On certain inequalities associated to curvature properties of the nonlinear PPH reconstruction operator
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Abstract
In this paper we study the curvature term for the Lagrange and PPH (Amat et al. in Found. Comput. Math. 6:193–225, 2006 and Ortiz and Trillo in Preprint, arXiv:1811.10566, 2018) reconstruction operators in uniform and nonuniform meshes. We also make a comparison between both curvature terms in order to obtain an inequality which clearly shows that the PPH reconstruction presents a lower curvature term. Presenting a low curvature term is crucial in some applications, such as smoothing splines.
Keywords
Interpolation Approximation Curvature Reconstruction operator Nonlinearity Nonuniform gridsMSC
41A05 41A10 65D171 Introduction
Reconstruction and subdivision operators have been studied, analyzed and implemented in computer aided geometric design, giving rise to interesting applications in different fields of science. Subdivision schemes provide easy and fast algorithms for the generation of curves and surfaces from a coarse initial set of control points. They are closely related to reconstruction operators.
PPH reconstruction was firstly defined in [1], although as subdivision scheme was already introduced in [4]. Later the PPH reconstruction operator was extended to allow for the use of nonuniform meshes [6], which is needed to link this reconstruction with general splines. This reconstruction is inherently a nonlinear interpolatory technique that has some remarkable characteristics. We mention those that are attractive for our purposes. In particular, a fixed centered stencil is used to build each polynomial piece, fourthorder accuracy is reached in smooth convex regions, reduction to secondorder occurs at the vicinity of singularities but the approximation order is not completely lost as it happens in the linear case, and Gibb’s effect is avoided. Also we especially remark two more properties which are going to be crucial for this reconstruction: convexity preservation when dealing with initial discrete set of convex data [6] and a low curvature term. The latter property about the curvature is part of what is going to be proven in the following sections. More precisely, we study the curvature term of the functional (1) for the Lagrange and PPH reconstructions, in the uniform and nonuniform cases. Then, due to these suitable properties, we think that connecting the PPH reconstruction with smoothing splines could result in very interesting applications.
The paper is organized as follows: In Sect. 2, we analyze the curvature term for the Lagrange and PPH reconstruction on uniform meshes. In Sect. 3, we study the case of nonuniform meshes. Finally, in Sect. 4, we present some conclusions and future perspectives.
2 Study of the curvature term in uniform meshes
From now on, we will use the following definition of the local curvature term.
Definition 1
2.1 Curvature term for the Lagrange reconstruction
2.2 Curvature term for the PPH reconstruction
Let now \(p_{H}(x)\) be the PPH polynomial (see [1]). This fourthorder reconstruction based also on the data \((x_{j+s},f_{j+s})\), \(s=1,0,1,2\), basically proceeds as follows: firstly, a modification of either \(f_{j1}\) or \(f_{j+2}\) is carried out in order to avoid the bad influence of a potential singularity at \([x_{j1},x_{j}]\) or \([x_{j+1},x_{j+2}]\), respectively; secondly, a thirdorder Lagrange interpolation is applied to the modified data. Then, due to this intrinsically nonlinear nature, we need to consider two different cases to carry out the curvature study for \(p_{H}(x)\). This is done in the following theorem.
Theorem 1
Proof
Depending on the absolute values of the secondorder divided differences \(D_{j}\) and \(D_{j+1}\) in (5), we analyze the following two cases:
Case 1.\(D_{j} \leq D_{j+1}\), i.e., a potential singularity lies at \([x_{j+1}, x_{j+2}]\).
