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Bivariate spline interpolation with optimal approximation order

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Abstract

Let Δ be a triangulation of some polygonal domain Ω ⊂ R2 and let S qr(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S q r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S q r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].

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... We make use of the nodal approach originated in the finite element method, see e.g. [12], and extended to the problems of spline spaces on general triangulations in [26] and more recently in [8811, 15, 16, 17]. We show that in the multivariate case the nodal smoothness conditions can be better localized than usual BernsteinnBe zier smoothness conditions [5, 20]. ...
... [24, 30]. The following estimation of the norm of H T f in the case of uniformly distributed points easily follows from the general results given in [13]; see also the proof of Lemma 3.9 in [16]. ...
... It is not difficult to see that the converse is also true. Note that for { # T 0 , Lemma 4.1 as well as its converse were given (in a slightly different form) in Theorem 4.1.2 of [11], and (in the bivariate case) in [16]. We now concentrate on the conditions (4.1) that involve the nodal functionals in the set N defined in Section 3. Namely, Lemma 4.1 implies that the following linear relations between the elements of N hold: ...
Article
We present an algorithm for constructing stable local bases for the spaces rd(Δ) of multivariate polynomial splines of smoothness r⩾1 and degree d⩾r2n+1 on an arbitrary triangulation Δ of a bounded polyhedral domain Ω⊂n, n⩾2.
... Proof. To show the estimate (6) for T * , we follow the proof of [8,Lemma 3.9]. We note that we only need to show that the interpolation scheme for pie-shaped triangles is a valid scheme, that is, we need to show that N P T is P 4 -unisolvent, and the rest of the proof can be done similar to that of [8,Lemma 3.9]. ...
... To show the estimate (6) for T * , we follow the proof of [8,Lemma 3.9]. We note that we only need to show that the interpolation scheme for pie-shaped triangles is a valid scheme, that is, we need to show that N P T is P 4 -unisolvent, and the rest of the proof can be done similar to that of [8,Lemma 3.9]. Recall that a set of functionals N is said to be P d -unisolvent if the only polynomial p ∈ P d satisfying η p = 0 for η ∈ N is the zero function. ...
... We note that the argument of the proof of [8,Lemma 3.9] applies to affine invariant interpolation schemes, that is the schemes that use the edge derivatives. As our scheme relies on the standard derivatives in the direction of the x, y axes, we need to express the edge derivatives as linear combinations of the x, y derivatives as follows. ...
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We develop a Hermite interpolation scheme and prove error bounds for $C^1$ bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary of curved domains enclosed by piecewise conics.
... Since (4.12) is a standard finite element interpolation scheme for bivariate polynomials of degree k − 1 (see, e.g., [57] or Lemma 3.7 in [25]), the polynomial s| is uniquely determined. We now show that the piecewise polynomial s constructed in this way lies in the space S k,r (T m ), i.e., it is r times differentiable. ...
... In view of (4.11) and Markov inequality (4.1), we have |ã e q,ξ | ≤ c δ −2q 2 s e,q L∞(e) , q = 0, . . . , r, and (4.8) follows by the properties of the interpolation problem (4.12), see Lemma 3.9 in [25]. ...
... = [u, v, w], := [u, v,w], e := [u, v], µ := [u, w],μ := [u,w], θ := ∠eµ,θ := ∠μe. The proof of the following lemma can be found in[17,25]. ...
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We study nonlinear n-term approximation in Lp(R2) (0 < p 1 ) from hierarchical sequences of stable local bases consisting of dierentiable (i.e., Cr with r 1) piece- wise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of R2, which allow arbitrarily sharp angles. To quantize nonlinear n- term spline approximation, we introduce and explore a collection of smoothness spaces (B-spaces). We utilize the B-spaces to prove companion Jackson and Bernstein esti- mates and then characterize the rates of approximation by interpolation. Even when applied on uniform triangulations with well-known families of basis functions such as box splines, these results give a more complete characterization of the approximation rates than the existing ones involving Besov spaces. Our results can easily be extended to properly defined multilevel triangulations in Rd, d > 2.
