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

Handle URI:
http://hdl.handle.net/10754/581502
Title:
Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines
Authors:
Barton, Michael ( 0000-0002-1843-251X ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
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.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Applied Mathematics and Computational Science Program; Earth Science and Engineering Program
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
Issue Date:
24-Oct-2015
DOI:
10.1016/j.cam.2015.09.036
Type:
Article
ISSN:
03770427
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0377042715004896
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Earth Science and Engineering Program

Full metadata record

DC FieldValue Language
dc.contributor.authorBarton, Michaelen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2015-11-01T11:11:32Zen
dc.date.available2015-11-01T11:11:32Zen
dc.date.issued2015-10-24en
dc.identifier.citationGaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines 2015 Journal of Computational and Applied Mathematicsen
dc.identifier.issn03770427en
dc.identifier.doi10.1016/j.cam.2015.09.036en
dc.identifier.urihttp://hdl.handle.net/10754/581502en
dc.description.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.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0377042715004896en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics, 24 October 2015. DOI: 10.1016/j.cam.2015.09.036en
dc.subjectGaussian quadratureen
dc.subjectB-splinesen
dc.subjectWell-constrained polynomial systemen
dc.subjectPolynomial homotopy continuationen
dc.titleGaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splinesen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentEarth Science and Engineering Programen
dc.identifier.journalJournal of Computational and Applied Mathematicsen
dc.eprint.versionPost-printen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorBarton, Michaelen
kaust.authorCalo, Victor M.en
kaust.authorCalo, Victor M.en
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