Piecewise Polynomial Functions Research Articles

On computing derivatives forC 1 interpolating schemes: an optimization

The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study alocal method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called acell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines theglobal Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed.

不確実性を持つ離散的センサデータに基づく空間経路生成法とそのロボット倣い作業への応用 システムコンセプトと経路関数生成法

A spatial path generation algorithm based on visual sensory data and its application to the seam tracking robotic system which can operate even under poor sensing conditions are presented. At first, the concept of the robotic system with a wrist-mounted high performance vision sensor is presented, then an accurate spatial path generation algorithm using sensory data with a reliability coefficient is described. In the algorithm, piecewise polynomial functions are generated from sensory data that is acquired ahead of an end-effector, and the spatial path is recovered by connecting the piecewise polynomial functions successively under the minimum error condition as well as the boundary conditions. Computer simulations and experiments on a spatial path on a curved surface show that the proposed system operates effectively even under disturbances such as sensing noise, irregular tack-welding-beads and a burst of lack of sensory data, often appearing in practical applications.

Canonical representation of piecewise-polynomial functions with nondegenerate linear-domain partitions

Piecewise-linear (PWL) functions are a widely used class of nonlinear approximate functions with applications in both mathematics and engineering. As an extension of this function class, piecewise-polynomial (PWP) functions with a linear-domain partition represents a more general function class than PWL functions, in that all function pieces in a partitioned domain are (instead of hyperplanes) hypersurfaces described by polynomials. Just like PWL functions, the global expression of PWP functions requires a so-called canonical representation, which is meaningful for practical applications. However, such a canonical representation is still unknown. Our study showed that it is not a straightforward extension of the canonical representation of PWL functions; instead, it has a more general form than the latter. In this paper, we discuss the canonical representation of PWP functions with nondegenerate linear-domain partitions. A canonical representation formula is derived and a sufficient condition for its existence is given. We show that under some degree constraints, the derived canonical formula reduces to the canonical formula of PWL functions. The consistent variation property, which is a sufficient and necessary condition for the canonical representation of PWL functions, is found to be less important for PWP functions.

Swapping Edgesof Arbitrary Triangulations to Achieve the Optimal Order of Approximation

In the representation of scattered data by smooth pp (:= piecewise polynomial) functions, perhaps the most important problem is to find an optimal triangulation of the given sample sites (called vertices). Of course, the notion of optimality depends on the desirable properties in the approximation or modeling problems. In this paper, we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions. We will only consider C1 pp approximation with r=1 and k=4. The main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites, such that these sample sites are the only vertices of the triangulation, and that for any discrete data given at these sample sites, there is a C1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation.

A Spectral Sequence for Splines

We define a complex R/J of graded modules on ad-dimensional simplicial complex Δ, so that the top homology module of this complex consists of piecewise polynomial functions (splines) of smoothnessron the cone of Δ. In a series of papers,4;5;6] used a similar approach to study the dimension of the spaces of splines on Δ, but with a complex substantially different from R/J. We obtain bounds on the dimension of the homology modulesHi(R/J) for alli<dand find a spectral sequence which relates these modules to the spline module. We use this to give simple conditions governing the projective dimension of the spline module. We also prove that if the spline module is free, then it is determined entirely by local data; that is, by the arrangements of hyperplanes incident to the various dimensional faces of Δ.

A reduced constraint $hp$ finite element method for shell problems

We propose and analyze an hp finite element method for the Nagdhi shell model, based on rectangular elements. We show that for the bending-dominated case, assuming sufficient smoothness on the solution, the method is locking free in terms of both h and p, as the thickness of the shell tends to zero. Our results are established under the assumption that the geometrical coefficients appearing in the model are piecewise polynomial functions.

Brunn-Minkowski inequality for multiplicities

This is a convex subset of the real vector space P⊗ZR. In fact, it is a convex polytope; see [Br], where this polytope is discussed from an algebraic point of view. It is known that the total mass of m is a polynomial in m for su ciently large m (denote by k its degree) and m −k m(m · ) weak −→ ( )d ; where d is the Lebesgue measure supported on and the density ( ) is a piecewise-polynomial function; we will not use this piecewise polynomiality in this paper. Recall that a real function f de ned on a convex subset U of a vector space V is called concave, if

Piecewise polynomial functions, convex polytopes and enumerative geometry

0. Introduction. This paper explores some of the connections between the objects of its title. It is based on a new approach to McMullen’s polytope algebra, and on its relation with equivariant cohomology of toric varieties. In particular, we give another proof of a recent result of Fulton and Sturmfels, which identifies the polytope algebra with the direct limit of all Chow rings of smooth, complete torus embeddings (see [14]). On the other hand, we obtain a version of the classical theorem of Bezout, which holds in any spherical homogeneous space. This generalizes a theorem of Bernstein and Kouchnirenko: The number of common points to d hypersurfaces in general position in a d-dimensional torus is d! times the mixed volume of the associated Newton polytopes (see [2], [18] and also [13] 5.5). Given a finite-dimensional vector space V over an ordered field K, there is a wellknown correspondence between convex polytopes in the dual space V ∗ and piecewise linear convex functions on V . Namely, to any convex polytope, we associate its support function. Denote by R the algebra generated by the support functions of all convex polytopes, in the algebra of continuous functions on V . In the first section of this paper, we study the algebra R when K is the field of rational numbers. It turns out (see 1.5) that R is the algebra of continuous, piecewise polynomial functions on V ; in particular, R contains the algebra of polynomial functions. We prove that any choice of coordinate functions on V defines a regular sequence in R; moreover, the quotient of R by the ideal generated by V ∗ is isomorphic to the rational polytope algebra of McMullen (see 1.3, 1.5; our proof is based on work of Morelli, see [21]). This explains the non-trivial grading of the polytope algebra, by the obvious grading of R. More generally, the quotients of R by powers of the ideal generated by V ∗, are isomorphic to the higher versions of the polytope algebra, considered recently by McMullen, Pukhlikov and Khovanskii; see [20], [24]. In fact, we study the algebra R as the direct limit of its subalgebras RΣ consisting of functions which are piecewise polynomial with respect to a fixed fan Σ. To such a fan is

Differential equations with locally perturbed coefficients: High order error estimates for function and derivative

High order error estimates are obtained for both function and derivative when the coefficients and right hand side of a given initial or boundary value problem are replaced by piece-wise polynomial functions. These estimates are given for partition points and also continuously on subintervals. Based on our results for linear case, we propose a new technique to solve nonlinear differential equations. Numerical examples demonstrate the accuracy of our estimates.

