A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

  • Joakim Beck
  • Fabio Nobile
  • Lorenzo Tamellini
  • Raúl TemponeEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 95)


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.


Interpolation Point Sparse Grid Permeability Field Polynomial Chaos Expansion Homogeneous Dirichlet Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 1.
    T. Arbogast, M. F. Wheeler, and I. Yotov. Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal., 34(2):828–852, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Review, 52(2):317–355, June 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    V. Barthelmann, E. Novak, and K. Ritter. High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math., 12(4):273–288, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bear and A.H.D. Cheng. Modeling Groundwater Flow and Contaminant Transport. Theory and Applications of Transport in Porous Media. Springer, 2010.CrossRefzbMATHGoogle Scholar
  5. 5.
    J. Beck, F. Nobile, L. Tamellini, and R. Tempone. On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Mathematical Models and Methods in Applied Sciences, 22(09), 2012.Google Scholar
  6. 6.
    F. Brezzi and M. Fortin. Mixed and hybrid finite element methods, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.Google Scholar
  7. 7.
    H.J Bungartz and M. Griebel. Sparse grids. Acta Numer., 13:147–269, 2004.Google Scholar
  8. 8.
    C.G. Canuto, Y. Hussaini, A. Quarteroni, and T.A. Zang. Spectral Methods: Fundamentals in Single Domains. Springer, 2006.Google Scholar
  9. 9.
    J. Charrier. Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal., 50(1), 2012.Google Scholar
  10. 10.
    J. Galvis and M. Sarkis. Approximating infinity-dimensional stochastic Darcy’s equations without uniform ellipticity. SIAM J. Numer. Anal., 47(5):3624–3651, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. Genz and B. D. Keister. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. J. Comput. Appl. Math., 71(2), 1996.Google Scholar
  12. 12.
    T. Gerstner and M. Griebel. Dimension-adaptive tensor-product quadrature. Computing, 71(1):65–87, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    C. J. Gittelson. Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci., 20(2):237–263, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    M. Griebel and S. Knapek. Optimized general sparse grid approximation spaces for operator equations. Math. Comp., 78(268):2223–2257, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Grigoriu. Stochastic Calculus: Applications in Science and Engineering. Birkhäuser Boston, 2002.Google Scholar
  16. 16.
    V. H. Hoang and C. Schwab. N-term galerkin wiener chaos approximations of elliptic pdes with lognormal gaussian random inputs. SAM-Report 2011–59, ETHZ, 2011.Google Scholar
  17. 17.
    A. S. Kronrod. Nodes and weights of quadrature formulas. Sixteen-place tables. Authorized translation from the Russian. Consultants Bureau, New York, 1965.Google Scholar
  18. 18.
    H. Li and D. Zhang. Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resources Research, 43(9), 2007.Google Scholar
  19. 19.
    G. Lin and A.M. Tartakovsky. An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Advances in Water Resources, 32(5):712–722, 2009.CrossRefGoogle Scholar
  20. 20.
    M. Loève. Probability theory. II. Springer-Verlag, New York, fourth edition, 1978. Graduate Texts in Mathematics, Vol. 46.Google Scholar
  21. 21.
    F. Müller, P. Jenny, and D.W. Meyer. Probabilistic collocation and lagrangian sampling for advective tracer transport in randomly heterogeneous porous media. Advances in Water Resources, 34(12):1527–1538, 2011.CrossRefGoogle Scholar
  22. 22.
    S.P. Neuman, M. Riva, and A. Guadagnini. On the geostatistical characterization of hierarchical media. Water Resources Research, 44(2), 2008.Google Scholar
  23. 23.
    F. Nobile, R. Tempone, and C.G. Webster. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal., 46(5):2411–2442, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    T. N. L. Patterson. The optimum addition of points to quadrature formulae. Math. Comp. 22 (1968), 847–856; addendum, ibid., 22(104):C1–C11, 1968.Google Scholar
  25. 25.
    C. Schwab and R. A. Todor. Karhunen-Loève approximation of random fields by generalized fast multipole methods. Journal of Computational Physics, 217(1):100 – 122, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    I. H. Sloan and H. Woźniakowski. When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity, 14(1):1–33, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    L. Tamellini. Polynomial approximation of PDEs with stochastic coefficients. PhD thesis, Politecnico di Milano, 2012.Google Scholar
  28. 28.
    D. Xiu and J.S. Hesthaven. High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput., 27(3):1118–1139, 2005.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Joakim Beck
    • 1
  • Fabio Nobile
    • 2
    • 3
  • Lorenzo Tamellini
    • 2
    • 3
  • Raúl Tempone
    • 1
    Email author
  1. 1.Applied Mathematics and Computational ScienceKAUSTThuwalSaudi Arabia
  2. 2.CSQI – MATHICSEEcole Politechnique Fédérale LausanneLausanneSwitzerland
  3. 3.MOX, Department of Mathematics “F. Brioschi”Politecnico di MilanoMilanItaly

Personalised recommendations