High-order numerical integration on self-affine sets

TL;DR

This paper develops high-order numerical integration methods for smooth functions over self-affine sets, leveraging self-similarity to compute weights and analyze errors, with practical numerical validation.

Contribution

It introduces a novel interpolatory cubature rule tailored for self-affine sets, utilizing algebraic characterization and self-similarity for efficient weight computation.

Findings

Successful construction of high-order cubature rules

Error analysis demonstrating convergence properties

Numerical experiments validating the methods

Abstract

We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an $h$-version and a $p$-version of the cubature, present an error analysis and conduct numerical experiments.