Lattice-supported splines on polytopal complexes

DiPasquale, Michael

doi:10.1016/j.aam.2013.12.002

Cited by 6 publications

(16 citation statements)

References 20 publications

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“…Since reg(C α (P)) in particular bounds the degrees of generators of C α (P) (see Remark 5.2), this construction indicates that a bound on reg(C α (P)) will need to be at least as large as the maximal sum of smoothness parameters over codimension one faces occurring in any facet of P (or at least boundary facets -see Conjecture 9.1). This example generalizes the construction in [11,Theorem 5.7].…”

Section: Regularitysupporting

confidence: 74%

“…Lattice-Supported Splines. In [11] certain subalgebras LS r,k (P) ⊂ C r (P) are constructed as approximations to C r (P). This construction carries over directly to mixed splines; we will denote the corresponding submodules by LS α,k (P).…”

Section: Splines and Lattice-supported Splinesmentioning

confidence: 99%

“…Theorem 4.3 of [11] makes precise the sense in which LS r,k (P) is an approximation to C r (P). This result and its proof extend directly to mixed splines, so we state the result in this context.…”

Section: Splines and Lattice-supported Splinesmentioning

confidence: 99%

“…This statement is made precise in Proposition 5.8 and Theorem 6.2. We take as our approximation certain locally-supported subalgebras of splines introduced in [11]. This could be viewed as an algebraic analogue of locally-supported bases used in [18,19].…”

Section: Introductionmentioning

confidence: 99%

“…In § 2 we give some background on the spline algebra C α (P), in particular the algebraic approach pioneered by Billera [6] and Billera and Rose [7]. We recall the construction of lattice-supported splines LS α,k (P) introduced in [11]. These will provide approximations to C α (P).…”

Section: Introductionmentioning

confidence: 99%

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Regularity of Mixed Spline Spaces

DiPasquale

2014

Preprint

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We derive bounds on the regularity of the algebra C α (P) of mixed splines over a central polytopal complex P ⊂ R 3 . As a consequence we bound the largest integer d (the postulation number) for which the Hilbert polynomial HP (C α (P), d) disagrees with the Hilbert function HF (C α (P), d) = dim C α (P) d . The polynomial HP (C α (P), d) has been computed in [12], building on [16,20]. Hence the regularity bounds obtained indicate when a known polynomial gives the correct dimension of the spline space C α (P) d . In the simplicial case with all smoothness parameters equal, we recover a bound originally due to Hong [18] and Ibrahim and Schumaker [19].

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Section: Regularitysupporting

confidence: 74%

Section: Splines and Lattice-supported Splinesmentioning

confidence: 99%

Section: Splines and Lattice-supported Splinesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity of Mixed Spline Spaces

DiPasquale

2014

Preprint

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Algebraic methods in approximation theory

Schenck

2016

Computer Aided Geometric Design

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Abstract. This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in R k , and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree. The objects appearing here may be computed using the spline package of the Macaulay2 software system.

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Associated primes of spline complexes

DiPasquale

2016

Journal of Symbolic Computation

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Abstract. The spline complex R/J [Σ] whose top homology is the algebra C α (Σ) of mixed splines over the fan Σ ⊂ R n+1 was introduced by SchenckStillman in [26] as a variant of a complex R/I[Σ] of Billera [5]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of C α (Σ), including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of Σ, generalizing the computations in [14,19]. The second is a description of the fourth coefficient of the Hilbert polynomial of HP (C α (Σ)) for simplicial fans Σ. We use this to derive the result of Alfeld, Schumaker, and Whiteley on the generic dimension of C 1 tetrahedral splines for d 0 [3] and indicate via an example how this description may be used to give the fourth coefficient in particular nongeneric configurations.

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Lattice-supported splines on polytopal complexes

Cited by 6 publications

References 20 publications

Regularity of Mixed Spline Spaces

Regularity of Mixed Spline Spaces

Algebraic methods in approximation theory

Associated primes of spline complexes

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