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Fast Sparse Grid Operations Using the Unidirectional Principle: A Generalized and Unified Framework

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Sparse Grids and Applications - Munich 2018

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 144))

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Abstract

Sparse grids enable accurate function approximation in higher-dimensional settings with a moderate number of grid points. In this paper, we introduce generalized versions of various efficient sparse grid algorithms in a unified notation: For local basis functions, transformations between function values at grid points and interpolation coefficients can be efficiently realized using the well-known unidirectional principle (Balder and Zenger, SIAM Journal on Scientific Computing, 17(3):631–646, May 1996; Bungartz, Finite Elements of Higher Order on Sparse Grids, Universität München, Aachen, November 1998). For general basis functions without spatial adaptivity, Avila and Carrington show in (The Journal of Chemical Physics, 143(21):214108, 2015) and follow-up work that the unidirectional principle can instead be applied to an equivalent incremental basis, and we show that a second application of the unidirectional principle can be used to compute interpolation coefficients for the original, non-incremental basis. Our Open Source implementation for the latter case is able to compute interpolation coefficients for up to 20 million sparse grid points in one second on a single CPU core. We also discuss the efficient application of linear operators occurring in derivative evaluations of interpolants and in the solution of partial differential equations on sparse grids with finite elements, e.g., using the UpDown scheme (Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung, PhD thesis, Technische Universität München, 1992; Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Verlag Dr. Hut, München, February 2010; Zeiser, Journal of Scientific Computing, 47(3):328–346, 2011; Bungartz et al., Journal of Computational and Applied Mathematics, 236(15):3741–3750, 2012).

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Notes

  1. 1.

    Proof: In a decomposition \(\tilde {M}^{(1)} = LU\), both L and U need to be invertible. Hence, their diagonal entries are nonzero and thus, \(\tilde {M}^{(1)}_{11}\) is nonzero, a contradiction.

  2. 2.

    https://github.com/dholzmueller/fast_sparse_interpolation.

  3. 3.

    https://doi.org/10.18419/darus-1779.

  4. 4.

    Since the runtime of the algorithm does not depend on the particular choice of global basis functions, we simply used monomials for our runtime measurements.

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Acknowledgements

We thank Tucker Carrington for many helpful discussions on this topic. Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2075 - 390740016. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting David Holzmüller.

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Authors and Affiliations

  1. Institute for Stochastics and Applications, University of Stuttgart, Stuttgart, Germany

    David Holzmüller

  2. Institute for Parallel and Distributed Systems, University of Stuttgart, Stuttgart, Germany

    Dirk Pflüger

Authors
  1. David Holzmüller
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  2. Dirk Pflüger
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Correspondence to Dirk Pflüger .

Editor information

Editors and Affiliations

  1. Institut für Informatik, TU München, Garching, Germany

    Hans-Joachim Bungartz

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

    Jochen Garcke

  3. Institut für Parallele und Verteilte Systeme (IPVS), Universität Stuttgart, Stuttgart, Germany

    Dirk Pflüger

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Holzmüller, D., Pflüger, D. (2021). Fast Sparse Grid Operations Using the Unidirectional Principle: A Generalized and Unified Framework. In: Bungartz, HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. Lecture Notes in Computational Science and Engineering, vol 144. Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_4

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