Polynomial splines of non-uniform degree on triangulations: Combinatorial bounds on the dimension
•We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown...
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| Veröffentlicht in: | Computer aided geometric design Jg. 75; S. 101763 |
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| Sprache: | Englisch |
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01.11.2019
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| ISSN: | 0167-8396, 1879-2332 |
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| Abstract | •We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown.•Example applications are presented; accompanying Macaulay2 package is publicly available.
For T a planar triangulation, let Rmr(T) denote the space of bivariate splines on T such that f∈Rmr(T) is Cr(τ) smooth across an interior edge τ and, for triangle σ in T, f|σ is a polynomial of total degree at most m(σ)∈Z≥0. The map m:σ↦Z≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of Rmr(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of Rmr(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rmr(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013). |
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| AbstractList | •We study spaces of bivariate polynomial splines on planar triangulations.•The splines are composed of pieces with different polynomial degrees on different triangles.•We provide combinatorial upper and lower bounds on the dimension of such splines.•Stability of the dimension in high degree is shown.•Example applications are presented; accompanying Macaulay2 package is publicly available.
For T a planar triangulation, let Rmr(T) denote the space of bivariate splines on T such that f∈Rmr(T) is Cr(τ) smooth across an interior edge τ and, for triangle σ in T, f|σ is a polynomial of total degree at most m(σ)∈Z≥0. The map m:σ↦Z≥0 is called a non-uniform degree distribution on the triangles in T, and we consider the problem of computing (or estimating) the dimension of Rmr(T) in this paper. Using homological techniques, developed in the context of splines by Billera (1988), we provide combinatorial lower and upper bounds on the dimension of Rmr(T). When all polynomial degrees are sufficiently large, m(σ)≫0, we prove that the number of splines in Rmr(T) can be determined exactly. In the special case of a constant map m, the lower and upper bounds are equal to those provided by Mourrain and Villamizar (2013). |
| ArticleNumber | 101763 |
| Author | Toshniwal, Deepesh Hughes, Thomas J.R. |
| Author_xml | – sequence: 1 givenname: Deepesh surname: Toshniwal fullname: Toshniwal, Deepesh email: d.toshniwal@tudelft.nl organization: Delft Institute of Applied Mathematics, Delft University of Technology, the Netherlands – sequence: 2 givenname: Thomas J.R. surname: Hughes fullname: Hughes, Thomas J.R. organization: Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, USA |
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| References | Sederberg, Zheng, Song (br0260) 2003; 20 Billera, Rose (br0040) 1991; 6 Schenck (br0170) 2003 Schenck, Stillman (br0200) 1997; 19 Alfeld, Schumaker (br0020) 1990; 57 Schenck (br0180) 2016; 45 Toshniwal, Speleers, Hiemstra, Hughes (br0280) 2017; 316 Grayson, Stillman (br0100) 2019 Schumaker (br0230) 1984; 14 Billera (br0030) 1988; 310 Piegl, Tiller (br0150) 2012 Toshniwal (br0290) 2019 Sederberg, Zheng, Bakenov, Nasri (br0250) 2003; 22 Alfeld, Schumaker (br0010) 1987; 3 Cox, Little, O'shea (br0050) 2007 Giannelli, Jüttler, Speleers (br0090) 2012; 29 Lai, Schumaker (br0120) 2007 Mourrain, Villamizar (br0140) 2013; 50 Schenck, Stillman (br0210) 1997; 117 Geramita, Schenck (br0080) 1998; 204 Schenck (br0160) 1997; 19 Schenck, Sorokina (br0190) 2018; 47 Schumaker (br0240) 2007 Strang (br0270) 1974 