Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines
KAUST DepartmentNumerical Porous Media SRI Center (NumPor)
Applied Mathematics and Computational Science Program
Earth Science and Engineering Program
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AbstractWe 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.
CitationGaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines 2015 Journal of Computational and Applied Mathematics