... For interpolation by bivariate polynomials, we refer to the survey of Gasca and Sauer [74]. Finally, we note that a characterization (diierent from Theorem 2.1) of the smoothness of polynomial pieces on adjacent triangles, without using BÃ ezier–Bernstein techniques, was proved by Davydov et al. [59]. ...
... In particular, a basis of star-supported splines (i.e. splines whose supports are at most the set of triangles surrounding a vertex, i.e., a cell (for interior vertices)) of S r q (); q¿3r + 2, is constructed (see also [59]). Recently, Alfeld and Schumaker [10] showed that such a basis does not exist in general if q ¡ 3r + 2 and r¿1. ...
... Recently, Alfeld and Schumaker [10] showed that such a basis does not exist in general if q ¡ 3r + 2 and r¿1. Thus, local arguments as in [36,59,80,82] fail in these cases. The problem of ÿnding an explicit formula for the dimension of S r q (); q ¡ 3r + 2; r¿1, remains open in general. ...
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The aim of this survey is to describe developments in the field of interpolation by bivariate splines. We summarize results on the dimension and the approximation order of bivariate spline spaces, and describe interpolation methods for these spaces. Moreover, numerical examples are given.
... Such spline spaces have been studied by many researchers in the literature, e.g., in [19] and the references therein. According to [4] and [5], there are spline functions s ∈ S r+1 d (△) with d ≥ 3r + 5 satisfying the interpolation conditions (1.1). Thus, the existence of a Hermite interpolatory spline can be easily understood. ...
... So we shall often make use of smoothness conditions to calculate one coefficient of a spline in terms of others. Recall from [4] and [5] that for any given sufficiently smooth function f ∈ Ω, there exists a quasi-interpolatory operator Q mapping f to S r d (△) with d ≥ 3r + 2 which achieves the optimal approximation order of S r d (△). That is Theorem 2.2 (cf. ...
... Recall from [4] and [5] that for any given sufficiently smooth function f ∈ Ω, there exists a quasi-interpolatory operator Q mapping f to S r d (△) with d ≥ 3r + 2 which achieves the optimal approximation order of S r d (△). That is Theorem 2.2 (cf. [5] ) Let r ≥ 1 and d ≥ 3r + 2. Suppose f ∈ C m (Ω) with m ≥ 2r. Then there exists a spline function Q f ∈ S r d (△) satisfying (1.1) and ...
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Given a set of scattered data with derivatives values, we use a minimal energy method to find Hermite interpolation based on bivariate spline spaces over a triangulation of the scattered data locations. We show that the minimal energy method produces a unique Hermite spline interpolation of the given scattered data with derivative values. Also we show that the Hermite spline interpolation converges to a given sufficiently smooth function f if the data values are obtained from this f. That is, the surface of the Hermite spline interpolation resembles the given set of derivative values. Some numerical examples are presented to demonstrate our method.
... [14]), and hence the approximation properties of the model are preserved by an argument of weak-interpolation (cf. [31]). It remains to determine the Bernstein-Bézier coefficient a v Q at the center v Q of Q (magenta dot in Fig. 5). ...
... [14]) and hence the approximation properties of the model are preserved by an argument of weak-interpolation type (cf. [31]). Now all the coefficients of the spline s are set appropriately. ...
Conference Paper
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... As a first step towards defining , we begin by sorting the cubes in ♦ into five different classes. This step is motivated by our earlier work on Lagrange interpolation with both bivariate and trivariate splines [18]- [19], [21]- [26]. ...
... Remark 7. For the bivariate case, the problem of constructing a Lagrange interpolating pair utilizing smooth splines defined on triangulations has been studied in a number of recent papers [18]- [19], [21]- [22], [24]- [26]. Our most recent paper [19] (see also [18]) deals with the general case of arbitrary smoothness and given initial triangulations. ...
