Positivity of cubic polynomials on intervals and positive spline interpolation

Schmidt, Jochen; Heb, Walter

doi:10.1007/bf01934097

Cited by 138 publications

(60 citation statements)

References 7 publications

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“…It is known (cf. [10,17]) that a necessary and sufficient condition for the quadratic polynomial d 2 to be nonnegative for 0 ≤ ξ ≤ 1 is that c 13 , c 14 ≥ 0 and c 10 ≥ − √ c 13 c 14 . In view of (3.10) and (3.…”

Section: Nonnegativity Of a Cubic Ct-macro-element Patchmentioning

confidence: 99%

Nonnegativity preserving macro-element interpolation of scattered data

Schumaker

Speleers

2010

Computer Aided Geometric Design

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Abstract. Nonnegative bivariate interpolants to scattered data are constructed using some C 1 macro-element spline spaces. The methods are local, and rely on adjusting gradients at the data points to insure nonnegativity of the spline when the original data is nonnegative. More general range-restricted interpolation is also considered.Keywords: spline interpolation, shape preservation, nonnegative surfaces §1. IntroductionInterpolating scattered data is a common problem in a wide range of application areas. It is often required that the interpolant preserves known shape properties inherent to the data. Common shape properties are convexity, monotonicity and nonnegativity. In this paper we consider the nonnegativity-preserving interpolation problem.Problem 1.1. Given a set of scattered data points, find a function s defined on Ω such that s(x i , y i ) = f i for i = 1, . . . , n, and s(x, y) ≥ 0 for all (x, y) ∈ Ω.A standard approach for solving this problem is to work with either polynomial splines [2,5,6,9,16] or rational splines [1,8,11,12,15]. These splines are defined on a triangulation with its vertices at the data points. Problem 1.1 is often solved as an optimization problem in order to find a visually pleasing, nonnegative spline that interpolates the data. Such methods are mostly global in nature, i.e., the spline coefficients globally depend on all of the data. Our aim here is to focus on the construction of local and easy to use methods based on standard C 1 macro-element spline spaces.The paper is organized as follows. In Sects. 2 and 3 we describe nonnegativitypreserving interpolation methods using the C 1 Powell-Sabin and Clough-Tocher macro-elements, respectively. We discuss the approximation order of the methods in Sect. 4, and in the following section we explain how to treat certain rangerestricted interpolation problems. Sect. 6 is devoted to some numerical examples. We conclude with some remarks.

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Section: Nonnegativity Of a Cubic Ct-macro-element Patchmentioning

confidence: 99%

Nonnegativity preserving macro-element interpolation of scattered data

Schumaker

Speleers

2010

Computer Aided Geometric Design

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“…Typically, the reconstruction profiles are not continuous on the edges of the grid cells. Sufficient conditions for positive spline interpolation in form of inequalities on the interpolation's coefficients have been derived in Schmidt and Heß (1988).…”

Section: Time Precipitation Ratementioning

confidence: 99%

A conservative reconstruction scheme for the interpolation of extensive quantities in the Lagrangian particle dispersion model FLEXPART

2018

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Abstract. Lagrangian particle dispersion models require interpolation of all meteorological input variables to the position in space and time of computational particles. The widely used model FLEXPART uses linear interpolation for this purpose, implying that the discrete input fields contain point values. As this is not the case for precipitation (and other fluxes) which represent cell averages or integrals, a preprocessing scheme is applied which ensures the conservation of the integral quantity with the linear interpolation in FLEXPART, at least for the temporal dimension. However, this mass conservation is not ensured per grid cell, and the scheme thus has undesirable properties such as temporal smoothing of the precipitation rates. Therefore, a new reconstruction algorithm was developed, in two variants. It introduces additional supporting grid points in each time interval and is to be used with a piecewise linear interpolation to reconstruct the precipitation time series in FLEXPART. It fulfils the desired requirements by preserving the integral precipitation in each time interval, guaranteeing continuity at interval boundaries, and maintaining non-negativity. The function values of the reconstruction algorithm at the sub-grid and boundary grid points constitute the degrees of freedom, which can be prescribed in various ways. With the requirements mentioned it was possible to derive a suitable piecewise linear reconstruction. To improve the monotonicity behaviour, two versions of a filter were also developed that form a part of the final algorithm. Currently, the algorithm is meant primarily for the temporal dimension. It was shown to significantly improve the reconstruction of hourly precipitation time series from 3-hourly input data. Preliminary considerations for the extension to additional dimensions are also included as well as suggestions for a range of possible applications beyond the case of precipitation in a Lagrangian particle model. MotivationIn numerical models, extensive variables (those being proportional to the volume or area that they represent, e.g. mass and energy) are usually discretised as grid-cell integral values so that conservation properties can be fulfilled. A typical example is the precipitation flux in a meteorological forecasting model. Usually, one is interested in the precipitation at the surface, and thus the quantity of interest is a two-dimensional horizontal integral (coordinates x and y) over each grid cell. Furthermore, it is accumulated over time t during the model run and written out at distinct intervals together with other variables, such as wind and temperature. After the trivial postprocessing step of de-accumulation, each value thus represents an integral in x, y, t space -the total amount of precipitation which fell on a discrete grid cell during a discrete time interval.In Lagrangian particle dispersion models (LPDMs) (Lin et al., 2013), it is necessary to interpolate the field variables obtained from a meteorological model to particle positions in (three-dimen...

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“…Indeed for some data set, our scheme is better. (ii) Our scheme not involving any knots insertion as appear in the works of Lahtinen [15], Lam [16], Schumaker [21], Fristch and Carlson [8], Schmidt and Hess [20] and Butt and Brodlie [5]. (iii) Our scheme has two degree freedom meanwhile there are no degree freedom in the works of Sarfraz [18], Delbourgo and Gregory [6,7] and Gregory [9].…”

Section: Introductionmentioning

confidence: 99%

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity

Karim

2017

MATEC Web Conf.

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Abstract. Monotonicity preserving interpolation is very important in many sciences and engineering based problems. This paper discuss the monotonicity preserving interpolation for monotone data set by using 2 C rational cubic spline interpolant (cubic/quadratic) with three parameters. The data dependent sufficient conditions for the monotonicity are derived with two degree freedom. Numerical results suggests that the proposed 2 C rational cubic spline preserves the monotonicity of the data and outperform the performance of the other rational cubic spline schemes in term of visually pleasing.

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Positivity of cubic polynomials on intervals and positive spline interpolation

Cited by 138 publications

References 7 publications

Nonnegativity preserving macro-element interpolation of scattered data

Nonnegativity preserving macro-element interpolation of scattered data

A conservative reconstruction scheme for the interpolation of extensive quantities in the Lagrangian particle dispersion model FLEXPART

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity

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Positivity of cubic polynomials on intervals and positive spline interpolation

Cited by 138 publications

References 7 publications

Nonnegativity preserving macro-element interpolation of scattered data

Nonnegativity preserving macro-element interpolation of scattered data

A conservative reconstruction scheme for the interpolation of extensive quantities in the Lagrangian particle dispersion model FLEXPART

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C2 Continuity

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Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity