Multiscale empirical interpolation for solving nonlinear PDEs

Handle URI:
http://hdl.handle.net/10754/563887
Title:
Multiscale empirical interpolation for solving nonlinear PDEs
Authors:
Calo, Victor M. ( 0000-0002-1805-4045 ) ; Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Galvis, Juan; Ghommem, Mehdi
Abstract:
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to represent the solution on a coarse grid with multiscale basis functions computed offline. Computing the GMsFEM solution involves calculating the system residuals and Jacobians on the fine grid. We use empirical interpolation concepts to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one. The empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. The proposed multiscale empirical interpolation techniques: (1) divide computing the nonlinear function into coarse regions; (2) evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution; and (3) introduce multiscale proper-orthogonal-decomposition techniques to find appropriate interpolation vectors. We demonstrate the effectiveness of the proposed methods on several nonlinear multiscale PDEs that are solved with Newton's methods and fully-implicit time marching schemes. Our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale PDEs on a coarse grid with bounded error and significant computational cost reduction.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Applied Mathematics and Computational Science Program; Earth Science and Engineering Program; Physical Sciences and Engineering (PSE) Division; Environmental Science and Engineering Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Elsevier BV
Journal:
Journal of Computational Physics
Issue Date:
Dec-2014
DOI:
10.1016/j.jcp.2014.07.052
ARXIV:
arXiv:1407.0103
Type:
Article
ISSN:
00219991
Sponsors:
YE's work is 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 by FA9550-11-1-0341 from the Air Force Office of Scientific Research.
Additional Links:
http://arxiv.org/abs/arXiv:1407.0103v1
Appears in Collections:
Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorCalo, Victor M.en
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorGalvis, Juanen
dc.contributor.authorGhommem, Mehdien
dc.date.accessioned2015-08-03T12:18:24Zen
dc.date.available2015-08-03T12:18:24Zen
dc.date.issued2014-12en
dc.identifier.issn00219991en
dc.identifier.doi10.1016/j.jcp.2014.07.052en
dc.identifier.urihttp://hdl.handle.net/10754/563887en
dc.description.abstractIn this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to represent the solution on a coarse grid with multiscale basis functions computed offline. Computing the GMsFEM solution involves calculating the system residuals and Jacobians on the fine grid. We use empirical interpolation concepts to evaluate these residuals and Jacobians of the multiscale system with a computational cost which is proportional to the size of the coarse-scale problem rather than the fully-resolved fine scale one. The empirical interpolation method uses basis functions which are built by sampling the nonlinear function we want to approximate a limited number of times. The coefficients needed for this approximation are computed in the offline stage by inverting an inexpensive linear system. The proposed multiscale empirical interpolation techniques: (1) divide computing the nonlinear function into coarse regions; (2) evaluate contributions of nonlinear functions in each coarse region taking advantage of a reduced-order representation of the solution; and (3) introduce multiscale proper-orthogonal-decomposition techniques to find appropriate interpolation vectors. We demonstrate the effectiveness of the proposed methods on several nonlinear multiscale PDEs that are solved with Newton's methods and fully-implicit time marching schemes. Our numerical results show that the proposed methods provide a robust framework for solving nonlinear multiscale PDEs on a coarse grid with bounded error and significant computational cost reduction.en
dc.description.sponsorshipYE's work is 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 by FA9550-11-1-0341 from the Air Force Office of Scientific Research.en
dc.publisherElsevier BVen
dc.relation.urlhttp://arxiv.org/abs/arXiv:1407.0103v1en
dc.subjectDiscrete empirical interpolation methoden
dc.subjectGeneralized multiscale finite element methodsen
dc.subjectModel reductionen
dc.subjectNonlinear PDEsen
dc.titleMultiscale empirical interpolation for solving nonlinear PDEsen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentEarth Science and Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEnvironmental Science and Engineering Programen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalJournal of Computational Physicsen
dc.contributor.institutionDepartment of Mathematics and Institute for Scientific Computation (ISC), Texas AandM UniversityCollege Station, TX, United Statesen
dc.contributor.institutionDepartamento de Matemáticas, Universidad Nacional de Colombia, Carrera 45 No 26-85 Edificio Uriel GutierrézBogotá D.C., Colombiaen
dc.identifier.arxividarXiv:1407.0103en
kaust.authorCalo, Victor M.en
kaust.authorEfendiev, Yalchin R.en
kaust.authorGhommem, Mehdien
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