Skip to main content
SpringerLink
Log in

Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case

  • Conference paper
  • First Online:
Sparse Grids and Applications - Munich 2012

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 97))

Abstract

We investigate the stochastic collocation method for parametric, elliptic partial differential equations (PDEs) with lognormally distributed random parameters in mixed formulation. Such problems arise, e.g., in uncertainty quantification studies for flow in porous media with random conductivity. We show the analytic dependence of the solution of the PDE w.r.t. the parameters and use this to show convergence of the sparse grid stochastic collocation method. This work fills some remaining theoretical gaps for the application of stochastic collocation in case of elliptic PDEs where the diffusion coefficient is not strictly bounded away from zero w.r.t. the parameters. We illustrate our results for a simple groundwater flow problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. I. Babuška, R. Tempone, G.E. Zouraris, Galerkin finite element approximations of stochastic elliptical partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. I. Babuška, F. Nobile, R. Tempone, A stochastic collocation method for elliptical partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Beck, R. Tempone, F. Nobile, L. Tamellini, On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods. Math. Models Methods Appl. Sci. 22(9), 1250023.1–1250023.33. doi: 10.1142/S0218202512500236 (2012)

    Article  MathSciNet  Google Scholar 

  4. A. Bespalov, C.E. Powell, D. Silvester, A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data. SIAM J. Numer. Anal. 50(4), 2039–2063 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, Berlin, 1991)

    Book  MATH  Google Scholar 

  6. J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50(1), 216–246 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Cliffe, O.G. Ernst, B. Sprungk, E. Ullmann, K.G. van den Boogaart, Probabilistic uncertainty quantification using PDEs with random data: a case study. (2014, in preparation)

    Google Scholar 

  8. A. Cohen, R. DeVore, C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Anal. Appl. 9, 11–47 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  10. O.G. Ernst, A. Mugler, H.-J. Starkloff, E. Ullmann, On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal. 46(2), 317–339 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. R.A. Freeze, A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res. 11(5), 725–741 (1975)

    Article  Google Scholar 

  12. B. Ganis, H. Klie, M.F. Wheeler, T. Wildey, I. Yotov, D. Zhang, Stochastic collocation and mixed finite elements for flow in porous media. Comput. Methods Appl. Mech. Eng. 197, 3547–3559 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach (Springer, New York, 1991)

    Book  MATH  Google Scholar 

  14. F. Hartogs, Zur Theorie der analytischen Funktionen mehrer Veränderlicher, insbesondere über die Darstellung derselben durch Reihen, welche Potenzen einer Veränderlichen fortschreiten. Math. Ann. 62, 1–88 (1906)

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Hille, Contributions to the theory of Hermitian series. Duke Math. J. 5, 875–936 (1939)

    Article  MathSciNet  Google Scholar 

  16. E. Hille, Contributions to the theory of Hermitian series. II. The representation problem. Trans. Am. Math. Soc. 47, 80–94 (1940)

    MathSciNet  Google Scholar 

  17. O. Kallenberg, Foundations of Modern Probability (Springer, Berlin, 2002)

    Book  MATH  Google Scholar 

  18. F. Nobile, R. Tempone, 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)

    Article  MATH  MathSciNet  Google Scholar 

  19. F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Schwab, C.J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. L.N. Trefethen, Approximation Theory and Approximation Practice (SIAM, Philadelphia, 2012)

    Google Scholar 

  22. G.W. Wasilkowski, H. Woźniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 11, 1–56 (1995)

    Article  MATH  Google Scholar 

  23. D. Xiu, J.S. Hesthaven, High-order collocation methods differential equations with random inputs. SIAM J. Sci. Comput. 37(3), 1118–1139 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support from the DFG Priority Programme 1324 as well as helpful discussions with Lorenzo Tamellini, Fabio Nobile and Alexei Bespalov.

Author information

Authors and Affiliations

  1. Department of Mathematics, TU Chemnitz, Reichenhainer Str. 41, 09126, Chemnitz, Germany

    Oliver G. Ernst & Björn Sprungk

Authors
  1. Oliver G. Ernst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Björn Sprungk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oliver G. Ernst .

Editor information

Editors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Jochen Garcke

  2. Institute for Parallel and Distributed Systems, Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Ernst, O.G., Sprungk, B. (2014). Stochastic Collocation for Elliptic PDEs with Random Data: The Lognormal Case. In: Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2012. Lecture Notes in Computational Science and Engineering, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-04537-5_2

Download citation

Publish with us

Policies and ethics

Search

Navigation