Academic literature on the topic 'Splines simpliciales'

A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).

AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

We analyze the convergence rate of a multilevel quasi-Monte Carlo (MLQMC) Finite Element Method (FEM) for a scalar diffusion equation with log-Gaussian, isotropic coefficients in a bounded, polytopal domain D ⊂ ℝd. The multilevel algorithm QL* which we analyze here was first proposed, in the case of parametric PDEs with sequences of independent, uniformly distributed parameters in Kuo et al. (Found. Comput. Math. 15 (2015) 411–449). The random coefficient is assumed to admit a representation with locally supported coefficient functions, as arise for example in spline- or multiresolution representations of the input random field. The present analysis builds on and generalizes our single-level analysis in Herrmann and Schwab (Numer. Math. 141 (2019) 63–102). It also extends the MLQMC error analysis in Kuo et al. (Math. Comput. 86 (2017) 2827–2860), to locally supported basis functions in the representation of the Gaussian random field (GRF) in D, and to product weights in QMC integration. In particular, in polytopal domains D ⊂ ℝd, d=2,3, our analysis is based on weighted function spaces to describe solution regularity with respect to the spatial coordinates. These spaces allow GRFs and PDE solutions whose realizations become singular at edges and vertices of D. This allows for non-stationary GRFs whose covariance operators and associated precision operator are fractional powers of elliptic differential operators in D with boundary conditions on ∂D. In the weighted function spaces in D, first order, Lagrangian Finite Elements on regular, locally refined, simplicial triangulations of D yield optimal asymptotic convergence rates. Comparison of the ε-complexity for a class of Matérn-like GRF inputs indicates, for input GRFs with low sample regularity, superior performance of the present MLQMC-FEM with locally supported representation functions over alternative representations, e.g. of Karhunen–Loève type. Our analysis yields general bounds for the ε-complexity of the MLQMC algorithm, uniformly with respect to the dimension of the parameter space.

Abstract We consider spline functions over simplicial meshes in $\mathbb {R}^{n}$ ℝ n . We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper, we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.

Les courbes de Bezier et les courbes splines ont trouve un cadre de présentation simple et naturel avec la notion de floraison d'une fonction polynomiale, telle qu'elle a été présentée dans les travaux de Lyle Ramshaw (1987). Notre but a consiste a étendre cette présentation au cas des surfaces et aussi des variétés de dimension supérieure. Les splines simpliciales sont une généralisation naturelle des b-splines au cas de plusieurs variables. Nous présentons leurs principales propriétés ainsi qu'une définition de différences divisées pour fonctions de plusieurs variables. Un algorithme d'évaluation d'une spline simpliciale est propose et teste. Floraisons et splines simpliciales sont les éléments essentiels d'un nouveau schéma de b-splines introduit par Dahmen, Micchelli et Seidel (1992). Ce schéma est étudié et ses principales propriétés sont présentées. Une grande similarité avec l'étude des courbes est retrouvée

Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.