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A trivariate Lagrange interpolation method based on C1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The in- terpolation method is local and stable, provides optimal order approximation, and has linear complexity.
... There is a fairly extensive history of Lagrange interpolation with bivariate splines, although even in this case, it is quite difficult to construct Lagrange interpolating pairs using C 1 splines. See [3][4][5][6]8,9]. ...
... Because of the optimal approximation order of the interpolation methods, it is natural to choose the partitions such that | 1 | 2 is about | 0 | 4 . For numerical results based on this idea in the bivariate setting, see [9]. ...
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... In [3], Nürnberger, Zeilfelder and Schumaker constructed a local Lagrange interpolation set for S 1 3 ( Q ), where Q is a kind of triangulated convex quadrangulations. In [4], Nürnberger and Zeilfelder used a coloring algorithm to divide all the triangles in into two kinds: white triangles and black triangles, and got a new triangulation CT through refining all the white triangles by the Clough-Tocher refinement, then gave a local Lagrange interpolation set for S 1 3 ( CT ). For r = 2, Nürnberger et al. constructed a local Lagrange interpolation set for S 2 7 ( CT ), where the CT is the triangulation by refining some of triangles in with the Clough-Tocher split; see [5]. ...
... In this paper, by using the same black and white coloring algorithm as [4], we shall first color all the triangles in an arbitrary regular triangulation and get DCT by refining some white triangles with the double Clough-Tocher refinement [6]. Then we consider the local Lagrange interpolation using a spline space S ⊂ S 2,3 5 ( DCT ) and construct a Lagrange interpolation pair on DCT . ...
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... With polynomials of degrees 2 k 4, spaces of C 1 continuous piecewise polynomials can be proved to provide optimal approximation when the triangulation is of some special structures, such as the Powell-Sabin triangulation [34], Hsieh-Clough-Tocher triangulation [8], Sander-Veubeke triangulation [15,36]. The restrictions on the grids can be weakened, but are generally needed on at least part of the triangulation [6,31,32]. On general triangulations, as is proved by [11], and illustrated by a counter example on a very regular triangulation [12,13], optimal approximation can not be expected with C 1 continuous piecewise polynomials of degree k < 5. It is illustrated in [1] that all the basis functions can not be determined locally on general grids. ...
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... We start with three lemmas that will also be used in the next section. Note that we use nodal smoothness conditions (see e.g. [10, 17]) in the proofs although the Bernstein-B` ezier techniques employed in [18]–[20] and [9] could have been applied with equal success. In the figures below we employ usual symbols (dots, circles and arrows) to indicate nodal degrees of freedom (see [2]). ...
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On arbitrary polygonal domains W Ì \RR2\Omega \subset \RR^2, we construct C1C^1 hierarchical Riesz bases for Sobolev spaces Hs(W)H^s(\Omega). In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from s Î (2,\frac52)s \in (2,\frac{5}{2}) to s Î (1,\frac52)s \in (1,\frac{5}{2}). Since the latter range includes s=2s=2, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned.
... For C 2 -splines of degree q = 5,6,7, Alfeld [2]' Gao [27]' Laghchim-Lahlou and Sablonniere [34,35]' Sablonniere [50] and Wang [57] defined Hermite interpolation schemes of finite element type. We note that our Hermite interpolation schemes are different from those for S;(~), q~3r + 2 in Davydov, Nürnberger & Zeilfelder [21]. Quasi interpolation methods were developed by Chui & Hong [11,12] for sl(~) and by Lai & Schumaker [38] for Sg(~) (see also [39]) for certain classes of triangulations ß. ...
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We describe a general method for constructing triangulations Δ which are suitable for interpolation by , where Sqr(Δ) denotes the space of splines of degree q and smoothness r. The triangulations Δ are obtained inductively by adding a subtriangulation of locally chosen scattered points in each step. By using Bézier–Bernstein techniques, we determine the dimension and construct Lagrange and Hermite interpolation sets for . The Hermite interpolation sets are obtained as limits of the Lagrange interpolation sets. The interpolating splines can be computed locally by passing from triangle to triangle. Several numerical results on interpolation of functions and scattered data are given.
