Bivariate hierarchical Hermite spline quasi--interpolation

TL;DR

This paper explores the extension of Hermite spline quasi-interpolation to hierarchical spline spaces for efficient bivariate function approximation, analyzing convergence and comparing performance with tensor-product methods.

Contribution

It introduces a hierarchical extension of Hermite spline quasi-interpolation, providing convergence analysis and numerical comparisons with traditional tensor-product schemes.

Findings

Hierarchical scheme converges effectively for bivariate approximation.

Numerical results show improved efficiency over tensor-product methods.

Hierarchical approach offers adaptive approximation capabilities.

Abstract

Spline quasi-interpolation (QI) is a general and powerful approach for the construction of low cost and accurate approximations of a given function. In order to provide an efficient adaptive approximation scheme in the bivariate setting, we consider quasi-interpolation in hierarchical spline spaces. In particular, we study and experiment the features of the hierarchical extension of the tensor-product formulation of the Hermite BS quasi-interpolation scheme. The convergence properties of this hierarchical operator, suitably defined in terms of truncated hierarchical B-spline bases, are analyzed. A selection of numerical examples is presented to compare the performances of the hierarchical and tensor-product versions of the scheme.