An always successful method in univariate convex $C^2$ interpolation

Schmidt, Jochen; Heß, Walter

doi:10.1007/s002110050143

Cited by 26 publications

(10 citation statements)

References 25 publications

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“…For further programming details and numerical experiments we refer to [23]. Several experiments concerning convex Lagrange interpolation using splines on refined grids have been described in [10,14,21]. In general, the optimization procedure leads to visually improved interpolants.…”

Section: Summary and Numerical Experimentsmentioning

confidence: 99%

“…Convex interval interpolation of C 2 smoothness can also be successfully treated using splines from the space S 2 3 (∆ 2 ) of cubic splines on two-fold refined grids, while C 3 smoothness is achieved in the space S 3 4 (∆ 3 ) of quartic splines on three-fold refined grids. For a description of these splines we refer to [21] and [10], respectively.…”

Section: Convex Quartic C 2 Splines On Refined Gridsmentioning

confidence: 99%

“…For overviews we refer to [9,25], and for recent results and references concerning convexity preserving interpolation up to smoothness C 3 to the survey [18] as well as [10,14,21]. By now it is well known that nonlinear spline manifolds have to be utilized in order to achieve the preservation of convexity for all strictly convex data [13,18].…”

Section: Introductionmentioning

confidence: 99%

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Convex interval interpolation using a three-term staircase algorithm

Mulansky

Schmidt

1999

Numerische Mathematik

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Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation, the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants. In particular, we refer to methods based on polynomial splines defined on grids with additional knots. Classification (1991): 65D05, 65D07, 41A15 Mathematics Subject

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Section: Summary and Numerical Experimentsmentioning

confidence: 99%

Section: Convex Quartic C 2 Splines On Refined Gridsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convex interval interpolation using a three-term staircase algorithm

Mulansky

Schmidt

1999

Numerische Mathematik

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“…. , n, too; see [2]. From [4] results that Proposition 1 even holds true in s:(A,) if the functionals and p3,i are replaced by…”

Section: Constrained Interpolations In the Univariate Spline Spacesmentioning

confidence: 99%

Interpolation with tensor product splines subject to derivative constraints

Walther

1998

Z Angew Math Mech

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In this paper restricted interpolations of data sets (xi, yj, zi, j), i = 0,…, n, j = 0,…, m, on the rectangular grid Δ × Σ = {x0 <… < xn} × {yo <… < ym} are handled using biquadratic C1 and bicubic C2 splines on refined grids. The constraints on the first order derivatives are piecewise constant with respect to the original grid Δ × Σ. If the refinement of the grids are chosen adaptively, the existence of restricted interpolants can be assured for bounds strictly compatible with the data.

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“…For a given grid I 0 =[t 1 , ..., t n ], where a=t 1 <t 2 < } } } <t n =b, (1.1) methods for the construction of such interpolation functions are characterized by different demands with respect to the degree of smoothness and by local or global constructions, see, for example, Akima [1], Fritsch and Carlson [8], and Schmidt and Hess [19]. The authors considered in several investigations also some kind of optimality conditions generalizing the Holladay property of the classical cubic spline, cf.…”

Section: Introductionmentioning

confidence: 99%

On the Construction of Optimal Monotone Cubic Spline Interpolations

Fredenhagen¹,

Oberle²,

Opfer³

1999

Journal of Approximation Theory

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In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call the local case, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs. Academic Press

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An always successful method in univariate convex $C^2$ interpolation

Cited by 26 publications

References 25 publications

Convex interval interpolation using a three-term staircase algorithm

Convex interval interpolation using a three-term staircase algorithm

Interpolation with tensor product splines subject to derivative constraints

On the Construction of Optimal Monotone Cubic Spline Interpolations

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