... The result of Theorem 10 can also be established with the weak-interpolation methods described in [17]. ...
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... Finally, we note that according to our experience (quasi-) interpolating bivariate and trivariate splines can be computed efficiently, where the piecewise B-form of the splines turns out to be very useful for the various purposes in the different applications since the Bernstein-Bézier techniques known from Computer Aided Geometric Design can be fully exploited. Concerning numerical examples for the reconstruction of surfaces and volumetric objects involving data of various type, we refer the interested reader to [5,12,[17][18][19]. ...
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... They further improved the method without imposing any constraint on the first derivatives and thus avoid any unwanted undulations when interpolating irregular triangulations [13]. Nürnberger and Zeilfelder presented [14] a local Lagrange interpolation scheme for C 1 -splines of degree q ≥ 3 on arbitrary triangulations. This interpolating spline yields optimal approximation order and can be computed with linear complexity. ...
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... On the other hand, it has only been recently that algorithms for the efficient interpolation and approximation of general bivariate data sets appearing in certain real-world settings have been developed which take many of the above requirements into account. These methods (see [10,15,19,26,27,30,32], and the survey article [31] as well as the references therein) are based on bivariate splines, i.e. piecewise polynomials satisfying smoothness conditions which are defined w.r.t. planar and three-dimensional triangulations. ...
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... It is well known that for arbitrary triangulations , the space S r q ( ) has optimal approximation order if q 3r + 2 (cf. [7,16,10]). For q < 3r + 2, the approximation order of S r q ( ) for arbitrary triangulations is not optimal (cf. ...
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... [12]) that the spline space S r d () possesses optimal approximation property: In particular, Qf may be chosen to be an interpolatory spline with optimal approximation property (cf. [5,6]). Such quasi-interpolatory operator Q will be used in the following sections. ...
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... With polynomials of degrees 2 k 4, spaces of C 1 continuous piecewise polynomials can be shown to provide optimal approximation when the triangulation is of some special structures, such as the Powell-Sabin and Powell-Sabin-Heindl triangulations [31,44,45], criss-cross triangulations [57], Hsieh-Clough-Tocher triangulation [12], and Sander-Veubeke triangulation [20,48]. The conditions on the grids can be relaxed, but they are generally required on at least some part of the triangulation [10,41,42]. On general triangulations, as is shown in [18] and illustrated by a counterexample on a regular triangulation [15,16], optimal approximation cannot be obtained with C 1 continuous piecewise polynomials of degree k < 5. ...
Preprint
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... Recall from [1,2] that for any given function f ∈ L 1 (Ω), there exists a quasi-interpolatory operator Q mapping f ∈ L 1 (Ω) to S r d () with d ≥ 3r + 2, which achieves the optimal approximation order of S r d (). The results are summarized below. ...
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... In the last few years, a series of papers have appeared on local Lagrange interpolation using bivariate splines (see [11,8,9,7,6]). On the other hand, only a few results are known for this problem in the trivariate case (see [3,4,12,10]). ...
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We develop a Hermite interpolation scheme and prove error bounds for \(C^1\) bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary of curved domains enclosed by piecewise conics.
Chapter
This chapter summarizes results on splines over triangulations. It describes survey results on Bernstein-Bezier techniques, which are important for computer-aided geometric design (CAGD) applications. In the context of bivariate splines, these techniques provide a useful tool for analyzing the structure of these spaces. The results for spline spaces that were given for arbitrary triangulations and for classes of triangulations are summarized. The chapter discusses the dimension of bivariate spline spaces. It discusses interpolation by bivariate spline spaces. It also discusses classical finite elements and their modern generalizations and the macro element method, which lead to Hermite interpolation by super splines. Hermite interpolation methods for bivariate splines for low degree splines are based on a suitable splitting procedure applied to every triangle or quadrilateral. The chapter also summarizes Hermite and Lagrange interpolation methods by spline spaces. Most of these methods have been developed in particular local Lagrange interpolation methods for C 1 spline spaces. A different method for analyzing splines over triangulations that is based on multivariate B-splines is also described.
