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    Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines

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    1-s2.0-S0377042715004896-main.pdf
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    Description:
    Accepted Manuscript
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    Type
    Article
    Authors
    Barton, Michael cc
    Calo, Victor M. cc
    KAUST Department
    Numerical Porous Media SRI Center (NumPor)
    Applied Mathematics and Computational Science Program
    Earth Science and Engineering Program
    Date
    2015-10-24
    Permanent link to this record
    http://hdl.handle.net/10754/581502
    
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    Abstract
    We introduce a new concept for generating optimal quadrature rules for splines. To generate an optimal quadrature rule in a given (target) spline space, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, while preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by changing the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with C1C1 cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C2C2 cubic spline spaces where the rule was only conjectured to date. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.
    Citation
    Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines 2015 Journal of Computational and Applied Mathematics
    Publisher
    Elsevier BV
    Journal
    Journal of Computational and Applied Mathematics
    ISSN
    03770427
    DOI
    10.1016/j.cam.2015.09.036
    Additional Links
    http://linkinghub.elsevier.com/retrieve/pii/S0377042715004896
    ae974a485f413a2113503eed53cd6c53
    10.1016/j.cam.2015.09.036
    Scopus Count
    Collections
    Articles; Applied Mathematics and Computational Science Program; Earth Science and Engineering Program

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