# A linear approach to shape preserving spline approximation

Research output: Book/ReportReportProfessional

### 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 language Undefined Enschede Universiteit Twente 21 0169-2690 Published - 1998

### Publication series

Name Memorandum / Faculty of Mathematical Sciences, University of Twente ; no. 1450 Universiteit Twente 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.
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).