Quadrature rules and distribution of points on manifolds

TL;DR

This paper investigates the accuracy of quadrature rules on compact manifolds by analyzing point distribution discrepancies and function variations, providing precise estimates for Sobolev class functions.

Contribution

It introduces sharp quantitative bounds for quadrature errors on manifolds considering discrepancy and generalized variation, extending classical inequalities.

Findings

Derived sharp error estimates for quadrature rules on manifolds.

Connected discrepancy of points with quadrature accuracy.

Extended Koksma-Hlawka type inequalities to manifold settings.

Abstract

We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.