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A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

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Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 95))

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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.

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Acknowledgements

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.

Author information

Authors and Affiliations

  1. Applied Mathematics and Computational Science, KAUST, Thuwal, Saudi Arabia

    Joakim Beck & Raúl Tempone

  2. CSQI – MATHICSE, Ecole Politechnique Fédérale Lausanne, Lausanne, Switzerland

    Fabio Nobile & Lorenzo Tamellini

  3. MOX, Department of Mathematics “F. Brioschi”, Politecnico di Milano, Milan, Italy

    Fabio Nobile & Lorenzo Tamellini

Authors
  1. Joakim Beck
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  2. Fabio Nobile
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  3. Lorenzo Tamellini
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  4. Raúl Tempone
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Corresponding author

Correspondence to Raúl Tempone .

Editor information

Editors and Affiliations

  1. Institut Polytechnique de Bordeaux, Université de Bordeaux, Pessac, France

    Mejdi Azaïez

  2. Ecole Nationale d'Ingénieurs de Tunis, Université de Tunis El Manar, Tunis, Tunisia

    Henda El Fekih

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Beck, J., Nobile, F., Tamellini, L., Tempone, R. (2014). A Quasi-optimal Sparse Grids Procedure for Groundwater Flows. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. http://doi.org/10.1007/978-3-319-01601-6_1

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