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In many practical problems, it has been detected that the scattered data are usually arranged in parallel lines. Thus, it is necessary to construct the bivariate spline interpolation function over the triangulation lattice. In this paper, we present a new approach to construct bivariate rational spline interpolation over triangulation, based on scattered data in parallel lines. The main advantage of the method is that the interpolation function has a simple and explicit mathematical representation with parameters α and β, and the shape of the interpolating surface can be modified by using the suitable parameters for the unchanged interpolating data. Moreover, a shape control method is employed to control the shape of surfaces, and numerical examples are presented to show the performance of the method. 1548-7741/
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We investigate the construction of local quasi-interpolation schemes based on a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r-1. These splines are defined on triangulations with Powell–Sabin refinement, and they can be represented in terms of locally supported basis functions that form a convex partition of unity. With the aid of the blossoming technique, we first derive a Marsden-like identity representing polynomials of degree 3r-1 in such a spline form. Then we present a general recipe to construct various families of smooth quasi-interpolation schemes involving values and/or derivatives of a given function.
Book
In this work, we construct two trivariate local Lagrange interpolation methods which yield optimal approximation order and Cr macro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition. The first interpolation method is based on cubic C1 splines over type-4 cube partitions, for which numerical tests are given. The other one is the first trivariate Lagrange interpolation method using C2 splines. It is based on arbitrary tetrahedral partitions using splines of degree nine. In order to obtain this method, several new results on C2 splines over partial Worsey- Farin splits are required. We construct trivariate macro-elements based on the Alfeld, where each tetrahedron is divided into four subtetrahedra, and theWorsey-Farin split, where each tetrahedron is divided into twelve subtetrahedra, of a tetrahedral partition. In order to obtain the macro-elements based on theWorsey-Farin split we construct minimal determining sets for Cr macro-elements over the Clough-Tocher split of a triangle, which are more variable than those in the literature. © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012. All rights are reserved.
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A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).
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We describe a method which can be used to interpolate function values at a set of scattered points in a planar domain using bivariate polynomial splines of any prescribed smoothness. The method starts with an arbitrary given triangulation of the data points, and involves refining some of the triangles with Clough-Tocher splits. The construction of the interpolating splines requires some additional function values at selected points in the domain, but no derivatives are needed at any point. Given n data points and a corresponding initial triangulation, the interpolating spline can be computed in just O(n) operations. The interpolation method is local and stable, and provides optimal order approximation of smooth functions.
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We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C 1 or C 2) splines on a uniform triangulation Δ (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Δ. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein–Bézier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.
Article
We develop the first local Lagrange interpolation scheme for C 1-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.
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In many practical problems, such as geological exploration, forging technology and medical imaging, among others, it has been detected that the scattered data are usually arranged in parallel lines. In this paper, a new approach to construct a bivariate rational interpolation over triangulation is presented, based on scattered data in parallel lines. The main advantage of this method comparing with the present interpolation methods have two points: (1) the interpolation function is carried out by a simple and explicit mathematical representation through the parameter α; (2) the shape of the interpolating surface can be modified by using the parameter for the unchanged interpolating data. Moreover, a local shape control method is employed to control the shape of surfaces. In the special case, the method of “Barycenter Value Control” is studied, and numerical examples are presented to show the performance of the method.
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We describe scattered data fitting by bivariate splines, i.e., splines defined w.r.t. triangulations in the plane. These spaces are powerful tools for the efficient approximation of large sets of scattered data which appear in many real world problems. Bernstein-Bézier techniques can be used for the efficient computation of bivariate splines and for analyzing the complex structure of these spaces. We report on the classical approaches and we describe interpolation and approximation methods for bivariate splines that have been developed recently. For the latter methods, we give illustrative examples treating sets of geodetic data (consisting of up to 10 6 points).
