Convex spline interpolants with minimal curvature

Burmeister, W.; He:Gb, W.; Schmidt, Jochen

doi:10.1007/bf02260507

Cited by 41 publications

(34 citation statements)

References 8 publications

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“…There the blocks are (1, 1), (1,4) or (4,1). In view of this structure gradient or Newton type methods should be used in solving (1.5) effectively.…”

Section: Numerical Aspects and Testsmentioning

confidence: 98%

“…This can be done by applying a dualization concept introduced by Burmeister/HeB/Schmidt [1] and Dietze/Schmidt [3] and used in further papers, e.g. in [11,12,13].…”

Section: Program Pamentioning

confidence: 99%

“…The corresponding grid on the interval [0, 1 ] shall be denoted by 3. For defining a smoothing cubic C 2-spline s, here the functional 1 …”

Section: Introductionmentioning

confidence: 99%

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A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Schmidt

Scholz²

1990

Numer. Math.

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Summary. In this paper the problem of smoothing a given data set by cubic C 2-splines is discussed. The spline may required to be convex in some parts of the domain and concave in other parts. Application of splines has the advantage that the smoothing problem is easily discretized. Moreover, the special structure of the arising finite dimensional convex program allows a dualization such that the resulting concave dual program is unconstrained. Therefore the latter program is treated numerically much more easier than the original program. Further, the validity of a return-formula is of importance by which a minimizer of the original program is obtained from a maximizer of the dual program.The theoretical background of this general approach is discussed and, above all, the details for applying the strategy to the present smoothing problem are elaborated. Also some numerical tests are presented.

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“…There the blocks are (1, 1), (1,4) or (4,1). In view of this structure gradient or Newton type methods should be used in solving (1.5) effectively.…”

Section: Numerical Aspects and Testsmentioning

confidence: 98%

“…This can be done by applying a dualization concept introduced by Burmeister/HeB/Schmidt [1] and Dietze/Schmidt [3] and used in further papers, e.g. in [11,12,13].…”

Section: Program Pamentioning

confidence: 99%

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A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Schmidt

Scholz²

1990

Numer. Math.

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“…The fact that we have a whole interval [m¡, m¡] of admissible parameters m¡ can be used to look for " visually pleasant" convex spline interpolants (see [2]). …”

Section: Holger Mettkementioning

confidence: 99%

Convex Interpolation by Splines of Arbitrary Degree

Mettke¹

1986

Mathematics of Computation

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Abstract. An algorithm is described for computing an interpolation spline of arbitrary but fixed degree which preserves the convexity of the given data set. Necessary and sufficient conditions for the solvability of the problem, some special cases and error estimations are given.1. Introduction. In some problems arising from science or engineering the solution of an interpolation problem with constraints is required. For example, if a convex data set is given then the interpolant should also be convex. Using standard techniques like polynomial or cubic spline interpolation the resulting interpolant is not convex in general (see [5]).In recent years convex interpolation by splines has been investigated in several publications (see [3]-[17]). Various possibilities were proposed to assure the convexity of an interpolation spline. They reach from additional conditions on the data set to additional knots of the interpolation spline.In this paper we consider convex interpolation by splines of arbitrary degree k > 3 with smoothness q, where 1 < q < [(k -l)/2]. As is known, the solvability of such an interpolation problem is equivalent to that of a special system of linear inequalities. Necessary and sufficient conditions are derived such that this system has a solution, and an algorithm is described to find all solutions of the system of inequalities. Furthermore, a sufficient condition is given which always assures the solvability of the system and which is easy to test. In this case, a solution can be found even with a reduced algorithm. The sufficient condition just mentioned is interpreted in different ways. Specifically, for a strictly convex data set it is possible to give a lower bound for the degree k such that the interpolation problem is solvable. For the developed method no additional spline knots are needed. In the last section, the important question of error estimation is investigated. Under certain assumptions, error bounds are derived for continuous and continuously differentiable underlying functions. The assumptions are related to the solvability of the convex interpolation problem, and in one case to the ratio of contiguous step sizes of the grid. In the differentiable case, the order of approximation is better than in the continuous one.

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“…In [1,2,8] constrained convex minimization problems in convex spline interpolations and in tridiagonal complementarity problems have been transformed into unconstrained convex minimization problems by Fenchel-Rockafellar dualization. The applicability of the decomposition-dualization technique considered in [1,2,8] is restricted to those constrained quadratic minimization problems where the Hesse matrix of the convex objective functional is tridiagonal. The aim of our investigations is to show that also more general constrained convex minimization problems can be effectively solved by decomposition and Fenchel-Rockafellar dualization, for instance problems in quadratic programming where the Hesse matrix of the objective functional is a H-matrix, e.g.…”

Section: Introductionmentioning

confidence: 99%

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

Krätzschmar

1989

Numer. Math.

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Summary.In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated by H-matrices. Numerical tests for solving obstacle problems in lR x discretized by using piecewise quadratic finite elements and in IR 2 by using the five-point difference approximation are presented.

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Convex spline interpolants with minimal curvature

Cited by 41 publications

References 8 publications

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

A dual algorithm for convex-concave data smoothing by cubicC 2-splines

Convex Interpolation by Splines of Arbitrary Degree

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

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