Numerical integration for fractal measures
This work provides a theoretical framework for numerical integration on fractals, which is important for researchers studying analysis and computation on fractal domains.
The authors extend the Koksma-Hlawka theorem to fractal measures on p.c.f. self-similar fractals, providing error estimates for numerical integration in terms of discrepancy and variance (energy or Laplacian norm). They compute discrepancies for sample sets on the Sierpiński gasket and other fractals.
We find estimates for the error in replacing an integral $\int f dμ$ with respect to a fractal measure $μ$ 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 $Δf$, 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 Sierpiński gasket, both for the standard self-similar measure and energy measures, and for other fractals.