刊

DOI: 10.1615/IntJMultCompEng.2017019851
pages 175-197

## BAYESIAN MULTISCALE FINITE ELEMENT METHODS. MODELING MISSING SUBGRID INFORMATION PROBABILISTICALLY

Yalchin Efendiev
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77840, USA; Multiscale Model Reduction Laboratory, North-Eastern Federal University, Yakutsk, Russia, 677980
Wing Tat Leung
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
S. W. Cheung
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas, 77843-3368, USA
N. Guha
Department of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas, 77843-3368, USA; Department of Statistics, Texas A&M University, College Station, Texas, 77843-3368, USA
V. H. Hoang
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371
B. Mallick
Department of Statistics, Texas A&M University, College Station, Texas, 77843-3368, USA

### ABSTRACT

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.

### Articles with similar content:

REITERATED MULTISCALE MODEL REDUCTION USING THE GENERALIZED MULTISCALE FINITE ELEMENT METHOD
International Journal for Multiscale Computational Engineering, Vol.14, 2016, issue 6
Wing Tat Leung, Maria Vasilyeva, Eric T. Chung, Yalchin Efendiev
GENERALIZED MULTISCALE FINITE ELEMENT METHODS: OVERSAMPLING STRATEGIES
International Journal for Multiscale Computational Engineering, Vol.12, 2014, issue 6
Michael Presho, Yalchin Efendiev, Guanglian Li, Juan Galvis
NONLINEAR NONLOCAL MULTICONTINUA UPSCALING FRAMEWORK AND ITS APPLICATIONS
International Journal for Multiscale Computational Engineering, Vol.16, 2018, issue 5
Eric T. Chung, Yalchin Efendiev, Wing T. Leung, Mary Wheeler
TRANSITIONAL ANNEALED ADAPTIVE SLICE SAMPLING FOR GAUSSIAN PROCESS HYPER-PARAMETER ESTIMATION
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 4
Alfredo Garbuno-Inigo, F. A. DiazDelaO, Konstantin M. Zuev
A HYBRID GENERALIZED POLYNOMIAL CHAOS METHOD FOR STOCHASTIC DYNAMICAL SYSTEMS
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 1
Michael Schick, Vincent Heuveline