Numerical differentiation on scattered data through multivariate polynomial interpolation

TL;DR

This paper introduces a new method for numerical differentiation on scattered multivariate data using local polynomial interpolation at Discrete Leja Points, providing error bounds and sensitivity analysis, with demonstrated accuracy through numerical tests.

Contribution

It presents a novel differentiation formula based on polynomial interpolation at Discrete Leja Points, including error bounds and sensitivity estimates, advancing numerical differentiation techniques for scattered data.

Findings

Accurate approximation of partial derivatives demonstrated

Error bounds established for the differentiation method

Sensitivity analysis to functional perturbations provided

Abstract

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.