Skip to main content
SpringerLink
Log in

Multilevel frames for sparse tensor product spaces

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

For Au = f with an elliptic differential operator \({A:\mathcal{H} \rightarrow \mathcal{H}'}\) and stochastic data f, the m-point correlation function \({{\mathcal M}^m u}\) of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A (m) of A. Sparse tensor products of hierarchic FE-spaces in \({\mathcal{H}}\) are known to allow for approximations to \({{\mathcal M}^m u}\) which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here how to achieve the fast computation of sparse approximations of \({{\mathcal M}^m u}\) for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balder, R.: Adaptive Verfahren für elliptische und parabolische Differentialgleichungen auf dünnen Gittern. Ph.D. Thesis, Institut für Informatik, TU München (1994)

  2. Balder R., Zenger C.: The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17, 631–646 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bebendorf M., Rjasanow S.: Adaptive low-rank approximation of collocation matrices. Computing 70, 1–24 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bramble J., Pasciak J., Xu J.: Parallel multilevel preconditioners. Math. Comput. 55, 1–22 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bungartz H.J.: A multigrid algorithm for higher order finite elements on sparse grids. ETNA Electron. Trans. Numer. Anal. 6, 63–77 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Bungartz, H.J., Dornseifer, T.: Sparse grids: Recent developments for elliptic partial differential equations. Multigrid methods V. In: Hackbusch, W. et al. (eds.) Proceedings of the 5th European multigrid conference, held in Stuttgart, Germany, October 1–4, 1996. Lecture Notes in Computer Science Engneering, vol. 3, pp. 45–70. Springer, Berlin (1998)

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

    Article  MathSciNet  Google Scholar 

  8. Christiansen O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Boston (2002)

    Google Scholar 

  9. Dahlke S., Fornasier M., Raasch T.: Adaptive frame methods for elliptic operator equations Adv. Comput. Math. 27, 27–63 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Dahlke, S., Fornasier, M., Raasch, T., Stevenson, R., Werner, M.: Adaptive frame methods for elliptic operator equations: the steepest descent approach. IMA J. Numer. Math. 27, 717–740 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  12. Daubechies, I.: Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61. SIAM, Philadephia (1992)

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Doppel K., Hochmuth R.: On the regularity of solutions of a homogeneous Dirichlet problem for a nonhypoelliptic linear partial differential operator. Ann. Acad. Sci. Fenn. A.I. Math. 16, 183–198 (1991)

    MATH  MathSciNet  Google Scholar 

  15. Greengard L., Rokhlin V.: A fast algorithm for particle simulation. J. Comput. Phys. 73, 325–348 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Griebel M.: Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Comput. 15, 547–565 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Griebel, M.: Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen. Teubner Skripten zur Numerik. B.G. Teubner, Stuttgart (1994)

  18. Griebel M., Oswald P.: On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66, 449–463 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Griebel M., Oswald P.: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4, 171–206 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  20. Harbrecht, H., Schneider, R., Schwab, C.: Sparse Second Moment Analysis for Elliptic Problems in Stochastic Domains. Numer. Math. 109, 167–188 (2008)

    Google Scholar 

  21. Hackbusch W.: A sparse matrix arithmetic based on \({\mathcal{H}}\) -matrices, Part I: Introduction to \({\mathcal{H}}\) -matrices. Computing 64, 89–108 (1999)

    Article  MathSciNet  Google Scholar 

  22. Hochmuth R.: An inhomogeneous Dirichlet problem for a non-hypoelliptic linear partial differential operator. Ann. Acad. Sci. Fenn. Math. 21, 179–187 (1996)

    MATH  MathSciNet  Google Scholar 

  23. Hochmuth, R.: Wavelets in Numerical Analysis and Restricted Nonlinear Approximation. Habilitation-Thesis (1999)

  24. Hochmuth R.: Nonlinear anisotropic boundary value problems—regularity results and multiscale discretizations. Nonlinear Anal. Ser. A. Theory Methods 46, 1–18 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kaasschieter E.F.: Preconditioned conjugate gradients for solving singular systems. J. Comput. Appl. Math. 24, 265–275 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  26. Oswald, P.: Multilevel finite element approximation. Theory and applications. Teubner Skripten zur Numerik. B.G. Teubner, Stuttgart (1994)

  27. Oswald P.: On function spaces related to Finite Element Approximation Theory. Z. Anal. Anwend. 9, 43–64 (1990)

    MATH  MathSciNet  Google Scholar 

  28. 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 

  29. Pfaffinger, A.: Funktionale Beschreibung und Parallelisierung von Algorithmen auf dünnen Gittern. Ph.D. Thesis, Institut für Informatik, TU München (1997)

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Schwab C., Todor R.: Sparse finite elements for stochastic elliptic problems-higher order moments. Computing 71, 43–63 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  32. Stevenson R.: Adaptive solution of operator quations using wavelet frames. SIAM J. Numer. Anal. 41, 1074–1100 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zenger, C.: Sparse grids. Parallel algorithms for partial differential equations. In: Proceedings of 6th GAMM- Seminar. Kiel/Ger, 1990. Notes Numerical Fluid Mechanics, vol. 31, pp. 241–251. Vieweg, Braunschweig (1991)

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Helmut Harbrecht

  2. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Reinhold Schneider

  3. Seminar für Angewandte Mathematik, Eidgenössische Technische Hochschule Zürich, Rämistr. 101, 8092, Zürich, Switzerland

    Christoph Schwab

Authors
  1. Helmut Harbrecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhold Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christoph Schwab
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut Harbrecht.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Harbrecht, H., Schneider, R. & Schwab, C. Multilevel frames for sparse tensor product spaces. Numer. Math. 110, 199–220 (2008). https://doi.org/10.1007/s00211-008-0162-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0162-x

Mathematics Subject Classification (2000)

Search

Navigation