Depending on the sign of the product \(D_{j} D_{j+1}\), the parameter \(B_{H}\) takes a different expression, and therefore the same happens for the curvature term \(C_{H}\) defined by \(C_{H}:=C(p_{H})\), according to expression (6). We consider now the following new two cases:
Case 1.1. \(D_{j} D_{j+1} > 0\).
We study now the other case.
Case 1.2. \(D_{j} D_{j+1} \leq 0\).
Again we see that also in this case the curvature term \(C_{H}\) for the PPH reconstruction is lower than the corresponding curvature \(C_{L}\) for the Lagrange polynomial.
Case 2.1. \(D_{j} D_{j+1} > 0\).
We have just seen that for data in uniform grids, the curvature term in equation (1) associated to PPH reconstruction operator remains below the value of the curvature associated to Lagrange operator.
3 Study of the curvature term in nonuniform meshes
3.1 Curvature term for the Lagrange reconstruction in nonuniform meshes
3.2 Curvature term for the PPH reconstruction in nonuniform meshes
The PPH reconstruction in nonuniform meshes is defined in the interval \([x_{j}, x_{j+1}]\) by using the data \(f_{j1}\), \(f_{j}\), \(f_{j+1}\), \(f_{j+2}\) at the abscissas \(x_{j1}\), \(x_{j}\), \(x_{j+1}\), \(x_{j+2}\) in the following way: depending on the relative size of \(D_{j}\) an \(D_{j+1}\), we substitute either \({f}_{j1}\) for \(\widetilde{f}_{j1}\) or \({f}_{j+2}\) for \(\widetilde{f}_{j+2}\). After this replacement, Lagrange reconstruction is applied to the new set of data. We remark that the initial substitution is made in order to adapt to the presence of potential singularities and, at the same time, maintain the fourthorder accuracy of Lagrange reconstruction in smooth convex areas.
Theorem 2
 (1.1)If\(D_{j}\leq D_{j+1} \ \& \ D_{j} D_{j+1}>0\),\(C_{L}C_{H}\geq 0\).$$\begin{aligned} C_{L}C_{H} =& \frac{12 h_{j+1}^{3} w_{j}w_{j+1}^{2}}{(2h_{j}+h_{j+1})^{2}} \biggl( \frac{w _{j} D_{j+1} + w_{j+1} D_{j}+ D_{j}}{(w_{j} D_{j+1} + w_{j+1} D_{j})^{2}} \biggr) (D _{j+1}D_{j})^{3} \\ &{}+ 4 h_{j+1} \bigl(M_{j}^{2} \widetilde{V}_{j}^{2}\bigr). \end{aligned}$$
 (1.2)If\(D_{j}\leq D_{j+1} \ \& \ D_{j} D_{j+1}\leq 0\),\(C_{L}C_{H}\geq 0\), under one of these natural conditions:$$\begin{aligned} C_{L}C_{H}=\frac{8 M_{j} h_{j+1}}{(2h_{j}+h_{j+1})^{2}}\bigl(2 \bigl(h_{j}^{2}+h _{j} h_{j+1} + h_{j+1}^{2}\bigr)M_{j}3h_{j+1}^{2} D_{j}\bigr), \end{aligned}$$
 (1.2.1)
If\(M_{j}\)and\(D_{j}\)have different sign.
 (1.2.2)If \(M_{j}\) and \(D_{j}\) have the same sign and$$ \frac{M_{j}}{D_{j}}> \frac{3 h_{j+1}^{2}}{2(h_{j}^{2}+h_{j}h_{j+1}+h _{j+1}^{2})}. $$
 (1.2.1)
 (2.1)If\(D_{j}> D_{j+1} \ \& \ D_{j} D_{j+1}>0\),\(C_{L}C_{H}\geq 0\).$$\begin{aligned} C_{L}C_{H} =&\frac{12 h_{j+1}^{3} w_{j}^{2}w_{j+1}}{(h_{j+1}+2 h _{j+2})^{2}}\biggl(\frac{w_{j} D_{j+1} + w_{j+1} D_{j}+ D_{j+1}}{(w_{j} D _{j+1} + w_{j+1} D_{j})^{2}} \biggr) (D_{j}D_{j+1})^{3} \\ &{}+ 4 h_{j+1} \bigl(M_{j}^{2} \widetilde{V}_{j}^{2}\bigr), \end{aligned}$$
 (2.2)If\(D_{j}> D_{j+1} \ \& \ D_{j} D_{j+1}\leq 0\),\(C_{L}C_{H}\geq 0\), under one of these natural conditions:$$\begin{aligned} C_{L}C_{H}=\frac{8 M_{j} h_{j+1}}{(h_{j+1}+2 h_{j+2})^{2}}\bigl(2 \bigl(h_{j+1} ^{2}+h_{j+1} h_{j+2} + h_{j+2}^{2}\bigr)M_{j}3h_{j+1}^{2} D_{j+1}\bigr), \end{aligned}$$
 (2.2.1)
If\(M_{j}\)and\(D_{j+1}\)have different sign.
 (2.2.2)If \(M_{j}\) and \(D_{j+1}\) have the same sign and$$ \frac{M_{j}}{D_{j+1}}> \frac{3 h_{j+1}^{2}}{2(h_{j+1}^{2}+h_{j+1}h _{j+2}+h_{j+2}^{2})}. $$
 (2.2.1)
Proof
We need to consider two main cases.
On the one hand, as the sign of \(D_{j}\) equals to the sign of \(D_{j+1}\), we get \(M_{j}^{2} \geq \widetilde{V}_{j}^{2}\).
So \(C_{L}C_{H}\) will be positive if \(M_{j} \) and \(2(h_{j}^{2}+h _{j} h_{j+1} + h_{j+1}^{2})M_{j}3h_{j+1}^{2} D_{j}\) have the same sign. This happens in the following cases:
Case 1.2.1. \(M_{j}\) and \(D_{j}\) (the lower divided difference in absolute value) have different sign.
Case 1.2.2. \(M_{j}\) and \(D_{j}\) have the same sign and \(\frac{M_{j}}{D _{j}}> \frac{3 h_{j+1}^{2}}{2(h_{j}^{2}+h_{j}h_{j+1}+h_{j+1}^{2})}\).

