34 research outputs found
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Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-BÂŽezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
On multivariate polynomials in BernsteinâBĂ©zier form and tensor algebra
AbstractThe BernsteinâBĂ©zier representation of polynomials is a very useful tool in computer aided geometric design. In this paper we make use of (multilinear) tensors to describe and manipulate multivariate polynomials in their BernsteinâBĂ©zier form. As an application we consider Hermite interpolation with polynomials and splines
On C2 cubic quasi-interpolating splines and their computation by subdivision via blossoming
We discuss the construction of C2 cubic spline quasi-interpolation schemes defined on a
refined partition. These schemes are reduced in terms of degrees of freedom compared to
those existing in the literature. Namely, we provide a rule for reducing them by imposing
super-smoothing conditions while preserving full smoothness and cubic precision. In
addition, we provide subdivision rules by means of blossoming. The derived rules are
designed to express the B-spline coefficients associated with a finer partition from those
associated with the former one."Maria de Maeztu" Excellence Unit IMAG (University of Granada, Spain) CEX2020-001105-MICIN/AEI/10.13039/501100011033University of Granada
University of Granada/CBU
The apolar bilinear form in geometric modeling
Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives
Handling convexity-like constraints in variational problems
We provide a general framework to construct finite dimensional approximations
of the space of convex functions, which also applies to the space of c-convex
functions and to the space of support functions of convex bodies. We give
estimates of the distance between the approximation space and the admissible
set. This framework applies to the approximation of convex functions by
piecewise linear functions on a mesh of the domain and by other
finite-dimensional spaces such as tensor-product splines. We show how these
discretizations are well suited for the numerical solution of problems of
calculus of variations under convexity constraints. Our implementation relies
on proximal algorithms, and can be easily parallelized, thus making it
applicable to large scale problems in dimension two and three. We illustrate
the versatility and the efficiency of our approach on the numerical solution of
three problems in calculus of variation : 3D denoising, the principal agent
problem, and optimization within the class of convex bodies.Comment: 23 page
Bernstein operators for exponential polynomials
Let be a linear differential operator with constant coefficients of order
and complex eigenvalues . Assume that the set
of all solutions of the equation is closed under complex
conjugation. If the length of the interval is smaller than , where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\}
, then there exists a basis %, , of the space with
the property that each has a zero of order at and a zero of
order at and each is positive on the open interval
Under the additional assumption that and
are real and distinct, our first main result states that there exist points and positive numbers %,
such that the operator \begin{equation*}
B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies
, for The second main result
gives a sufficient condition guaranteeing the uniform convergence of
to for each .Comment: A very similar version is to appear in Constructive Approximatio
Some properties of LR-splines
Recently a new approach to piecewise polynomial spaces generated by B-spline has been presented by T. Dokken, T. Lyche and H.F. Pettersen, namely Locally Refined splines. In their recent work (Dokken et al., 2013) they define the LR B-spline collection and provide tools to compute the space dimension. Here different properties of the LR-splines are analyzed: in particular the coefficients for polynomial representations and their relation with other properties such as linear independence and the number of B-splines covering each element. © 2013 Elsevier B.V
Jacobi polynomials in Bernstein form
AbstractThe paper describes a method to compute a basis of mutually orthogonal polynomials with respect to an arbitrary Jacobi weight on the simplex. This construction takes place entirely in terms of the coefficients with respect to the so-called BernsteinâBĂ©zier form of a polynomial