Hughes, Cottrell, Bazilevs (br0110) 2005; 194 de Visser, Chu, Mulder (br0060) 2009; 45 Dokken, Lyche, Pettersen (br0070) 2013; 30 Li, Huang, Liu (br0130) 2012; 27 Schumaker (br0220) 1979 Schenck (10.1016/j.cagd.2019.07.002_br0170) 2003 Alfeld (10.1016/j.cagd.2019.07.002_br0010) 1987; 3 Schenck (10.1016/j.cagd.2019.07.002_br0190) 2018; 47 Sederberg (10.1016/j.cagd.2019.07.002_br0260) 2003; 20 Grayson (10.1016/j.cagd.2019.07.002_br0100) Cox (10.1016/j.cagd.2019.07.002_br0050) 2007 Toshniwal (10.1016/j.cagd.2019.07.002_br0280) 2017; 316 Strang (10.1016/j.cagd.2019.07.002_br0270) 1974 Schenck (10.1016/j.cagd.2019.07.002_br0160) 1997; 19 Schumaker (10.1016/j.cagd.2019.07.002_br0220) 1979 Dokken (10.1016/j.cagd.2019.07.002_br0070) 2013; 30 Giannelli (10.1016/j.cagd.2019.07.002_br0090) 2012; 29 de Visser (10.1016/j.cagd.2019.07.002_br0060) 2009; 45 Piegl (10.1016/j.cagd.2019.07.002_br0150) 2012 Billera (10.1016/j.cagd.2019.07.002_br0030) 1988; 310 Lai (10.1016/j.cagd.2019.07.002_br0120) 2007 Hughes (10.1016/j.cagd.2019.07.002_br0110) 2005; 194 Schenck (10.1016/j.cagd.2019.07.002_br0200) 1997; 19 Alfeld (10.1016/j.cagd.2019.07.002_br0020) 1990; 57 Schenck (10.1016/j.cagd.2019.07.002_br0210) 1997; 117 Li (10.1016/j.cagd.2019.07.002_br0130) 2012; 27 Schumaker (10.1016/j.cagd.2019.07.002_br0230) 1984; 14 Schumaker (10.1016/j.cagd.2019.07.002_br0240) 2007 Geramita (10.1016/j.cagd.2019.07.002_br0080) 1998; 204 Billera (10.1016/j.cagd.2019.07.002_br0040) 1991; 6 Mourrain (10.1016/j.cagd.2019.07.002_br0140) 2013; 50 Sederberg (10.1016/j.cagd.2019.07.002_br0250) 2003; 22 Toshniwal (10.1016/j.cagd.2019.07.002_br0290) Schenck (10.1016/j.cagd.2019.07.002_br0180) 2016; 45 |
| References_xml | – volume: 204 start-page: 116 year: 1998 end-page: 128 ident: br0080 article-title: Fat points, inverse systems, and piecewise polynomial functions publication-title: J. Algebra – start-page: 396 year: 1979 end-page: 412 ident: br0220 article-title: On the dimension of spaces of piecewise polynomials in two variables publication-title: Multivariate Approximation Theory – volume: 22 start-page: 477 year: 2003 end-page: 484 ident: br0250 article-title: T-splines and T-NURCCs publication-title: ACM Trans. Graph. (TOG) – volume: 57 start-page: 651 year: 1990 end-page: 661 ident: br0020 article-title: On the dimension of bivariate spline spaces of smoothness publication-title: Numer. Math. – volume: 30 start-page: 331 year: 2013 end-page: 356 ident: br0070 article-title: Polynomial splines over locally refined box-partitions publication-title: Comput. Aided Geom. Des. – volume: 14 start-page: 251 year: 1984 end-page: 264 ident: br0230 article-title: Bounds on the dimension of spaces of multivariate piecewise polynomials publication-title: Rocky Mt. J. Math. – year: 2003 ident: br0170 article-title: Computational Algebraic Geometry, vol. 58 – volume: 20 start-page: 455 year: 2003 end-page: 468 ident: br0260 article-title: Knot intervals and multi-degree splines publication-title: Comput. Aided Geom. Des. – volume: 29 start-page: 485 year: 2012 end-page: 498 ident: br0090 article-title: THB-splines: the truncated basis for hierarchical splines publication-title: Comput. Aided Geom. Des. – year: 2012 ident: br0150 article-title: The NURBS Book – year: 2007 ident: br0120 article-title: Spline Functions on Triangulations, vol. 110 – year: 2007 ident: br0050 article-title: Ideals, Varieties, and Algorithms, vol. 3 – year: 2019 ident: br0290 article-title: SimplicialMDSplines: splines of mixed degree and smoothness on simplicial complexes – year: 2019 ident: br0100 article-title: Macaulay2, a software system for research in algebraic geometry – volume: 45 start-page: 14 year: 2016 end-page: 31 ident: br0180 article-title: Algebraic methods in approximation theory publication-title: Comput. Aided Geom. Des. – volume: 