Analysis of a New Nonlinear Interpolatory Subdivision Scheme on σ Quasi-Uniform Grids

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.

Stirling numbers and Gregory coefficients for the factorization of Hermite subdivision operators

In this paper we present a factorization framework for Hermite subdivision schemes refining function values and first derivatives, which satisfy a spectral condition of high order. In particular we show that spectral order $d$ allows for $d$ factorizations of the subdivision operator with respect to the Gregory operators: a new sequence of operators we define using Stirling numbers and Gregory coefficients. We further prove that the $d$th factorization provides a ‘convergence from contractivity’ method for showing $C^d$-convergence of the associated Hermite subdivision scheme. Gregory operators are derived by explicitly solving a recursion based on the Taylor operator and iterated vector scheme factorizations. The explicit expression of these operators allows one to compute the $d$th factorization directly from the mask of the Hermite scheme. In particular, it is not necessary to compute intermediate factorizations, which simplifies the procedures used up to now.

Improvement of friction estimation along wellbores using multi-scale smoothing of trajectories

Drilling monitoring aims at anticipating and detecting any drill string failures during well construction. A key element for the monitoring activity is the estimation of friction along the wellbore trajectory. Friction models require the evaluation of the actual wellbore trajectory. This evaluation is performed applying any of various reconstruction methods available in the industry to discrete deviation measurements. Although all these methods lead to nearly identical bit location, friction estimations are highly dependent on reconstruction methods due to huge dierences in the trajectory derivatives. To control this instability, a new reliable estimation of wellbore friction using a nonlinear trajectory smoothing process is introduced. This process uses a multi-scale approach and a specic nonlinear smoothing through subdivision schemes and their related decimation schemes. Two smoothing processes are compared: one using an interpolatory subdivision operator, and the other, a non-interpolatory subdivision operator. Validation has been performed on a synthetic plane noisy trajectory. The non-interpolatory process provides trajectory derivatives estimate much closer to those of the initial trajectory. Both processes have been applied to a real three-dimensional wellbore trajectory, improving signicantly the friction estimates.

Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.

Approximation by Multiple Refinable Functions

We consider the shift-invariant space, 𝕊(Φ), generated by a set Φ = {Φ1,..., Φr} of compactly supported distributions on R when the vector of distributions ϕ:= {Φ1,..., Φr} T satisfies a system of refinement equations expressed in matrix form aswhere a is a finitely supported sequence of r x r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets.The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in 𝕊(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lp-approximation order of 𝕊(Φ) when the functions in (Φ) belong to Lp(ℝ) (see Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of Φ.