An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient

Handle URI:
http://hdl.handle.net/10754/622131
Title:
An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient
Authors:
Nobile, Fabio; Tamellini, Lorenzo; Tesei, Francesco; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process. We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e.g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.
KAUST Department:
Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Nobile F, Tamellini L, Tesei F, Tempone R (2016) An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient. Sparse Grids and Applications - Stuttgart 2014: 191–220. Available: http://dx.doi.org/10.1007/978-3-319-28262-6_8.
Publisher:
Springer Science + Business Media
Journal:
Lecture Notes in Computational Science and Engineering
Conference/Event name:
3rd Workshop on Sparse Grids and Applications, SGA 2014
Issue Date:
18-Mar-2016
DOI:
10.1007/978-3-319-28262-6_8
Type:
Conference Paper
ISSN:
1439-7358; 2197-7100
Appears in Collections:
Conference Papers

Full metadata record

DC FieldValue Language
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTamellini, Lorenzoen
dc.contributor.authorTesei, Francescoen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-01-02T08:10:20Z-
dc.date.available2017-01-02T08:10:20Z-
dc.date.issued2016-03-18en
dc.identifier.citationNobile F, Tamellini L, Tesei F, Tempone R (2016) An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient. Sparse Grids and Applications - Stuttgart 2014: 191–220. Available: http://dx.doi.org/10.1007/978-3-319-28262-6_8.en
dc.identifier.issn1439-7358en
dc.identifier.issn2197-7100en
dc.identifier.doi10.1007/978-3-319-28262-6_8en
dc.identifier.urihttp://hdl.handle.net/10754/622131-
dc.description.abstractIn this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process. We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e.g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.en
dc.publisherSpringer Science + Business Mediaen
dc.titleAn Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficienten
dc.typeConference Paperen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalLecture Notes in Computational Science and Engineeringen
dc.conference.date2014-09-01 to 2014-09-05en
dc.conference.name3rd Workshop on Sparse Grids and Applications, SGA 2014en
dc.conference.locationStuttgart, DEUen
dc.contributor.institutionSB-MATHICSE-CSQI-EPFL, Station 8, Lausanne, Switzerlanden
kaust.authorTempone, Raulen
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