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Let u be a function defined on a triangulated bounded domain Ω in R2. In this paper, we study a recursive method for the construction of a Hermite spline interpolant uk of class Ck on Ω, defined by some data scheme Dk (u). We show that when Dr - 1 (u) ⊂ Dr (u) for all 1 ≤ r ≤ k, the spline function uk can be decomposed as a sum of (k + 1) simple elements. As application, we give the decomposition of the Ženišeck polynomial spline of class Ck and degree 4 k + 1, and we illustrate our results by an example.
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The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.
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We show how two recent algorithms for computing C1 quartic interpolating splines can be stabilized to ensure that, for smooth functions, they provide full approximation power with approximation constants depending only on the smallest angle in the triangulation [C. K. Chui and D. Hong, Math. Comp., 65 (1996), pp. 85--98; C. K. Chui and D. Hong, SIAM J. Numer. Anal., 34 (1997), pp. 1472--1482; D. Hong, Approximation Theory VIII, Vol.1: Approximation and Interpolation, C. K. Chui and L. L. Schumaker, eds., World Scientific, Singapore, 1995, pp. 249--256].
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We give a survey of recent methods to construct Lagrange interpolation points for splines of arbitrary smoothness r and degree q on general crosscut partitions in R&sup2;. For certain regular types of partitions, also results on Hermite interpolation sets and on the approximation order of the corresponding interpolating splines are given.
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Full-text available
. Let Delta be a triangulation of some polygonal domain in R 2 and S r k (Delta), the space of all bivariate C r piecewise polynomials of total degree k on Delta. In this paper, we construct a local basis of some subspace of the space S r k (Delta), where k 3r+2, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of Delta with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimalorder approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their B-net representations is derived for this purpose. 1. Introduction The objective of this paper is to describe the approximation properties of certain bivariate spline spaces over arbitrary triangulations of a poly...
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A basis for the space of $C^1$ piecewise polynomials in two variables of degree $n \geqslant 5$ is constructed. The basis is parametrized by "nodal variables," namely, the values and derivatives of the basis functions at a discrete set of points.
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Let V be a vector space of continuous functions in a region D in RS, s ≥ 1, with dimension N, and AN = {x1,…, xN} be a set of N distinct points in D.
We describe an algorithm which determines a C 1 cubic spline interpolating a function and its first derivatives at points regularly distributed on a non uniform type-2 triangulation of a rectangular domain. We give upper bounds of the norm of the operator and associated error. We conclude by some numerical examples concerning the error and the convergence order.
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. We construct a locally linearly independent basis for thespace S1q (\Delta) (q 5). Bases with this property were available only forsome subspaces of smooth bivariate splines.x1. IntroductionLet\Omegaae IR2be a simply connected polygonal domain, and let \Delta denote atriangulationof\Omegaconsisting of N triangles, V vertices and E edges. Given0 r ! q, consider the linear space of bivariate polynomial splines of degreeq and smoothness r,Srq (\Delta) := fs 2 C...
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Super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations, where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges. Dimension formulae for super spline spaces are established, and the construction of explicit bases of locally supported splines is presented. In addition, it is shown that many classical finite-element spaces can be interpreted as super spline spaces, providing a link between spline theory and the highly applicable finite-element theory.
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In two recent papers by Nürnberger & Riessinger algorithms were developed for constructing point sets at which unique Lagrange interpolation by spaces of bivariate splines of arbitrary-degree and smoothness on uniform-type triangulations is possible. Furthermore in Nürnberger (1996 J. Approx. Theory 87, 117-136) we proved that similar Hermite interpolation sets yield (nearly) optimal approximation order. This was done for differentiable splines of degree at least four defined on domains divided into subrectangles with one diagonal. In this paper, we analyze the error of Hermite interpolation by differentiable splines of arbitrary degree, where to each subrectangle of the partition two diagonals are added, and show that this method yields (nearly) optimal approximation order. The method of proof is different from that used in Nürnberger (1996). Finally, numerical examples and applications to data fitting are given.
