Skip to main content
SpringerLink
Log in

Tensor structured evaluation of singular volume integrals

  • Published:
Computing and Visualization in Science

Abstract

In this article, we introduce a new method for the accurate and fast computation of singular integrals over cuboids in three-dimensional space. Using a straightforward geometric parametrisation of the domain of integration, we interpret the integral as a smooth function on a high-dimensional parameter space. A standard interpolation scheme then leads to a high-dimensional tensor to which an approximation in the data-sparse hierarchical tensor format is applied. Once this approximation is available, the evaluation of an integral value becomes an easy task which does no longer require the treatment of singular terms. Numerical experiments illustrate the potential of the proposed approach for typical examples.

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. Ballani, J.: Fast evaluation of singular BEM integrals based on tensor approximations. Numer. Math. 121(3), 433–460 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical Tucker format. Linear Algebra Appl. 438(2), 639–657 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertoglio, C., Khoromskij, B.N.: Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels. Comput. Phys. Comm. 183(4), 904–912 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Börm, S., Hackbusch, W.: Hierarchical quadrature of singular integrals. Computing 74, 75–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hackbusch, W.: Direct integration of the Newton potential over cubes. Computing 68(3), 193–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hackbusch, W.: Entwicklungen nach Exponentialsummen. Preprint 4/2005. Max Planck Institute for Mathematics in the Sciences (2005)

  8. Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  9. Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals. J. Numer. Math. 18(2), 91–117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Meszmer, P.: Hierarchical quadrature for multidimensional singular integrals—part II. Preprint 71/2012. Max Planck Institute for Mathematics in the Sciences (2012)

  12. Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31(5), 3744–3759 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tobler, C.: Low-rank tensor methods for linear systems and eigenvalue problems. Ph.D. thesis, ETH Zürich (2012)

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 , Leipzig, Germany

    Jonas Ballani & Peter Meszmer

Authors
  1. Jonas Ballani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Meszmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonas Ballani.

Additional information

Communicated by: Wolfgang Hackbusch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ballani, J., Meszmer, P. Tensor structured evaluation of singular volume integrals. Comput. Visual Sci. 15, 75–86 (2012). https://doi.org/10.1007/s00791-013-0199-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-013-0199-y

Keywords

Search

Navigation