A quasi-optimal sparse grids procedure for groundwater flows

Joakim Beck; Fabio Nobile; Lorenzo Tamellini; Raúl Tempone

doi:10.1007/978-3-319-01601-6_1

A quasi-optimal sparse grids procedure for groundwater flows

Joakim Beck, Fabio Nobile, Lorenzo Tamellini, Raúl Tempone^*

^*Corresponding author for this work

Computer, Electrical and Mathematical Sciences and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

14 Scopus citations

Abstract

In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work “On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods” to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectiveness on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.

Original language	English (US)
Title of host publication	Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers
Editors	Mejdi Azaïez, Jan S. Hesthaven, Henda El Fekih
Publisher	Springer Verlag
Pages	1-16
Number of pages	16
ISBN (Electronic)	9783319016009
DOIs	https://doi.org/10.1007/978-3-319-01601-6_1
State	Published - 2014
Event	9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012 - Gammarth, Tunisia Duration: Jun 25 2012 → Jun 29 2012

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	95
ISSN (Print)	1439-7358

Other

Other	9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012
Country/Territory	Tunisia
City	Gammarth
Period	06/25/12 → 06/29/12

Bibliographical note

Funding Information:
The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems”. Support from the VR project “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) AEA project “Predictability and Uncertainty Quantification for Models of Porous Media” is also acknowledged. The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.

Publisher Copyright:
© Springer International Publishing Switzerland 2014.

ASJC Scopus subject areas

Modeling and Simulation
Engineering(all)
Discrete Mathematics and Combinatorics
Control and Optimization
Computational Mathematics

Access to Document

10.1007/978-3-319-01601-6_1

Cite this

Beck, J., Nobile, F., Tamellini, L., & Tempone, R. (2014). A quasi-optimal sparse grids procedure for groundwater flows. In M. Azaïez, J. S. Hesthaven, & H. E. Fekih (Eds.), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers (pp. 1-16). (Lecture Notes in Computational Science and Engineering; Vol. 95). Springer Verlag. https://doi.org/10.1007/978-3-319-01601-6_1

Beck, Joakim ; Nobile, Fabio ; Tamellini, Lorenzo et al. / A quasi-optimal sparse grids procedure for groundwater flows. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers. editor / Mejdi Azaïez ; Jan S. Hesthaven ; Henda El Fekih. Springer Verlag, 2014. pp. 1-16 (Lecture Notes in Computational Science and Engineering).

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abstract = "In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work “On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods” to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectiveness on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.",

author = "Joakim Beck and Fabio Nobile and Lorenzo Tamellini and Ra{\'u}l Tempone",

note = "Funding Information: The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems”. Support from the VR project “Effektiva numeriska metoder f{\"o}r stokastiska differentialekvationer med till{\"a}mpningar” and King Abdullah University of Science and Technology (KAUST) AEA project “Predictability and Uncertainty Quantification for Models of Porous Media” is also acknowledged. The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2014.; 9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012 ; Conference date: 25-06-2012 Through 29-06-2012",

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Beck, J, Nobile, F, Tamellini, L & Tempone, R 2014, A quasi-optimal sparse grids procedure for groundwater flows. in M Azaïez, JS Hesthaven & HE Fekih (eds), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers. Lecture Notes in Computational Science and Engineering, vol. 95, Springer Verlag, pp. 1-16, 9th International Conference on Spectral and High Order Methods, ICOSAHOM 2012, Gammarth, Tunisia, 06/25/12. https://doi.org/10.1007/978-3-319-01601-6_1

A quasi-optimal sparse grids procedure for groundwater flows. / Beck, Joakim; Nobile, Fabio; Tamellini, Lorenzo et al.
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers. ed. / Mejdi Azaïez; Jan S. Hesthaven; Henda El Fekih. Springer Verlag, 2014. p. 1-16 (Lecture Notes in Computational Science and Engineering; Vol. 95).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N1 - Funding Information: The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project n. RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems”. Support from the VR project “Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) AEA project “Predictability and Uncertainty Quantification for Models of Porous Media” is also acknowledged. The fourth author is a member of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. Publisher Copyright: © Springer International Publishing Switzerland 2014.

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AB - In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work “On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods” to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectiveness on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.

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Beck J, Nobile F, Tamellini L, Tempone R. A quasi-optimal sparse grids procedure for groundwater flows. In Azaïez M, Hesthaven JS, Fekih HE, editors, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2012 - ICOSAHOM conference, Selected papers. Springer Verlag. 2014. p. 1-16. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-319-01601-6_1

A quasi-optimal sparse grids procedure for groundwater flows

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