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generalized multiscale finite element *** Elliptic Equations

Communications in Computational Physics 2014年第3期15卷 733-755页

作者： Yalchin Efendiev Juan Galvis Guanglian Li Michael Presho Center for Numerical Porous Media(NumPor) King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Kingdom of Saudi Arabia Department of Mathematics&Institute for Scientific Computation(ISC) Texas A&M UniversityCollege StationTexas 77843USA Departamento de Matem´aticas Universidad Nacional de ColombiaBogot´a D.C.Colombia

In this paper we use the generalized multiscale finite element method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast *** proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem *** this convention,we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin(CG)or discontinuous Galerkin(DG)global ***,we use Symmetric Interior Penalty Discontinuous Galerkin *** methods yield a predictable error decline that depends on the respective coarse space dimension,and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

关键词： generalized multiscale finite element method nonlinear equations high-contrast

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Multilevel Markov Chain Monte Carlo method for High-Contrast Single-Phase Flow Problems

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Communications in Computational Physics 2015年第1期17卷 259-286页

作者： Yalchin Efendiev Bangti Jin Michael Presho Xiaosi Tan Department of Mathematics Texas A&M UniversityCollege StationTX 77843-3368USA SRI Center for Numerical Porous Media(NumPor) King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Kingdom of Saudi Arabia Department of Computer Science University College LondonGower StreetLondon WC1E 6BTUK

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow *** is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)*** 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 *** 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 *** 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 *** 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 ***,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 *** 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.

关键词： generalized multiscale finite element method multilevel Monte Carlo method multilevel Markov chain Monte Carlo uncertainty quantification

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