The PCHIP subdivision scheme

Aràndiga, Francesc; Donat, Rosa; Santágueda, Maria

doi:10.1016/j.amc.2015.07.071

Cited by 12 publications

(31 citation statements)

References 22 publications

(55 reference statements)

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“…In what follows, we set the notation for the remainder of the paper and briefly review those tools used in the analysis of stationary subdivision schemes (linear or not) that are relevant in our development. The interested reader is referred to [4,12] for a more complete description of the theory of linear subdivision schemes, and to [2,3,8,9,16] for the relevant theory of nonlinear subdivision schemes that can be written as a nonlinear perturbation of a convergent linear scheme. In the following we use the letter T only for linear schemes, while S shall denote a general subdivision scheme (linear or not).…”

Section: Convergence Of Stationary Subdivision Schemesmentioning

confidence: 99%

“…Then, convergence can be proven using the following result from [3] (notice the similarity with (9)).…”

Section: Convergence Of Stationary Subdivision Schemesmentioning

confidence: 99%

“…It is well known that the approximation order of a subdivision scheme after one step determine the order of approximation of the scheme, provided the scheme is stable (Theorem 2.4.10 of [17]). The order of approximation and the stability of nonlinear schemes are often studied together [2,3,9].…”

Section: Stability and Approximation Ordermentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

Donat

López-Ureña

2019

Adv Comput Math

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In this paper we define a family of nonlinear, stationary, interpolatory subdivision schemes with the capability of reproducing conic shapes including polynomials upto second order. Linear, nonstationary, subdivision schemes do also achieve this goal, but different conic sections require different refinement rules to guarantee exact reproduction. On the other hand, with our construction, exact reproduction of different conic shapes can be achieved using exactly the same nonlinear scheme. Convergence, stability, approximation and shape preservation properties of the new schemes are analyzed. In addition, the conditions to obtain C 1 limit functions are also studied.

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Section: Convergence Of Stationary Subdivision Schemesmentioning

confidence: 99%

“…Then, convergence can be proven using the following result from [3] (notice the similarity with (9)).…”

Section: Convergence Of Stationary Subdivision Schemesmentioning

confidence: 99%

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Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

Donat

López-Ureña

2019

Adv Comput Math

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“…Pchip stands for piecewise cubic hermite interpolating polynomial, which is an interpolation method in which a cubic polynomial approximation is assumed over each subinterval. Aràndiga et al (2016) describe this interpolation scheme in detail together with its advantages, mainly that it is both accurate (preserves values at the nodes) and preserves monotonicity. Pchip was selected for the present study because (i) the fitted curve passes through observed values at inflexion points unlike spline or quadratic methods, for example, and (ii) it does not require re-fitting when the period of application is extended as each subinterval is treated separately.…”

Section: Temperature Datamentioning

confidence: 99%

Historical gridded reconstruction of potential evapotranspiration for the UK

Tanguy

Prudhomme

Smith

et al. 2018

Earth Syst. Sci. Data

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Abstract. Potential evapotranspiration (PET) is a necessary input data for most hydrological models and is often needed at a daily time step. An accurate estimation of PET requires many input climate variables which are, in most cases, not available prior to the 1960s for the UK, nor indeed most parts of the world. Therefore, when applying hydrological models to earlier periods, modellers have to rely on PET estimations derived from simplified methods. Given that only monthly observed temperature data is readily available for the late 19th and early 20th century at a national scale for the UK, the objective of this work was to derive the best possible UK-wide gridded PET dataset from the limited data available.To that end, firstly, a combination of (i) seven temperature-based PET equations, (ii) four different calibration approaches and (iii) seven input temperature data were evaluated. For this evaluation, a gridded daily PET product based on the physically based Penman-Monteith equation (the CHESS PET dataset) was used, the rationale being that this provides a reliable "ground truth" PET dataset for evaluation purposes, given that no directly observed, distributed PET datasets exist. The performance of the models was also compared to a "naïve method", which is defined as the simplest possible estimation of PET in the absence of any available climate data. The "naïve method" used in this study is the CHESS PET daily long-term average (the period from 1961 to 1990 was chosen), or CHESS-PET daily climatology.The analysis revealed that the type of calibration and the input temperature dataset had only a minor effect on the accuracy of the PET estimations at catchment scale. From the seven equations tested, only the calibrated version of the McGuinness-Bordne equation was able to outperform the "naïve method" and was therefore used to derive the gridded, reconstructed dataset. The equation was calibrated using 43 catchments across Great Britain.The dataset produced is a 5 km gridded PET dataset for the period 1891 to 2015, using the Met Office 5 km monthly gridded temperature data available for that time period as input data for the PET equation. The dataset includes daily and monthly PET grids and is complemented with a suite of mapped performance metrics to help users assess the quality of the data spatially.This dataset is expected to be particularly valuable as input to hydrological models for any catchment in the UK.The data can be accessed at https://doi.org/10.5285/17b9c4f7-1c30-4b6f-b2fe-f7780159939c.Published by Copernicus Publications.

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“…Nonlinear subdivision schemes like ENO, WENO, PPH and PCHIP [1][2][3][4][5][6][7][8][9], were introduced during last several years to address Gibbs phenomenon. The arithmetic mean of second differences was replaced by their harmonic mean in a linear subdivision scheme to change it to a nonlinear scheme in [1,2].…”

Section: Introductionmentioning

confidence: 99%

A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C2 Smoothness

Aslam

2018

MCA

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Abstract:The occurrence of the Gibbs phenomenon near irregular initial data points is a widely known fact in curve generation by interpolating subdivision schemes. In this article, we propose a family of 5-point nonlinear ternary interpolating subdivision schemes. We provide the convergence analysis and prove that this family of subdivision schemes is C 2 continuous. Numerical results are presented to show that nonlinear schemes reduce the Gibbs phenomenon significantly while keeping the same order of smoothness.

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The PCHIP subdivision scheme

Cited by 12 publications

References 22 publications

Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

Nonlinear stationary subdivision schemes reproducing hyperbolic and trigonometric functions

Historical gridded reconstruction of potential evapotranspiration for the UK

A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C2 Smoothness

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