High-Order Numerical Integration on Self-Affine Sets

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 [Formula: see text]-version and a [Formula: see text]-version of the cubature, present an error analysis, and conduct numerical experiments. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/zmoitier/IFSCubature and in the supplementary materials ( IFSCubature-main.zip [2.84MB]). [Formula: see text]