316 start-page: 1005 year: 2017 end-page: 1061 ident: br0280 article-title: Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis publication-title: Comput. Methods Appl. Mech. Eng. – volume: 6 start-page: 107 year: 1991 end-page: 128 ident: br0040 article-title: A dimension series for multivariate splines publication-title: Discrete Comput. Geom. – volume: 310 start-page: 325 year: 1988 end-page: 340 ident: br0030 article-title: Homology of smooth splines: generic triangulations and a conjecture of Strang publication-title: Trans. Am. Math. Soc. – volume: 194 start-page: 4135 year: 2005 end-page: 4195 ident: br0110 article-title: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement publication-title: Comput. Methods Appl. Mech. Eng. – volume: 45 start-page: 2903 year: 2009 end-page: 2909 ident: br0060 article-title: A new approach to linear regression with multivariate splines publication-title: Automatica – year: 2007 ident: br0240 article-title: Spline Functions: Basic Theory – volume: 19 start-page: 183 year: 1997 end-page: 199 ident: br0160 article-title: A spectral sequence for splines publication-title: Adv. Appl. Math. – volume: 47 start-page: 237 year: 2018 end-page: 247 ident: br0190 article-title: Subdivision and spline spaces publication-title: Constr. Approx. – volume: 27 start-page: 841 year: 2012 end-page: 850 ident: br0130 article-title: A geometric approach for multi-degree splines publication-title: J. Comput. Sci. Technol. – volume: 3 start-page: 189 year: 1987 end-page: 197 ident: br0010 article-title: The dimension of bivariate spline spaces of smoothness publication-title: Constr. Approx. – volume: 50 start-page: 564 year: 2013 end-page: 577 ident: br0140 article-title: Homological techniques for the analysis of the dimension of triangular spline spaces publication-title: J. Symb. Comput. – volume: 117 start-page: 535 year: 1997 end-page: 548 ident: br0210 article-title: Local cohomology of bivariate splines publication-title: J. Pure Appl. Algebra – start-page: 144 year: 1974 end-page: 152 ident: br0270 article-title: The dimension of piecewise polynomial spaces, and one-sided approximation publication-title: Conference on the Numerical Solution of Differential Equations – volume: 19 start-page: 169 year: 1997 end-page: 182 ident: br0200 article-title: A family of ideals of minimal regularity and the Hilbert series of publication-title: Adv. Appl. Math. – ident: 10.1016/j.cagd.2019.07.002_br0100 – volume: 14 start-page: 251 issue: 1 year: 1984 ident: 10.1016/j.cagd.2019.07.002_br0230 article-title: Bounds on the dimension of spaces of multivariate piecewise polynomials publication-title: Rocky Mt. J. Math. doi: 10.1216/RMJ-1984-14-1-251 – volume: 29 start-page: 485 year: 2012 ident: 10.1016/j.cagd.2019.07.002_br0090 article-title: THB-splines: the truncated basis for hierarchical splines publication-title: Comput. Aided Geom. Des. doi: 10.1016/j.cagd.2012.03.025 – volume: 47 start-page: 237 issue: 2 year: 2018 ident: 10.1016/j.cagd.2019.07.002_br0190 article-title: Subdivision and spline spaces publication-title: Constr. Approx. doi: 10.1007/s00365-017-9367-5 – volume: 50 start-page: 564 year: 2013 ident: 10.1016/j.cagd.2019.07.002_br0140 article-title: Homological techniques for the analysis of the dimension of triangular spline spaces publication-title: J. Symb. Comput. doi: 10.1016/j.jsc.2012.10.002 – volume: 194 start-page: 4135 year: 2005 ident: 10.1016/j.cagd.2019.07.002_br0110 article-title: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement publication-title: Comput. Methods Appl. Mech. Eng. doi: 10.1016/j.cma.2004.10.008 – volume: 45 start-page: 2903 issue: 12 year: 2009 ident: 10.1016/j.cagd.2019.07.002_br0060 article-title: A new approach to linear regression with multivariate splines publication-title: Automatica doi: 