Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.     

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C ¹ smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots—that are not considered by the authors of the corresponding spline basis—thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.     

Exponentials Reproducing Subdivision Schemes

This paper is concerned with a family of nonstationary, interpolatory subdivision schemes that have the capability of reproducing functions in a finite-dimensional subspace of exponential polynomials. We give conditions for the existence and uniqueness of such schemes, and analyze their convergence and smoothness. It is shown that the refinement rules of an even-order exponentials reproducing scheme converge to the Dubuc—Deslauriers interpolatory scheme of the same order, and that both schemes have the same smoothness. Unlike the stationary case, the application of a nonstationary scheme requires the computation of a different rule for each refinement level. We show that the rules of an exponentials reproducing scheme can be efficiently derived by means of an auxiliary orthogonal scheme , using only linear operations. The orthogonal schemes are also very useful tools in fitting an appropriate space of exponential polynomials to a given data sequence.     

A combinatorial Lefschetz fixed point formula

We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdⁿK →K is expanding directions preserving.     

From Hermite to stationary subdivision schemes in one and several variables

Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the “renormalization” of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so–called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the “zero at π” condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations.     

Nonlinear Subdivision Schemes on Irregular Meshes

The present article deals with convergence and smoothness analysis of geometric, nonlinear subdivision schemes in the presence of extraordinary points. We discuss when the existence of a proximity condition between a linear scheme and its nonlinear analogue implies convergence of the nonlinear scheme (for dense enough input data). Furthermore, we obtain C ¹ smoothness of the nonlinear limit function in the vicinity of an extraordinary point over Reif’s characteristic parametrization. The results apply to the geometric analogues of well-known subdivision schemes such as Doo–Sabin or Catmull–Clark schemes.     

Degree estimates for C k‐piecewise polynomial subdivision surfaces

Piecewise polynomial subdivision surfaces are considered which consist of tri‐ or quadrilateral patches in a mostly regular arrangement with finitely many irregularities. A sharp estimate on the lowest possible degree of the polynomial patches is given. It depends on the smoothness and flexibility of the underlying subdivision scheme. This revised version was published online in June 2006 with corrections to the Cover Date.     

Subdivision schemes inL p spaces

Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes inL _p spaces (1≤p≤∞). We characterize theL _p -convergence of a subdivision scheme in terms of thep-norm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of the limit function of a subdivision scheme, such as stability, linear independence, and smoothness.     

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).     

Hermite Subdivision Schemes and Taylor Polynomials

We propose a general study of the convergence of a Hermite subdivision scheme ℋ of degree d>0 in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is C ⁰ and ℋ is C ^d . We apply this result to two families of Hermite subdivision schemes. The first one is interpolatory; the second one is a kind of corner cutting. Both of them use the Tchakalov-Obreshkov interpolation polynomial.      

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

Elliptic grids, rational functions, and the Padé interpolation

We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of the Padé interpolation with prescribed poles and zeros on the elliptic grids. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21.     

Extended finite operator calculus—an example of algebraization of analysis

“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus of Rota—Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of sequences in its afterwards elaborated form—a ψ-calculus. The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota—Mullin or equivalently—of umbral calculus of Roman and Rota. At the same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials. Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one. This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]).     

de Rham transforms for subdivision schemes

For any subdivision scheme, we define its de Rham transform, which generalizes the de Rham and Chaikin corner cutting. The main property of the de Rham transform is that it preserves a sum rule. This allows comparison of the Hölder regularity of a given subdivision scheme with that of its de Rham transform. A graphical comparison is made for three different families of subdivision schemes, the last one being the generalized four-point scheme.     

Convergent Vector and Hermite Subdivision Schemes

Hermite subdivision schemes have been studied by Merrien, Dyn, and Levin and they appear to be very different from subdivision schemes analyzed before since the rules depend on the subdivision level. As suggested by Dyn and Levin, it is possible to transform the initial scheme into a uniform stationary vector subdivision scheme which can be handled more easily.With this transformation, the study of convergence of Hermite subdivision schemes is reduced to that of vector stationary subdivision schemes. We propose a first criterion for C⁰-convergence for a large class of vector subdivision schemes. This gives a criterion for C¹-convergence of Hermite subdivision schemes. It can be noticed that these schemes do not have to be interpolatory. We conclude by investigating spectral properties of Hermite schemes and other necessary/sufficient conditions of convergence.     

A relax-and-cut framework for Gomory mixed-integer cuts

Gomory mixed-integer cuts (GMICs) are widely used in modern branch-and-cut codes for the solution of mixed-integer programs. Typically, GMICs are iteratively generated from the optimal basis of the current linear programming (LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a different scheme for the generation of rank-1 GMICs read from a basis of the original LP—the one before the addition of any cut. We adopt a relax-and-cut approach where the generated GMICs are not added to the current LP, but immediately relaxed in a Lagrangian fashion. Various elaborations of the basic idea are presented, that lead to very fast—yet accurate—variants of the basic scheme. Very encouraging computational results are presented, with a comparison with alternative techniques from the literature also aimed at improving the GMIC quality. We also show how our method can be integrated with other cut generators, and successfully used in a cut-and-branch enumerative framework.     

A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes

In the recent literature on subdivision methods for approximation of manifold-valued data, a certain “proximity condition” comparing a nonlinear subdivision scheme to a linear subdivision scheme has proved to be a key analytic tool for analyzing regularity properties of the scheme. This proximity condition is now well known to be a sufficient condition for the nonlinear scheme to inherit the regularity of the corresponding linear scheme (this is called smoothness equivalence). Necessity, however, has remained an open problem. This paper introduces a smooth compatibility condition together with a new proximity condition (the differential proximity condition). The smooth compatibility condition makes precise the relation between nonlinear and linear subdivision schemes. It is shown that under the smooth compatibility condition, the differential proximity condition is both necessary and sufficient for smoothness equivalence. It is shown that the failure of the proximity condition corresponds to the presence of resonance terms in a certain discrete dynamical system derived from the nonlinear scheme. Such resonance terms are then shown to slow down the convergence rate relative to the convergence rate of the corresponding linear scheme. Finally, a super-convergence property of nonlinear subdivision schemes is used to conclude that the slowed decay causes a breakdown of smoothness. The proof of sufficiency relies on certain properties of the Taylor expansion of nonlinear subdivision schemes, which, in addition, explain why the differential proximity condition implies the proximity conditions that appear in previous work.     

Designs in Product Association Schemes

A number of important families of association schemes—such as the Hamming and Johnson schemes—enjoy the property that, in each member of the family, Delsarte t-designs can be characterised combinatorially as designs in a certain partially ordered set attached to the scheme. In this paper, we extend this characterisation to designs in a product association scheme each of whose components admits a characterisation of the above type. As a consequence of our main result, we immediately obtain linear programming bounds for a wide variety of combinatorial objects as well as bounds on the size and degree of such designs analogous to Delsarte's bounds for t-designs in Q-polynomial association schemes.     

Viscosity Solutions of an Infinite-Dimensional Black—Scholes—Barenblatt Equation

We study an infinite-dimensional Black—Scholes—Barenblatt equation which is a Hamilton—Jacobi—Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black—Scholes—Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton—Jacobi—Bellman equations.