Dimension of mixed splines on polytopal cells

DiPasquale, Michael

doi:10.1090/mcom/3224

Dimension of mixed splines on polytopal cells
HTML articles powered by AMS MathViewer

by Michael DiPasquale PDF

Math. Comp. 87 (2018), 905-939 Request permission

Abstract:

The dimension of planar splines on polygonal subdivisions of degree at most $d$ is known to be a degree two polynomial for $d\gg 0$. For planar $C^r$ splines on triangulations this formula is due to Alfeld and Schumaker; the formulas for planar splines with varying smoothness conditions across edges on convex polygonal subdvisions are due to Geramita, McDonald, and Schenck. In this paper we give a bound on how large $d$ must be for the known polynomial formulas to give the correct dimension of the spline space. Bounds are given for central polytopal complexes in three dimensions, or polytopal cells, with varying smoothness across two-dimensional faces. In the case of tetrahedral cells with uniform smoothness $r$ we show that the known polynomials give the correct dimension for $d\ge 3r+2$; previously Hong and separately Ibrahim and Schumaker had shown that this bound holds for planar triangulations. All bounds are derived using techniques from computational commutative algebra.

References

Peter Alfeld, Bruce Piper, and L. L. Schumaker, An explicit basis for $C^1$ quartic bivariate splines, SIAM J. Numer. Anal. 24 (1987), no. 4, 891–911. MR 899711, DOI 10.1137/0724058
Peter Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothness $r$ for degree $d\geq 4r+1$, Constr. Approx. 3 (1987), no. 2, 189–197. MR 889554, DOI 10.1007/BF01890563
Peter Alfeld and Larry L. Schumaker, On the dimension of bivariate spline spaces of smoothness $r$ and degree $d=3r+1$, Numer. Math. 57 (1990), no. 6-7, 651–661. MR 1062372, DOI 10.1007/BF01386434
Peter Alfeld and Larry L. Schumaker, Nonexistence of star-supported spline bases, SIAM J. Math. Anal. 31 (1999/00), no. 2, 455–465. MR 1740946, DOI 10.1137/S0036141098342349
Louis J. Billera, Homology of smooth splines: generic triangulations and a conjecture of Strang, Trans. Amer. Math. Soc. 310 (1988), no. 1, 325–340. MR 965757, DOI 10.1090/S0002-9947-1988-0965757-9
Louis J. Billera and Lauren L. Rose, A dimension series for multivariate splines, Discrete Comput. Geom. 6 (1991), no. 2, 107–128. MR 1083627, DOI 10.1007/BF02574678
Louis J. Billera and Lauren L. Rose, Modules of piecewise polynomials and their freeness, Math. Z. 209 (1992), no. 4, 485–497. MR 1156431, DOI 10.1007/BF02570848
Carl de Boor, A practical guide to splines, Revised edition, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York, 2001. MR 1900298
Michael DiPasquale, Lattice-supported splines on polytopal complexes, Adv. in Appl. Math. 55 (2014), 1–21. MR 3176714, DOI 10.1016/j.aam.2013.12.002
Michael DiPasquale, Associated primes of spline complexes, J. Symbolic Comput. 76 (2016), 158–199. MR 3461264, DOI 10.1016/j.jsc.2016.01.004
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Anthony V. Geramita and Henry K. Schenck, Fat points, inverse systems, and piecewise polynomial functions, J. Algebra 204 (1998), no. 1, 116–128. MR 1623949, DOI 10.1006/jabr.1997.7361
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Dong Hong, Spaces of bivariate spline functions over triangulation, Approx. Theory Appl. 7 (1991), no. 1, 56–75. MR 1117308
Adel Kh. Ibrahim and Larry L. Schumaker, Super spline spaces of smoothness $r$ and degree $d\geq 3r+2$, Constr. Approx. 7 (1991), no. 3, 401–423. MR 1120412, DOI 10.1007/BF01888166
Terry McDonald, Splines with boundary conditions, Comput. Math. Appl. 54 (2007), no. 9-10, 1234–1239. MR 2397674, DOI 10.1016/j.camwa.2006.10.034
Terry McDonald and Hal Schenck, Piecewise polynomials on polyhedral complexes, Adv. in Appl. Math. 42 (2009), no. 1, 82–93. MR 2475315, DOI 10.1016/j.aam.2008.06.001
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Ján Mináč and Stefan O. Tohǎneanu, From spline approximation to Roth’s equation and Schur functors, Manuscripta Math. 142 (2013), no. 1-2, 101–126. MR 3081001, DOI 10.1007/s00229-012-0594-6
Bernard Mourrain and Nelly Villamizar, Homological techniques for the analysis of the dimension of triangular spline spaces, J. Symbolic Comput. 50 (2013), 564–577. MR 2996896, DOI 10.1016/j.jsc.2012.10.002
Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29–41. MR 2199564, DOI 10.4310/MRL.2006.v13.n1.a3
Hal Schenck, A spectral sequence for splines, Adv. in Appl. Math. 19 (1997), no. 2, 183–199. MR 1459497, DOI 10.1006/aama.1997.0534
Hal Schenck and Mike Stillman, A family of ideals of minimal regularity and the Hilbert series of $C^r(\hat \Delta )$, Adv. in Appl. Math. 19 (1997), no. 2, 169–182. MR 1459496, DOI 10.1006/aama.1997.0533
Hal Schenck and Mike Stillman, Local cohomology of bivariate splines, J. Pure Appl. Algebra 117/118 (1997), 535–548. Algorithms for algebra (Eindhoven, 1996). MR 1457854, DOI 10.1016/S0022-4049(97)00026-1
Henry K. Schenck and Peter F. Stiller, Cohomology vanishing and a problem in approximation theory, Manuscripta Math. 107 (2002), no. 1, 43–58. MR 1892771, DOI 10.1007/s002290100222
Larry L. Schumaker, On the dimension of spaces of piecewise polynomials in two variables, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser, Basel-Boston, Mass., 1979, pp. 396–412. MR 560683
Ştefan O. Tohǎneanu, Smooth planar $r$-splines of degree $2r$, J. Approx. Theory 132 (2005), no. 1, 72–76. MR 2110576, DOI 10.1016/j.jat.2004.10.011
Walter Whiteley, A matrix for splines, Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 821–828. MR 1114817
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1