This book describes in detail the key algorithms needed for computing with spline functions and illustrates their use in solving several basic problems in numerical analysis, including function approximation, numerical quadrature, data fitting, and the numerical solution of PDEs. The focus is on computational methods for bivariate splines on triangulations in the plane and on the sphere, although both univariate and tensor-product splines are also discussed. The book contains numerous examples and figures to illustrate the methods and their performance. All of the algorithms in the book have been coded in a separate MATLAB package available for license. The package can be used to run all of the examples in the book and also provides readers with the essential tools needed to create software for their own applications. In addition to the included bibliography, a list of over 100 pages of additional references can be found on the book's website. Audience: This book is designed for mathematicians, engineers, scientists, and anyone else wanting to make use of spline functions for numerical computation.

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Spline Functions: Computational Methods

2D shape optimization under proximity constraints by CFD and response surface methodology

http://directory.umm.ac.id/Data%20Elmu/jurnal/J-a/Journal%20of%20Computational%20And%20Applied%20Mathematics/Vol103.Issue2/2707.pdf

Range restricted interpolation using Gregory's rational cubic splines

A C 1 -rational spline in range restricted interpolation of scattered data

This paper discusses the positivity preserving interpolation for positive surfaces data by extending the C1 rational cubic spline interpolant of Karim and Kong to the bivariate cases. The partially blended rational bicubic spline has 12 parameters in the descriptions where 8 of them are free parameters. The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail. Based on Root Mean Square Error (RMSE), our partially blended rational bicubic spline is on a par with the established methods.

/pdf/positivity-preserving-interpolation-using-rational-bicubic-71elt5d9m5.pdf

Positivity Preserving Interpolation Using Rational Bicubic Spline

This paper is concerned with the interpolation of gridded data subject to lower and upper bounds on the first order derivatives. We apply Gregory’s rational cubic C 1 splines and corresponding tensor products. The occurring rationality parameters can always be determined such that the considered interpolation problems turn out to be solvable.

Gregory’s Rational Cubic Splines in Interpolation Subject to Derivative Obstacles

This paper is concerned with numerical methods in range restricted histopolation. The proposal is to apply splines on refined grids. The ratios of the added split points are considered to be parameters. In this way, by choosing suitable spline classes, range restricted histosplines can always be constructed if the restrictions are compatible with the given histogram. We offer an algorithm for solving the bivariate problem on a rectangular grid which utilizes univariate results as well as tensor product techniques.

Marion Bastian-Walther

Papers

Range restricted interpolation using Gregory's rational cubic splines

Gregory’s Rational Cubic Splines in Interpolation Subject to Derivative Obstacles

Algorithm for constructing range restricted histosplines