Skip to main content
Log in

Sparse second moment analysis for elliptic problems in stochastic domains

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations. Assuming normal perturbations with small amplitude and known mean field and two-point correlation function, we derive, using a second order shape calculus, deterministic equations for the mean field and the two-point correlation function of the random solution for a model Dirichlet problem which are 3rd order accurate in the boundary perturbation size. Using a variational boundary integral equation formulation on the unperturbed, “nominal” boundary and a wavelet discretization, we present and analyze an algorithm to approximate the random solution’s mean and its two-point correlation function at essentially optimal order in essentially \({\mathcal{O}(N)}\) work and memory, where N denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Babuška, I., Nobile, F., Tempone, R.: Worst case scenario analysis for elliptic problems with uncertainty. Numer. Math. 101, 185–219 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bungartz, H.J., Griebel, M.: Sparse Grids. Acta Numer. 13, 147–269 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bogachev, V.I.: Gaussian Measures. AMS Mathematical Surveys and Monographs, vol. 62 (1998)

  4. Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)

    MathSciNet  Google Scholar 

  5. Dahmen, W., Harbrecht, H., Schneider, R.: Compression techniques for boundary integral equations—optimal complexity estimates. SIAM J. Numer. Anal. 43, 2251–2271 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delfour M., Zolesio J.-P. (2001). Shapes Geometries. SIAM, Philadelphia

    MATH  Google Scholar 

  7. Eppler, K.: Optimal shape design for elliptic equations via BIE-methods. J. Appl. Math. Comput. Sci. 10, 487–516 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Eppler, K.: Boundary integral representations of second derivatives in shape optimization. Discussiones Math. (Differential Inclusions, Control and Optimization) 20, 63–78 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Harbrecht, H., Schneider, R.: Wavelet Galerkin Schemes for Boundary Integral Equations—Implementation and Quadrature. SIAM J. Sci. Comput. 27, 1347–1370 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Harbrecht, H., Schneider, R.: Biorthogonal wavelet bases for the boundary element method. Math. Nachr. 269–270, 167–188 (2005)

    MathSciNet  Google Scholar 

  11. Hettlich, F., Rundell, W.: A second degree method for nonlinear inverse problems. SIAM J. Numer. Anal. 37, 587–620 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin (1991)

    MATH  Google Scholar 

  13. Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Pitman, Boston, 1985. Monographs and Studies in Mathematics, 23. Pitman Advanced Publishing Program. Boston-London-Melbourne: Pitman Publishing Inc. XIII, 344 p.

  14. Mc Lean, W.: Strongly Elliptic Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  15. Murat, F., Simon, J.: Étude de problèmes d’optimal design. In: CÉa, J. (eds) Optimization Techniques, Modeling and Optimization in the Service of Man, Lecture Notes Computer Science vol. 41, pp. 54–62. Springer, Berlin (1976)

    Google Scholar 

  16. NÉdÉlec, J.C., Planchard, J.P.: Une méthode variationelle d’élements finis pour la résolution numérique d’un problème extérieur dans \({\mathbb{R}^3}\) . RAIRO Anal. Numér. 7, 105–129 (1973)

    Google Scholar 

  17. von Petersdorff, T., Schwab, C.: Numerical solution of parabolic equations in high dimensions. M2AN Math. Model. Numer. Anal. 38, 93–127 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. von Petersdorff, T., Schwab, C.: Sparse wavelet methods for operator equations with stochastic data. Appl. Math. 51(2), 145–180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, New York (1983)

    Google Scholar 

  20. Potthast, R.: Fréchet-Differenzierbarkeit von Randintegraloperatoren und Randwertproblemen zur Helmholtzgleichung und den zeitharmonischen Maxwellgleichungen. PhD Thesis, Universität Göttingen (1994)

  21. Schmidlin, G., Lage, Ch., Schwab, Ch.: Rapid solution of first kind boundary integral equations in R 3. Eng. Anal. Boundary Elements 27(5), 469–490 (2003)

    Article  MATH  Google Scholar 

  22. Schneider, R.: Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur Lösung großer vollbesetzter Gleichungssysteme. B.G. Teubner, Stuttgart (1998)

    MATH  Google Scholar 

  23. Sauter, S.A., Schwab, C.: Randelementmethoden. B.G. Teubner, Stuttgart (2004)

    MATH  Google Scholar 

  24. Schwab, C., Todor, R.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Schwab, C., Wendland, W.L.: On the extraction technique in boundary integral equations. Math. Comput. 68, 91–122 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Simon, J.: Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2, 649–687 (1980)

    MathSciNet  MATH  Google Scholar 

  27. Sokolowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Springer, Berlin (1992)

    MATH  Google Scholar 

  28. Tausch, J., White, J.: Multiscale bases for the sparse representation of boundary integral operators on complex geometry. SIAM J. Sci. Stat. Comput. 24, 1610–1629 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Triebel, H.: Theory of function spaces 2nd edn. Joh. A. Barth Publ., Leipzig (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Helmut Harbrecht.

Additional information

This work was supported by the EEC Human Potential Programme under contract HPRN-CT-2002-00286, “Breaking Complexity.” Work initiated while HH visited the Seminar for Applied Mathematics at ETH Zürich in the Wintersemester 2005/06 and completed during the summer programme CEMRACS2006 “Modélisation de l’aléatoire et propagation d’incertitudes” in July and August 2006 at the C.I.R.M., Marseille, France.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harbrecht, H., Schneider, R. & Schwab, C. Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109, 385–414 (2008). https://doi.org/10.1007/s00211-008-0147-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0147-9

Mathematics Subject Classification (2000)

Navigation