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

Pedro Ortiz,Juan Carlos Trillo

doi:10.3390/math9121320

Abstract

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.

Highlights

Subdivision schemes are closely related to reconstruction operators
Let {Rk } be the sequence of nonlinear PPH reconstruction operators associated with a sequence of nested σ quasi-uniform grids { X k } with σ < 1 + 2M
We defined and analyzed the PPH subdivision scheme on nonuniform grids, which were derived from its associated reconstruction operator

Summary

Introduction

Subdivision schemes are closely related to reconstruction operators. They have been used in the last few decades in many applications ranging from the numerical solution of partial differential equations to image processing and computer-aided geometric design. Nonlinearity means data-dependent subdivision schemes, which may involve nonlinear operations in their definition By definition, they are designed to overcome certain drawbacks that appear when dealing with their linear counterparts, such as bad behavior in the presence of isolated discontinuities for instance. An example of these types of operators was defined in [1] and was named PPH (Piecewise Polynomial Harmonic).

A Nonlinear PPH Reconstruction Operator on Nonuniform Grids

Harten’s Interpolatory Multiresolution Setting

A Nonlinear PPH Subdivision Scheme on Nonuniform Grids

Main Properties of the PPH Subdivision Scheme in Nonuniform Meshes

Numerical Experiments

Conclusions