A linear approach to shape preserving spline approximation

    Research output: Book/ReportReportProfessional

    54 Downloads (Pure)

    Abstract

    This report deals with approximation of a given scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data. Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local {\em linear sufficient\/} conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the minimization problem is also linear, as the problem can be written as a linear programming problem then. A special linear approach based on constrained least squares is presented, which reduces the complexity of the problem in case of large data sets in contrast with the $\ell_\infty$ and the $\ell_1$-norms. An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversiteit Twente
    Number of pages21
    ISBN (Print)0169-2690
    Publication statusPublished - 1998

    Publication series

    NameMemorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450
    PublisherUniversiteit Twente
    No.1450

    Keywords

    • METIS-141109
    • IR-30469
    • MSC-41A15
    • MSC-65D07
    • monotonicity
    • linearisation of shape constraints
    • linear constraints
    • EWI-3270
    • MSC-65D15
    • Spline approximation
    • Convexity
    • MSC-41A29

    Cite this

    Kuijt, F., & van Damme, R. M. J. (1998). A linear approach to shape preserving spline approximation. (Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450; No. 1450). Enschede: Universiteit Twente.
    Kuijt, F. ; van Damme, Rudolf M.J. / A linear approach to shape preserving spline approximation. Enschede : Universiteit Twente, 1998. 21 p. (Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450; 1450).
    @book{6369242480434fe0934dddd626398683,
    title = "A linear approach to shape preserving spline approximation",
    abstract = "This report deals with approximation of a given scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data. Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local {\em linear sufficient\/} conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the minimization problem is also linear, as the problem can be written as a linear programming problem then. A special linear approach based on constrained least squares is presented, which reduces the complexity of the problem in case of large data sets in contrast with the $\ell_\infty$ and the $\ell_1$-norms. An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.",
    keywords = "METIS-141109, IR-30469, MSC-41A15, MSC-65D07, monotonicity, linearisation of shape constraints, linear constraints, EWI-3270, MSC-65D15, Spline approximation, Convexity, MSC-41A29",
    author = "F. Kuijt and {van Damme}, {Rudolf M.J.}",
    note = "Imported from MEMORANDA",
    year = "1998",
    language = "Undefined",
    isbn = "0169-2690",
    series = "Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450",
    publisher = "Universiteit Twente",
    number = "1450",

    }

    Kuijt, F & van Damme, RMJ 1998, A linear approach to shape preserving spline approximation. Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450, no. 1450, Universiteit Twente, Enschede.

    A linear approach to shape preserving spline approximation. / Kuijt, F.; van Damme, Rudolf M.J.

    Enschede : Universiteit Twente, 1998. 21 p. (Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450; No. 1450).

    Research output: Book/ReportReportProfessional

    TY - BOOK

    T1 - A linear approach to shape preserving spline approximation

    AU - Kuijt, F.

    AU - van Damme, Rudolf M.J.

    N1 - Imported from MEMORANDA

    PY - 1998

    Y1 - 1998

    N2 - This report deals with approximation of a given scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data. Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local {\em linear sufficient\/} conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the minimization problem is also linear, as the problem can be written as a linear programming problem then. A special linear approach based on constrained least squares is presented, which reduces the complexity of the problem in case of large data sets in contrast with the $\ell_\infty$ and the $\ell_1$-norms. An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.

    AB - This report deals with approximation of a given scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data. Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local {\em linear sufficient\/} conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the minimization problem is also linear, as the problem can be written as a linear programming problem then. A special linear approach based on constrained least squares is presented, which reduces the complexity of the problem in case of large data sets in contrast with the $\ell_\infty$ and the $\ell_1$-norms. An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.

    KW - METIS-141109

    KW - IR-30469

    KW - MSC-41A15

    KW - MSC-65D07

    KW - monotonicity

    KW - linearisation of shape constraints

    KW - linear constraints

    KW - EWI-3270

    KW - MSC-65D15

    KW - Spline approximation

    KW - Convexity

    KW - MSC-41A29

    M3 - Report

    SN - 0169-2690

    T3 - Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450

    BT - A linear approach to shape preserving spline approximation

    PB - Universiteit Twente

    CY - Enschede

    ER -

    Kuijt F, van Damme RMJ. A linear approach to shape preserving spline approximation. Enschede: Universiteit Twente, 1998. 21 p. (Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450; 1450).