High-order numerical integration on self-affine sets

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.

💡 Research Summary

This paper addresses the problem of numerically integrating smooth functions over self‑affine (more generally, self‑similar) fractal sets with respect to their natural invariant measure. The authors develop an interpolatory cubature rule that can achieve arbitrarily high algebraic order, extending far beyond the low‑order or Monte‑Carlo approaches that have dominated the literature on fractal integration.

The mathematical setting starts with an iterated function system (IFS) S = {Sℓ}ℓ=1^L on ℝⁿ, each map being a contraction Sℓ(x)=Aℓx+bℓ with spectral norm ρℓ<1. The Hutchinson operator H yields a unique compact attractor Γ, while a probability vector μ = (μ₁,…,μ_L) defines a linear operator M on probability measures. The fixed point of M is the invariant measure ν (denoted μ in the paper), whose support is exactly Γ. Under the open set condition (OSC) and for similarity IFS, ν coincides (up to normalization) with the d‑dimensional Hausdorff measure on Γ.

A key observation is the self‑similarity of the integral: ∫Γ f dν = Σ{ℓ=1}^L μ_ℓ ∫_Γ f∘S_ℓ dν, which can be written compactly using the Ruelle (or transfer) operator F