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Fast Approximation of the Discrete Gauss Transform in Higher Dimensions

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Abstract

We present a novel approach for the fast approximation of the discrete Gauss transform in higher dimensions. The algorithm is based on the dual-tree technique and introduces a new Taylor series expansion. It compares favorably to existing methods especially when it comes to higher dimensions and a broad range of bandwidths. Numerical results with different datasets in up to 62 dimensions demonstrate its performance.

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Notes

  1. We choose direct computation as the only option on leaf level, since it always outperformed Taylor approximation in our numerical experiments with our choice of the parameter q=30. Furthermore, we stop the recursion as soon as one of the nodes S or T is a leaf. In case the other node is still associated to a larger subtree, it might be favorable to further descend in this particular subtree only. However, in our test cases, the sources and targets coincide anyway. Therefore, the two trees are identical and such a general situation can not appear at all.

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  1. Institute for Numerical Simulation, University of Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Michael Griebel & Daniel Wissel

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  1. Michael Griebel
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  2. Daniel Wissel
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Correspondence to Daniel Wissel.

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Griebel, M., Wissel, D. Fast Approximation of the Discrete Gauss Transform in Higher Dimensions. J Sci Comput 55, 149–172 (2013). https://doi.org/10.1007/s10915-012-9626-3

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  • DOI: https://doi.org/10.1007/s10915-012-9626-3

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