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Convexity and Solvability for Compactly Supported Radial Basis Functions with Different Shapes

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Abstract

It is known that interpolation with radial basis functions of the same shape can guarantee a nonsingular interpolation matrix, whereas little was known when one uses various shapes. In this paper, we prove that functions from a class of compactly supported radial basis functions are convex on a certain region; based on this local convexity and other local geometrical properties of the interpolation points, we construct a sufficient condition which guarantees diagonally dominant interpolation matrices for radial basis functions interpolation with different shapes. The proof is constructive and can be used to design algorithms directly. Numerical examples show that the scheme has a low accuracy but can be implemented efficiently. It can be used for inaccurate models where efficiency is more desirable. Large scale 3D implicit surface reconstruction problems are used to demonstrate the utility and reasonable results can be obtained efficiently.

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Notes

  1. http://graphics.stanford.edu/data/3Dscanrep/.

  2. http://www.cc.gatech.edu/projects/large_models/ply.html.

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Acknowledgments

We thank the referees for valuable advice and suggestion on presenting the results in a more illustrative way, in particular, for one referee who pointed out the fact in Lemma 3, which simplifies the proof in Theorem 3.

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Authors and Affiliations

  1. Oxford Center for Collaborative and Applied Mathematics and Numerical Analysis Group, Mathematical Institute, The University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Shengxin Zhu

  2. Numerical Analysis Group, Mathematical Institute, The University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Andrew J. Wathen

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  1. Shengxin Zhu
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  2. Andrew J. Wathen
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Correspondence to Shengxin Zhu.

Additional information

This research is supported by Award no KUK-C1-013-04, made by King Abdullah University of Science of Technology.

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Zhu, S., Wathen, A.J. Convexity and Solvability for Compactly Supported Radial Basis Functions with Different Shapes. J Sci Comput 63, 862–884 (2015). https://doi.org/10.1007/s10915-014-9919-9

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  • DOI: https://doi.org/10.1007/s10915-014-9919-9

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