Numerical integration for fractal measures

TL;DR

This paper extends classical numerical integration error estimates to fractal measures, providing bounds for approximating integrals on fractals like the Sierpiński gasket using sample sets and weights.

Contribution

It develops a framework for error estimation in numerical integration over fractals, incorporating energy and Laplacian-based variances, with explicit examples on the Sierpiński gasket.

Findings

Derived discrepancy measures for fractal integration

Applied results to Sierpiński gasket with various measures

Provided error bounds for different variance choices

Abstract

We find estimates for the error in replacing an integral $\int f d\mu$ with respect to a fractal measure $\mu$ with a discrete sum $\sum_{x \in E} w(x) f(x)$ over a given sample set $E$ with weights $w$. Our model is the classical Koksma-Hlawka theorem for integrals over rectangles, where the error is estimated by a product of a discrepancy that depends only on the geometry of the sample set and weights, and variance that depends only on the smoothness of $f$. We deal with p.c.f self-similar fractals, on which Kigami has constructed notions of energy and Laplacian. We develop generic results where we take the variance to be either the energy of $f$ or the $L^1$ norm of $\Delta f$, and we show how to find the corresponding discrepancies for each variance. We work out the details for a number of interesting examples of sample sets for the Sierpi\'{n}ski gasket, both for the standard self-similar measure and energy measures, and for other fractals.