10.1016/j.automatica.2009.09.017 – volume: 45 start-page: 14 year: 2016 ident: 10.1016/j.cagd.2019.07.002_br0180 article-title: Algebraic methods in approximation theory publication-title: Comput. Aided Geom. Des. doi: 10.1016/j.cagd.2015.11.001 – year: 2012 ident: 10.1016/j.cagd.2019.07.002_br0150 – year: 2007 ident: 10.1016/j.cagd.2019.07.002_br0050 – volume: 19 start-page: 183 issue: 2 year: 1997 ident: 10.1016/j.cagd.2019.07.002_br0160 article-title: A spectral sequence for splines publication-title: Adv. Appl. Math. doi: 10.1006/aama.1997.0534 – volume: 316 start-page: 1005 year: 2017 ident: 10.1016/j.cagd.2019.07.002_br0280 article-title: Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis publication-title: Comput. Methods Appl. Mech. Eng. doi: 10.1016/j.cma.2016.11.009 – start-page: 144 year: 1974 ident: 10.1016/j.cagd.2019.07.002_br0270 article-title: The dimension of piecewise polynomial spaces, and one-sided approximation – volume: 27 start-page: 841 year: 2012 ident: 10.1016/j.cagd.2019.07.002_br0130 article-title: A geometric approach for multi-degree splines publication-title: J. Comput. Sci. Technol. doi: 10.1007/s11390-012-1268-2 – volume: 3 start-page: 189 issue: 1 year: 1987 ident: 10.1016/j.cagd.2019.07.002_br0010 article-title: The dimension of bivariate spline spaces of smoothness r for degree ≥4r+1 publication-title: Constr. Approx. doi: 10.1007/BF01890563 – volume: 204 start-page: 116 issue: 1 year: 1998 ident: 10.1016/j.cagd.2019.07.002_br0080 article-title: Fat points, inverse systems, and piecewise polynomial functions publication-title: J. Algebra doi: 10.1006/jabr.1997.7361 – volume: 117 start-page: 535 year: 1997 ident: 10.1016/j.cagd.2019.07.002_br0210 article-title: Local cohomology of bivariate splines publication-title: J. Pure Appl. Algebra doi: 10.1016/S0022-4049(97)00026-1 – year: 2007 ident: 10.1016/j.cagd.2019.07.002_br0240 – volume: 22 start-page: 477 year: 2003 ident: 10.1016/j.cagd.2019.07.002_br0250 article-title: T-splines and T-NURCCs publication-title: ACM Trans. Graph. (TOG) doi: 10.1145/882262.882295 – volume: 20 start-page: 455 year: 2003 ident: 10.1016/j.cagd.2019.07.002_br0260 article-title: Knot intervals and multi-degree splines publication-title: Comput. Aided Geom. Des. doi: 10.1016/S0167-8396(03)00096-7 – volume: 6 start-page: 107 issue: 2 year: 1991 ident: 10.1016/j.cagd.2019.07.002_br0040 article-title: A dimension series for multivariate splines publication-title: Discrete Comput. Geom. doi: 10.1007/BF02574678 – volume: 19 start-page: 169 issue: 2 year: 1997 ident: 10.1016/j.cagd.2019.07.002_br0200 article-title: A family of ideals of minimal regularity and the Hilbert series of Cr(Δ) publication-title: Adv. Appl. Math. doi: 10.1006/aama.1997.0533 – volume: 57 start-page: 651 issue: 1 year: 1990 ident: 10.1016/j.cagd.2019.07.002_br0020 article-title: On the dimension of bivariate spline spaces of smoothness r and degree =3r+1 publication-title: Numer. Math. doi: 10.1007/BF01386434 – year: 2007 ident: 10.1016/j.cagd.2019.07.002_br0120 – volume: 30 start-page: 331 year: 2013 ident: 10.1016/j.cagd.2019.07.002_br0070 article-title: Polynomial splines over locally refined box-partitions publication-title: Comput. Aided Geom. Des. doi: 10.1016/j.cagd.2012.12.005 – ident: 10.1016/j.cagd.2019.07.002_br0290 – year: 2003 ident: 10.1016/j.cagd.2019.07.002_br0170 – start-page: 396 year: 1979 ident: 10.1016/j.cagd.2019.07.002_br0220 article-title: On the dimension of spaces of piecewise polynomials in two variables – volume: 310 start-page: 325 issue: 1 year: 1988 ident: 10.1016/j.cagd.2019.07.002_br0030 article-title: Homology of smooth splines: generic triangulations and a conjecture of Strang publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1988-0965757-9 |
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