Dimension and Local Bases of Homogeneous Spline Spaces

Alfeld, Peter; Neamtu, Marian; Schumaker, Larry L.

doi:10.1137/s0036141094276275

Cited by 37 publications

(52 citation statements)

References 14 publications

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“…There is an extensive theory of splines defined on spherical triangulations which is remarkably similar to the theory of bivariate splines, see [1][2][3] and Chapters 13-14 of [13]. Such splines are piecewise spherical harmonics.…”

Section: Remarkmentioning

confidence: 99%

Computing bivariate splines in scattered data fitting and the finite-element method

Schumaker

2008

Numer Algor

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Abstract. A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at one time. Typically this involves solving a system of linear equations. Examples include several well-known methods for fitting scattered data, such as the minimal energy, least-squares, and penalized least-squares methods. Finite-element methods for solving boundary-value problems are also of this type. It is shown here that these types of globally-defined splines can be efficiently computed, provided we work with spline spaces with stable local minimal determining sets. §1. Introduction Bivariate splines defined over triangulations are important tools in several application areas including scattered data fitting and the numerical solution of boundaryvalue problems by the finite element method. Methods for computing spline approximations fall into two classes: 1) Local methods, where the coefficients of the spline are computed one at a time or in small groups, 2) Global methods, where all of the coefficients of the spline have to be computed simultaneously, usually as the solution of a single linear system of equations.In this paper we focus on global methods, and in particular those that arise from minimizing a quadratic form, possibly with some constraints. The purpose of this paper is to show how such minimization problems can be efficiently solved for spline spaces that possess stable local minimal determining sets (see Sect. 2). In particular, we show how our approach applies to three commmonly used scattered data fitting methods: the minimal energy method, the discrete least-squares method, and the penalized least-squares method. In addition, we discuss how it works for solving boundary-value problems involving partial differential equations. The standard approach to solving global variational problems involving piecewise polynomials on triangulations is to use Lagrange multipliers to enforce interpolation and smoothness conditions, see Remark 1. This results in a linear system

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Section: Remarkmentioning

confidence: 99%

Computing bivariate splines in scattered data fitting and the finite-element method

Schumaker

2008

Numer Algor

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“…Let H d denote the space of trivariate homogeneous polynomials of degree d. That is, 3 be a nondegenerate spherical triangle, i.e., assume that the area on the unit sphere bounded by the great circular arcs connecting v 1 and v 2 , v 1 and v 3 , and v 2 and v 3 is not zero. Let b 1 (v), b 2 (v), and b 3 (v) be the spherical barycentric coordinates of a point v ∈ S 2 , i.e.…”

Section: Preliminarymentioning

confidence: 99%

Spherical Splines for Data Interpolation and Fitting

Baramidze¹,

Lai²,

Shum³

2006

SIAM J. Sci. Comput.

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Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines.Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D171. Introduction. Contemporary research in atmospheric sciences, geodesy and geophysics requires the use of global data heterogeneously distributed in space around the Earth. For spherical data interpolation/approximation tensor products of univariate splines are not a good choice, since data locations are not usually spaced over a regular grid. Radial basis functions are not good candidates either, since the data values may have no rotational symmetry.Spherical Bernstein-Bézier splines (introduced in [1] and studied in [2] and [7]) are well suited for scattered data interpolation/approximation problems. The spherical spline functions have many properties in common with classical polynomial splines over planar triangulations. Moreover, many spline interpolation and approximation methods for planar scattered data problems have analogs in spherical setting [2].One of the disadvantages of using homogeneous spherical splines is that spline spaces of even and odd degrees have only zero function in common due to homogeneity of the basis. More explicitly, the spline space S

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“…We use Bernstein-Bézier techniques as in [1,2,3,4,5,6,7,8,9,10,11,12,13,16,17]. In particular, we represent polynomials p of degree d on a triangle T := v 1 , v 2 In this paper we are interested in subspaces S of S 0 d ( ) that satisfy additional smoothness conditions.…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, we represent polynomials p of degree d on a triangle T := v 1 , v 2 In this paper we are interested in subspaces S of S 0 d ( ) that satisfy additional smoothness conditions. Following [6], to describe smoothness we shall make use of smoothness functionals defined as follows.…”

Section: Preliminariesmentioning

confidence: 99%

“…In [3] it was noted that the classical C 1 Clough-Tocher and PowellSabin macro-elements have natural analogs in terms of spherical splines. Since the algebra of spherical splines is essentially the same as for bivariate splines [2], it is clear that the entire family of macro-elements constructed here can also be immediately carried over to the sphere.…”

Section: Remarksmentioning

confidence: 99%

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Smooth macro-elements on Powell-Sabin-12 splits

Schumaker

Sorokina

2005

Math. Comp.

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Abstract. Macro-elements of smoothness C r are constructed on PowellSabin-12 splits of a triangle for all r ≥ 0. These new elements complement those recently constructed on Powell-Sabin-6 splits and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary-value problems and for interpolation of Hermite data.

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Dimension and Local Bases of Homogeneous Spline Spaces

Cited by 37 publications

References 14 publications

Computing bivariate splines in scattered data fitting and the finite-element method

Computing bivariate splines in scattered data fitting and the finite-element method

Spherical Splines for Data Interpolation and Fitting

Smooth macro-elements on Powell-Sabin-12 splits

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