Case 1.2 will only take place around inflection points on the underlying function. Therefore, if we work with data corresponding to strictly convex or concave functions this case will never happen.

Case \(1.2.2\) will not occur around discontinuities except for extremely nonuniform grids where \(w_{j} \approx 1\), since \(M_{j}\) and \(D_{j}\) have the same sign if and only if \(\frac{D_{j+1}}{D_{j}}<\frac{w _{j}}{w_{j+1}}\).

Under the assumption that for the given data condition in Case \(1.2.1\) is not satisfied, although this is a rare situation, we can consider the replacement at this concrete interval of the original data \(f_{j1}\) by \(\tilde{f}_{j1}\) according to (27a) instead of \(f_{j+2}\) by \(\tilde{f}_{j+2}\) in order to attain \(C_{L} \geq C _{H}\). This observation is easily proven because we go directly to Case \(2.2.1\). Thus, we give priority to the minimization of the curvature instead to the adaption to possible singularities. Notice that as mentioned in the previous point, there should not be a singularity at the considered interval but for exceptional cases.
Case 2.2.1. \(M_{j}\) and \(D_{j+1}\) (the lower divided difference in absolute value) have different sign.
Case 2.2.2. \(M_{j}\) and \(D_{j+1}\) have the same sign and \(\frac{M_{j}}{D_{j+1}}> \frac{3 h_{j+1}^{2}}{2(h_{j+1}^{2}+h_{j+1}h _{j+2}+h_{j+2}^{2})}\).

Case 2.2 will only appear around inflection points. Therefore, the case is avoided if we consider only data corresponding to strictly convex or concave functions.

Case \(2.2.2\) will not occur around discontinuities except for extremely nonuniform grids where \(w_{j+1} \approx 1\), since \(M_{j}\) and \(D_{j+1}\) have the same sign if and only if \(\frac{D_{j}}{D_{j}+1}<\frac{w _{j+1}}{w_{j}}\).

Under the assumption that for the given data condition in Case \(2.2.1\) is not satisfied, albeit this is not a common situation, we can give priority, as it happened in Case \(1.2.2\), to the minimization of the curvature instead to the adaption to possible singularities. Then, we consider in this case the replacement at this particular interval of the original data \(f_{j+2}\) by \(\tilde{f}_{j+2}\) according to (27b) instead of \(f_{j1}\) by \(\tilde{f}_{j1}\) in order to attain \(C_{L} \geq C_{H}\). Again this observation is trivial to prove. □
4 Conclusions and perspectives
We have obtained some inequalities which demonstrate that PPH reconstruction operator behaves better than usual linear Lagrange reconstruction operator regarding curvature issues. This study complements other previous results [1, 2, 6] where it was proven that PPH reconstruction preserves also the convexity properties of the initial data. This property is also inherited by the associated subdivision scheme [5, 8].
This opens up a potential future work connecting PPH reconstruction with smoothing splines in order to obtain a PPHtype reconstruction of class \(C^{2}\) in the whole interval with interesting convexitypreserving properties and low curvature term. Notice that piecewise PPH reconstruction is only continuous at the joint nodes.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Authors’ contributions
Both two authors worked together, prepared, read and approved the manuscript.
Funding
The authors have been supported through the Programa de Apoyo a la investigación de la fundación SénecaAgencia de Ciencia y Tecnología de la Región de Murcia 19374/PI714 and through the national research project MTM201564382P (MINECO/FEDER).
Competing interests
The authors declare that they have no competing interests.
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