Unicity in piecewise polynomial 𝐿¹-approximation via an algorithm

Our main result shows that certain generalized convex functions on a real interval possess a unique best L 1 L^{1} approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [Uniqueness of optimal piecewise polynomial L 1 L_{1} approximations for generalized convex functions, from “Functional Analysis and Approximation”, Internat. Ser. Numer. Math., vol. 60 (1981), 421–432]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the L 2 L^{2} case in [J. Chow, On the uniqueness of best L 2 [ 0 , 1 ] L_{2}[0,1] approximation by piecewise polynomials with variable breakpoints, Math. Comp. 39 (1982), 571–585.]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [Optimal segmented polynomial L p L^{p} -approximation, Computing 26 (1981), 239–246.]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the L 2 L^{2} case.

On the Lagrange interpolation for a subset ofC k functions

We study the optimal order of approximation forC k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, ifx is away from the singular points, the local order of approximation atx can be improved byO(n −1). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for |x|3) in a much more general form.

Regularity for Hamilton-Jacobi equations via approximation

We prove new regularity results for solutions of first-order partial differential equations of Hamilton-Jacobi type posed as initial value problems on the real line. We show that certain spaces determined by quasinorms related to the solution's approximation properties in C(ℝ) by continuous, piecewise quadratic polynomial functions are invariant under the action of the differential equation. As a result, we show that solutions of Hamilton-Jacobi equations have enough regularity to be approximated well in C(ℝ) by moving-grid finite element methods. The preceding results depend on a new stability theorem for Hamilton-Jacobi equations in any number of spatial dimensions.

Accurate Reconstructions of Functions of Finite Regularity from Truncated Fourier Series Expansions

Knowledge of a truncated Fourier series expansion for a $2\pi$-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the $2\pi$-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiable up to the specified order.

Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions

Knowledge of a truncated Fourier series expansion for a 2 π 2\pi -periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the 2 π 2\pi -periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiable up to the specified order.

Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.

Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.

Canonical representation: from piecewise-linear function to piecewise-smooth functions

The canonical representation of piecewise-linear (PWL) functions provides a global compact formulation of continuous PWL functions, which has significant advantages in the research and applications concerning nonlinear systems. This work studies the generalization of the canonical representation from PWL functions to piecewise-smooth (PWS) functions. First a class of PWS functions, called the regular PWS functions, is defined as a generalization of the continuous PWL functions. An important example of the regular PWS functions is the continuous piecewise-polynomial function. The continuous PWL function with a PWL partition is also covered by the regular PWS function. Then the canonical representation of the PWS function is defined and the existence conditions are discussed. The PWS generalization of the canonical representation is significant in applications where a PWS scheme can improve the performance of a PWL scheme in the approximation of a nonlinear function, i.e., in approximating the input/output (I/O) relation of a nonlinear system or a mapping neural network or in nonlinear signal processing. >

Numerical Solution of Continuous-State Dynamic Programs Using Linear and Spline Interpolation

This paper demonstrates that the computational effort required to develop numerical solutions to continuous-state dynamic programs can be reduced significantly when cubic piecewise polynomial functions, rather than tensor product linear interpolants, are used to approximate the value function. Tensor product cubic splines, represented in either piecewise polynomial or B-spline form, and multivariate Hermite polynomials are considered. Computational savings are possible because of the improved accuracy of higher-order functions and because the smoothness of higher-order functions allows efficient quasi-Newton methods to be used to compute optimal decisions. The use of the more efficient piecewise polynomial form of the spline was slightly superior to the use of Hermite polynomials for the test problem and easier to program. In comparison to linear interpolation, use of splines in piecewise polynomial form reduced the CPU time to obtain results of equivalent accuracy by a factor of 250–330 for a stochastic 4-dimensional water supply reservoir problem with a smooth objective function, and factors ranging from 25–400 for a sequence of 2-, 3-, 4-, and 5-dimensional problems. As a result, a problem that required two hours to solve with linear interpolation was solved in a less than a minute with spline interpolation with no loss of accuracy.

Multivariate piecewise polynomials

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

An algorithm for determining the approximation orders of multivariate periodic spline spaces

We give an algorithm which computes the approximation order of spaces of periodic piece-wise polynomial functions, given the degree, the smoothness and tesselation. The algorithm consists of two steps. The first gives an upper bound and the second a lower bound on the approximation order. In all known cases the two bounds coincide.

Numerical Analysis of a Bidimensional Hencky Problem Approximated by a Discontinuous Finite Element Method

A finite element method for the solution of a bidimensional elastoplastic Hencky problem is formulated and analyzed. The discrete spaces consist of discontinuous piecewise polynomial functions. Convergence results for the discrete displacements are proved. Then, error estimates in ${\bf L}^2 $-norms are stated for the stress tensors, under a piecewise ${\bf H}^2 $-regularity hypothesis that is both weaker than the one commonly used and physically admissible.