Доступ предоставлен для: Guest
Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
International Journal for Multiscale Computational Engineering
Импакт фактор: 1.016 5-летний Импакт фактор: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Печать: 1543-1649
ISSN Онлайн: 1940-4352

Выпуски:
Том 17, 2019 Том 16, 2018 Том 15, 2017 Том 14, 2016 Том 13, 2015 Том 12, 2014 Том 11, 2013 Том 10, 2012 Том 9, 2011 Том 8, 2010 Том 7, 2009 Том 6, 2008 Том 5, 2007 Том 4, 2006 Том 3, 2005 Том 2, 2004 Том 1, 2003

International Journal for Multiscale Computational Engineering

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

Краткое описание

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:

GENERALIZED MULTISCALE FINITE ELEMENT METHODS: OVERSAMPLING STRATEGIES
International Journal for Multiscale Computational Engineering, Vol.12, 2014, issue 6
Juan Galvis, Michael Presho, Yalchin Efendiev, Guanglian Li
A MULTI-INDEX MARKOV CHAIN MONTE CARLO METHOD
International Journal for Uncertainty Quantification, Vol.8, 2018, issue 1
Ajay Jasra, Yan Zhou, Kengo Kamatani, Kody J. H. Law
A GRADIENT-BASED SAMPLING APPROACH FOR DIMENSION REDUCTION OF PARTIAL DIFFERENTIAL EQUATIONS WITH STOCHASTIC COEFFICIENTS
International Journal for Uncertainty Quantification, Vol.5, 2015, issue 1
Miroslav Stoyanov, Clayton G. Webster
OPTIMIZATION-BASED SAMPLING IN ENSEMBLE KALMAN FILTERING
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 4
Alexander Bibov, Heikki Haario, Antti Solonen, Johnathan M. Bardsley
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