Article
The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degreed withr continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super splines, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-Bézier method.
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A basis for the space of C piecewise polynomials in two variables of degree n > 5 is constructed. The basis is parametrized by nodal variables, namely, the values and derivatives of the basis functions at a discrete set of points.
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In [G. Nürnberger and Th. Riessinger,Numer. Math.71(1995), 91–119], we developed an algorithm for constructing point sets at which unique Lagrange interpolation by spaces of bivariate splines of arbitrary degree and smoothness on uniform type triangulations is possible. Here, we show that similar Hermite interpolation sets yield (nearly) optimal approximation order. This is shown for differentiable splines of degree at least four defined on non-rectangular domains subdivided in uniform type triangles. Therefore, in practice we use Lagrange configurations which are “close” to these Hermite configurations. Applications to data fitting problems and numerical examples are given.
Book
Splines play an important role in applied mathematics since they possess high flexibility to approximate efficiently, even nonsmooth functions which are given explicitly or only implicitly, e.g. by differential equations. The aim of this book is to analyse in a unified approach basic theoretical and numerical aspects of interpolation and best approximation by splines in one variable. The first part on "spaces of"" " "polynomials" serves as a basis for investigating the more complex structure of spline spaces. Given in the appendix are brief introductions to the theory of splines with "free knots" (an algorithm is described in the main part), to "splines in"" " "two variables" and to "spline " "collocation for differential equations."A large number of new results presented here cannot be found in earlier books on splines. Researchers will find several references to recent developments. The book is an indispensable aid for graduate courses on splines or approximation theory. Students with a basic knowledge of analysis and linear algebra will be able to read the text. Engineers will find various pactical interpolation and approximation methods.
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We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.
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A vertex spline basis of the super-spline subspace $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ of Sd r(), where d3r+2 and is an arbitrary triangulation inR 2, is constructed, so that the full approximation order ofd+1 can be achieved via an approximation formula using this basis.
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We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given.
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the meshsize, and with the distance between functions measured in the max-norm on some domain G contained in U 6. The exponent r depends on the smoothness 5cA off, in general. We are interested in determining its largest possible value under the assumption that f is sufficiently smooth. We call this number the approximation order of S and denote it (again) by r. It is well known that r = k + 1 for p < 0, but it is at present not clear what happens when p > 1. Because of results of ZeniSek [Z70], [-Z73], it is believed that the polynomial degree k must be at least 4p + 1 to obtain the full approximation order r=k+ 1. It is the purpose of this paper to show that actually a degree of 3 p + 2 suffices. Precisely, we prove the following.
Article
We describe algorithms for constructing point sets at which interpolation by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are selected in such a way that the grid points of the partition are contained in these sets, and no large linear systems have to be solved. Our method is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for certain univariate spline spaces such that a principle of degree reduction can be applied. In order to include the grid points in the interpolation sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This approach is completely different from the known interpolation methods for bivariate splines of degree at most three. Our method is illustrated by some numerical examples.
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Let S denote the space of bivariate piecewise polynomial functions of degree ⩽ k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For , k ≢ 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for , with Ω a convex subset of 2.
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By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sqr(Δ1) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q ⩾ 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.
Chapter
An attempt is made to analyze within reasonable limits the basic mathematical aspects of the finite element method. The information given should serve as an introduction to current research on this subject. Only actual problems are covered. Theorems are given to represent important results.
Article
We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces S dr(Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the L p norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain.
