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Sparse grids

Published online by Cambridge University Press:  04 August 2010

Hans-Joachim Bungartz
Affiliation:
IPVS, Universität Stuttgart, Universitdtsstrafie 38, D-70569 Stuttgart, Germany
Michael Griebel
Affiliation:
Institut für Numerische Simulation, Universität Bonn, Wegelerstrafie 6, D-53113 Bonn, Germany
Arieh Iserles
Affiliation:
University of Cambridge
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Summary

We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced in Zenger (1991), is based on a higherdimensional multiscale basis, which is derived from a one-dimensional multiscale basis by a tensor product construction. Discretizations on sparse grids involve O(N · (log N)d-1) degrees of freedom only, where d denotes the underlying problem's dimensionality and where N is the number of grid points in one coordinate direction at the boundary. The accuracy obtained with piecewise linear basis functions, for example, is O(N-2 · (log N)d-1) with respect to the L2- and L∞- norm, if the solution has bounded second mixed derivatives. This way, the curse of dimensionality, i.e., the exponential dependence O(Nd) of conventional approaches, is overcome to some extent. For the energy norm, only O(N) degrees of freedom are needed to give an accuracy of O(N-1). That is why sparse grids are especially well-suited for problems of very high dimensionality.

The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way.

We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining.

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Acta Numerica 2004 , pp. 147 - 270
Publisher: Cambridge University Press
Print publication year: 2004

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  • Sparse grids
    • By Hans-Joachim Bungartz, IPVS, Universität Stuttgart, Universitdtsstrafie 38, D-70569 Stuttgart, Germany, Michael Griebel, Institut für Numerische Simulation, Universität Bonn, Wegelerstrafie 6, D-53113 Bonn, Germany
  • Edited by Arieh Iserles, University of Cambridge
  • Book: Acta Numerica 2004
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569975.003
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  • Sparse grids
    • By Hans-Joachim Bungartz, IPVS, Universität Stuttgart, Universitdtsstrafie 38, D-70569 Stuttgart, Germany, Michael Griebel, Institut für Numerische Simulation, Universität Bonn, Wegelerstrafie 6, D-53113 Bonn, Germany
  • Edited by Arieh Iserles, University of Cambridge
  • Book: Acta Numerica 2004
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569975.003
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Sparse grids
    • By Hans-Joachim Bungartz, IPVS, Universität Stuttgart, Universitdtsstrafie 38, D-70569 Stuttgart, Germany, Michael Griebel, Institut für Numerische Simulation, Universität Bonn, Wegelerstrafie 6, D-53113 Bonn, Germany
  • Edited by Arieh Iserles, University of Cambridge
  • Book: Acta Numerica 2004
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569975.003
Available formats
×