Minimal Numerical Differentiation Formulas

TL;DR

This paper develops minimal numerical differentiation formulas on irregular grids, optimizing weights to improve accuracy and stability, with theoretical error bounds and practical algorithms tested in experiments.

Contribution

It introduces new weighted minimization approaches for numerical differentiation on irregular points, providing error bounds and effective formulas.

Findings

Weighted $ ext{l}_1$ minimization yields accurate differentiation formulas.

Error bounds depend on a growth function related to point geometry.

Numerical experiments demonstrate improved performance over existing methods.

Abstract

We investigate numerical differentiation formulas on irregular centers in two or more variables that are exact for polynomials of a given order and minimize an absolute seminorm of the weight vector. Error bounds are given in terms of a growth function that carries the information about the geometry of the centers. Specific forms of weighted $\ell_1$ and weighted least squares minimization are proposed that produce numerical differentiation formulas with particularly good performance in numerical experiments.