Tschebyscheff-approximation by regular splines with free knots

1976 ◽  
pp. 439-452 ◽  
Author(s):  
Helmut Werner ◽  
Henry Loeb
Keyword(s):  
Splines With Free Knots

scholarly journals Algorithms for piecewise polynomials and splines with free knots

1989 ◽  
Vol 53 (187) ◽  
pp. 235-235 ◽  
Author(s):  
G. Meinardus ◽  
G. N{ürnberger ◽  
M. Sommer ◽  
H. Strauss
Keyword(s):  
Piecewise Polynomials ◽  
Splines With Free Knots

scholarly journals Uniform approximation by generalized splines with free knots

1989 ◽  
Vol 59 (2) ◽  
pp. 150-169 ◽  
Author(s):  
G Nürnberger ◽  
L Schumaker ◽  
M Sommer ◽  
H Strauss
Keyword(s):  
Uniform Approximation ◽  
Splines With Free Knots

Constrained approximation by splines with free knots

1997 ◽  
Vol 37 (1) ◽  
pp. 105-137 ◽  
Author(s):  
Torsten Schütze ◽  
Hubert Schwetlick
Keyword(s):  
Constrained Approximation ◽  
Splines With Free Knots

Data smoothing by splines with free knots

2008 ◽  
Vol 5 (3) ◽  
pp. 324-327 ◽  
Author(s):  
N. D. Dikoussar ◽  
Cs. Török
Keyword(s):  
Data Smoothing ◽  
Splines With Free Knots

Approximation to Data by Splines with Free Knots

1978 ◽  
Vol 15 (2) ◽  
pp. 328-343 ◽  
Author(s):  
David L. B. Jupp
Keyword(s):  
Splines With Free Knots

scholarly journals Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1099
Author(s):  
Lakshman S. Thakur ◽  
Mikhail A. Bragin
Keyword(s):  
Linear Programming ◽  
Interpolation Problem ◽  
Compact Set ◽  
Data Interpolation ◽  
Fine Mesh ◽  
Quadratic Splines ◽  
Splines With Free Knots ◽  
Data Points ◽  
Derivatives Of ◽  
High Computational Efficiency

The problem of obtaining an optimal spline with free knots is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished, interpolation by splines of higher orders is far more challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined is discretized and continuous derivatives are replaced by their discrete counterparts. The l∞-norm used for maximum rth order curvature (a derivative of order r) is then linearized, and the problem to obtain a near-optimal spline becomes a linear programming (LP) problem, which is solved in polynomial time by using LP methods, e.g., by using the Simplex method implemented in modern software such as CPLEX. It is shown that, as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriately fine mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and the resulting spline provides a good approximation to the sought-for optimal spline.

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