Multilevel markov chain monte carlo method for high-contrast single-phase flow problems

Handle URI:
http://hdl.handle.net/10754/563926
Title:
Multilevel markov chain monte carlo method for high-contrast single-phase flow problems
Authors:
Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Jin, Bangti; Michael, Presho; Tan, Xiaosi
Abstract:
In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. © Global Science Press Limited 2015.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Global Science Press
Journal:
Communications in Computational Physics
Issue Date:
19-Dec-2014
DOI:
10.4208/cicp.021013.260614a
Type:
Article
ISSN:
18152406
Sponsors:
Y. Efendiev's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project. The research of B. Jin is partly supported by NSF Grant DMS-1319052.
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorJin, Bangtien
dc.contributor.authorMichael, Preshoen
dc.contributor.authorTan, Xiaosien
dc.date.accessioned2015-08-03T12:19:47Zen
dc.date.available2015-08-03T12:19:47Zen
dc.date.issued2014-12-19en
dc.identifier.issn18152406en
dc.identifier.doi10.4208/cicp.021013.260614aen
dc.identifier.urihttp://hdl.handle.net/10754/563926en
dc.description.abstractIn this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. © Global Science Press Limited 2015.en
dc.description.sponsorshipY. Efendiev's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project. The research of B. Jin is partly supported by NSF Grant DMS-1319052.en
dc.publisherGlobal Science Pressen
dc.subjectGeneralized multiscale finite element methoden
dc.subjectmultilevel Markov chain Monte Carloen
dc.subjectmultilevel Monte Carlo methoden
dc.subjectuncertainty quantificationen
dc.titleMultilevel markov chain monte carlo method for high-contrast single-phase flow problemsen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalCommunications in Computational Physicsen
dc.contributor.institutionDepartment of Mathematics, Texas A and M UniversityCollege Station, TX, United Statesen
dc.contributor.institutionDepartment of Computer Science, University College London, Gower StreetLondon, United Kingdomen
kaust.authorEfendiev, Yalchin R.en
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