Box Splines

Boor, Carl de; Höllig, Klaus; Riemenschneider, S. D.

doi:10.1007/978-1-4757-2244-4

Cited by 319 publications

(254 citation statements)

References 8 publications

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“…is introduced and studied (see also [2,4,38]). Here the mesh-points A rs that lie outside Ω should be thought of as if they were projected on ∂ Ω , i.e.…”

Section: )mentioning

confidence: 99%

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Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Lamberti

2009

Bit Numer Math

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In this paper we generate and study new cubature formulas based on spline quasi-interpolants defined as linear combinations of C 1 bivariate quadratic B-splines on a rectangular domain Ω , endowed with a non-uniform criss-cross triangulation, with discrete linear functionals as coefficients. Such B-splines have their supports contained in Ω and there is no data point outside this domain. Numerical results illustrate the methods.

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“…is introduced and studied (see also [2,4,38]). Here the mesh-points A rs that lie outside Ω should be thought of as if they were projected on ∂ Ω , i.e.…”

Section: )mentioning

confidence: 99%

“…In [4,38] and [2], p.72, the authors introduce two C 1 bivariate quadratic spline QIs, defined on uniform and nonuniform criss-cross partitions T mn of the rectangular domain Ω , using C 1 quadratic box-splines with octagonal support and simple knots [5,29] …”

mentioning

confidence: 99%

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

Lamberti

2009

Bit Numer Math

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“…The (symbolic) computation of the function c can be done using the following recurrence (see Chapter Quasi-interpolants and approximation power of the book [12]): c r = m r − r ≺r µ(m r−r ) c r c (0,...,0) = 1.…”

Section: Computing the Box Spline Representationmentioning

confidence: 99%

“…Given x, the index set I(x) associated with the box spline B M is I(x) = {i ∈ Z n | B M (x − i) = 0}. It can be proven that [12] I…”

Section: Computing the Box Spline Representationmentioning

confidence: 99%

Approximate Reachability Computation for Polynomial Systems

Dang

2006

Hybrid Systems: Computation and Control

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Abstract. In this paper we propose an algorithm for approximating the reachable sets of systems defined by polynomial differential equations. Such systems can be used to model a variety of physical phenomena. We first derive an integration scheme that approximates the state reachable in one time step by applying some polynomial map to the current state. In order to use this scheme to compute all the states reachable by the system starting from some initial set, we then consider the problem of computing the image of a set by a multivariate polynomial. We propose a method to do so using the Bézier control net of the polynomial map and the blossoming technique to compute this control net. We also prove that our overall method is of order 2. In addition, we have successfully applied our reachability algorithm to two models of a biological system.

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“…For k = 1 this follows from the definition of B(x | V n ) and V n (see [2]). Assume that (12) holds for some k ≥ n. Applying (3) and the fact that w k is periodic we obtain…”

Section: Introductionmentioning

confidence: 99%