Article
: We consider the spaces of bivariate C ¯ -splines of degree k defined over arbitrary triangulations of a polygonal domain. We get an explicit formula for the dimension of such spaces when k 3¯ + 2 and construct a local supported basis for them. The dimension formula is valid for any polygonal domain even it is complex connected , and the formula is sharp since it arrives at the lower-bound which is given by Schumaker in [11]. x1. Introduction As usual, letOmega be a subset of R 2 , and let Delta = fø i g N 1 be a collection of closed triangles such that i) For all i, j, if i 6= j, the intersection ø i " ø j is either empty, their common edge or their common vertex. ii)Omega = S N 1 ø i . Then we call Delta a triangulation of OmegaGamma Given a positive integer k, we denote by Pi k the space of all polynomials in two variables with total degree k. For a triangulation Delta of OmegaGamma let S k;Delta = fs; sj ø i 2 Pi k ; i = 1; :::; Ng be the linear space of s...
Article
Introduction It is the purpose of this note to show that the approximation order from the space # # k,# of all piecewise polynomial functions in C # of polynomial degree # k on a triangulation # of IR 2 is, in general, no better than k in case k < 3# + 2. This complements the result of [BH88] that the approximation order from # # k,# for an arbitrary mesh # is k + 1 if k # 3# + 2. Here, we define the approximation order of a space S of functions on IR 2 to be the largest real number r for which dist (f, # h S) # const f h r for any su#ciently smooth function f , with the d
Article
By using the algorithm of Nurnberger & Riessinger [11], we construct Hermite interpolation sets for spaces of bivariate splines S r q (Delta 1 ) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3:5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation. Introduction We investigate spline spaces of the following type. Let a rectangle R and a partition of R into uniform subrectangles be given. We add to each subrectangle the same diagonal and denote the resulting partition by Delta 1 . (If we add both diagonals, then the resulting partition is denoted by Delta 2 .) The space of bivariate splines of degree q and smoothness r with respect to the partition Delta i is denoted by S r q (Delta i ); i = 1; 2. Nurnberger and Riessinger [10], [11] developed a method for constructing point sets which admit unique...
Article
. A characterization of almost interpolation configurations of points in terms of supports of basis functions is presented. Moreover, we show that this characterization can be significantly simplified in the case of existence of a locally linearly independent basis, so that almost interpolation sets can be constructed by taking a point in a support of each basis function. Some further results, including several equivalent definitions of a locally linearly independent system of functions, are given. 1. Introduction and main results Let U denote a finite-dimensional space of real-valued functions defined on K ae R d . The problem of describing those configurations of points T = ft 1 ; : : : ; t n g ae K ; n = dimU ; (interpolation sets or I-sets), such that for any given real data fy 1 ; : : : ; y n g there exists a unique function u 2 U satisfying u(t i ) = y i ; i = 1; : : : ; n; has attracted considerable interest in recent years, especially for the case when d 2 and U is a...
Article
. A simple method for constructing almost interpolation sets in the case of existence of locally linearly independent systems of basis functions is presented. Various examples of such systems, including translates of box splines and finite-element splines, are considered. 1. Introduction In [16] we have shown how the well-known Schoenberg-Whitney condition for interpolation by univariate polynomial splines can be extended to multivariate splines or even to the general setting of real functions defined on a topological space. For this case it characterizes almost interpolation sets (AI --sets); i.e., those configurations T such that in every neighborhood of T there exists a configuration ~ T (I --set) which admits Lagrange interpolation. In practice it is clearly quite important to have algorithms of constructing I-sets. For instance, for a system fB 1 ; : : : ; Bn g of univariate polynomial B-splines it is well-known that any set T = ft 1 ; : : : ; t n g such that t i 2 ft 2 IR : B ...
On interpolation by S 2 1 (Δ m,n 2 )
  • Z Sha
  • Z. Sha
On interpolation by Si (Om,n) Approx
  • Z Sha
Z. SHA (1985): On interpolation by Si (Om,n) Approx. Theory Appl., 1:1-18.
Zeilfelder We also need the following univariate "weak interpolation" lemma (compare
  • G Davydov
  • Niimberger
Davydov, G. Niimberger, and F. Zeilfelder We also need the following univariate "weak interpolation" lemma (compare [21, Remark 5(ii)] and [12, Lemma 4]).
Bivariate spline-interpolation auf crosscut-partitionen
  • M H Adam